The Picard Group

of the Compactified Universal Jacobian

Dedicated to the memory of Torsten Ekedahl, with great admiration.

Margarida Melo and Filippo Viviani^{1}

Received: December 25, 2011 Revised: June 26, 2013

Communicated by Gavril Farkas

Abstract. We determine explicitly the Picard groups of the universal Jacobian stack and of its compactification over the stack of stable curves. Along the way, we prove some results concerning the gerbe structure of the universal Jacobian stack over its rigidification by the natural action of the multiplicative group and relate this with the existence of generalized Poincaré line bundles. We also compare our results with Kouvidakis-Fontanari computations of the divisor class group of the universal (compactified) Jacobian scheme.

2010 Mathematics Subject Classification: 14H10, 14H40, 14C22; 14A20, 14L24. Keywords and Phrases: Universal Jacobian stack and scheme, Picard group, 𝔾_{m}-gerbe, Brauer group, Poincaré line bundle.

1The first author was supported by the FCT project Espaços de Moduli em Geometria Algébrica (PTDC/MAT/111332/2009), by the FCT project Geometria Algébrica em Portugal (PTDC/MAT/099275/2008) and by the Fundação Calouste Gulbenkian program “Estímulo à investigação 2010”. The second author is a member of the research center CMUC (University of Coimbra) and he was supported by the FCT project Espaços de Moduli em Geometria Algébrica (PTDC/ MAT/111332/2009) and by MIUR–FIRB project Spazi di moduli e applicazioni.

1. introduction

Picard Group of the Compactified Universal Jacobian 459

The Picard group of a given moduli stack carries important informations on the geometry of the moduli problem one is dealing with. Since Mumford’s pioneer work in [Mum65], the subject has been widely developed and nowadays the literature on the computation of the Picard group of moduli stacks is quite vast. Remarkable examples are the Picard groups of the moduli stacks of curves possibly with level structures (see e.g. [AC87], [Cor91], [Kou94], [Jar01], [Mor01], [Cor07], [GV08], [Put12]) and of the moduli stacks of principal bundles over curves (see e.g. [DN89], [Kou91], [Kou93], [BL94], [KN97], [LS97], [BLS98], [Sor99], [Fal03], [BK05], [BH10]). The aim of this paper is to compute and give explicit generators for the Picard group of the degree-d universal Jacobian stack J ac_{d,g} over the moduli stack M_{g} of smooth curves of genus g and of its compactification \overline{J ac}_{d,g} over the moduli stack \overline{M}_{g} of stable curves of genus g, constructed by Caporaso in [Cap94] and [Cap05] and later generalized by the first author in [Mel09]. Moreover, we will compare our results with the computation of the divisor class group of the Caporaso’s universal compactified Jacobian scheme \overline{J}_{d,g}, carried out by Fontanari in [Fon05] (based upon the work of Kouvidakis in [Kou91]). The motivation for this work comes from the wish of understanding the (log)canonical model of \overline{J}_{d,g} and its relation to the different modular compactifications of the universal Jacobian. The Kodaira dimension and the Iitaka fibration of \overline{J}_{d,g} were computed by Farkas-Verra in [FV13] for d = g, by Bini, Fontanari and the second author in [BFV12] when \overline{J}_{d,g} has finite quotient singularities (which occurs exactly when d+g −1 and 2g −2 are coprime) and by Casalaina-Martin, Kass and the second author in [CMKVb] in the general case. An alternative compactification \overline{J}_{d,g}^{ps} of the universal Jacobian over Schubert’s moduli space \overline{M}_{g}^{ps} of pseudo-stable curves was recently found by G. Bini, F. Felici and the two authors in [BFMV] (see also [BMV12]). We expect that \overline{J}_{d,g}^{ps} is the first step towards the construction of the canonical model of \overline{J}_{d,g}, analogously to the fact that \overline{M}_{g}^{ps} is the first step towards the construction of the canonical model of \overline{M}_{g} (see [HH09]). Clearly, in order to verify this, one needs an explicit description of the (rational) Picard group of \overline{J}_{d,g}, which naturally embeds into the (rational) Picard group of the stack \overline{J ac}_{d,g}. Before describing our results, we need to briefly recall the definitions of the stacks J ac_{d,g} and \overline{J ac}_{d,g}, referring to Section 2 for more details. The degree-d universal Jacobian stack J ac_{d,g} is the (Artin) stack whose fiber over a scheme S consists of families of smooth curves C → S over S endowed with a line bundle L over C of relative degree d over S. The stack J ac_{d,g} is contained as a dense open substack in the degree-d compactified Jacobian stack \overline{J ac}_{d,g}, whose fiber over a scheme S consists of families of quasistable curves X → S endowed with a properly balanced line bundle over X of relative degree d over S (see 2.1 for the definitions). The stack \overline{J ac}_{d,g} is smooth and irreducible of dimension 4g − 4, and it is endowed with a (forgetful) universally closed surjective morphism Φ̃_{d} to the stack \overline{M}_{g} of stable curves. The stack \overline{J ac}_{d,g} is naturally endowed with the structure of a 𝔾_{m}-stack, since the group 𝔾_{m} naturally injects into the automorphism group of every object (C → S, L) ∈ \overline{J ac}_{d,g}(S) as multiplication by scalars on L. Therefore \overline{J ac}_{d,g} becomes a 𝔾_{m}-gerbe over the 𝔾_{m}-rigidification \overline{J}_{d,g} := \overline{J ac}_{d,g} 𝔾_{m}. We call ν_{d} : \overline{J ac}_{d,g} → \overline{J}_{d,g} the rigidification map. Analogously, J ac_{d,g} is a 𝔾_{m}-gerbe over its rigidification J_{d,g} := J ac_{d,g} 𝔾_{m} which is an open dense substack of \overline{J}_{d,g}. The stack \overline{J}_{d,g} is smooth and irreducible of dimension 4g − 3, and the morphism Φ̃_{d} : \overline{J ac}_{d,g} → \overline{M}_{g} factors through Φ_{d} : \overline{J}_{d,g} → \overline{M}_{g}, which is again a universally closed surjective morphism. Caporaso’s compactification \overline{J}_{d,g} of the universal Jacobian variety J_{d,g} over the moduli scheme \overline{M}_{g} of stable curves (see [Cap94]) is an adequate moduli space for \overline{J ac}_{d,g} and for \overline{J}_{d,g} (in the sense of [Alp2]) and even a good moduli space (in the sense of [Alp1]) if our base field k has characteristic zero. We will call it simply the moduli space for \overline{J ac}_{d,g} and for \overline{J}_{d,g}^{1}. The main result of this paper is a description of the Picard groups of the stacks J ac_{d,g} and J_{d,g} and of their compactifications \overline{J ac}_{d,g} and \overline{J}_{d,g}. Since J ac_{d,g} ⊂ \overline{J ac}_{d,g} and J_{d,g} ⊂ \overline{J}_{d,g} are open inclusions of smooth stacks, the natural restriction morphisms Pic(\overline{J ac}_{d,g}) → Pic(J ac_{d,g}) and Pic(\overline{J}_{d,g}) → Pic(J_{d,g}) are surjective. Moreover, since ν_{d} is a 𝔾_{m}-gerbe, the pull-back morphisms ν_{d}^{∗} : Pic(\overline{J}_{d,g}) → Pic(\overline{J ac}_{d,g}) and ν_{d}^{∗} : Pic(J_{d,g}) → Pic(J ac_{d,g}) are injective. Therefore, the above Picard groups are related by the following commutative diagram

(1.1)  \xymatrix{Pic(\overline{J ac}_{d,g}) \ar@{->>}[r] & Pic(J ac_{d,g}) \\ Pic(\overline{J}_{d,g}) \ar@{^{(}->}[u]^{ν_{d}^{∗}} \ar@{->>}[r] & Pic(J_{d,g}) \ar@{^{(}->}[u]^{ν_{d}^{∗}}}

in which the horizontal arrows are surjective and the vertical arrows are injective. We will prove that the four Picard groups of diagram (1.1) are generated by boundary line bundles and tautological line bundles, which we are now going to define.

1In the literature, the universal (resp. universal compactified) Jacobian stack is often called the universal (resp. universal compactified) Picard stack and it is denoted by Pic_{d,g} (resp. \overline{Pic}_{d,g}), see e.g. [Cap05], [Mel09], [BFV12]. Similarly the universal (resp. universal compactified) Jacobian scheme is often called the universal (resp. universal compactified)

Picard scheme and it is denoted by P_{d,g} (resp. \overline{P}_{d,g}), see e.g. [Cap94]. Following [CMKVa] and [BFMV], we prefer here to use the word universal (resp. universal compactified) Jacobian stack/scheme and consequently the symbols J ac_{d,g}, \overline{J ac}_{d,g}, J_{d,g} and \overline{J}_{d,g} for two reasons: (i) the word Jacobian stack/scheme is used only for curves while the word Picard stack/scheme is used also for varieties of higher dimensions and therefore it is more ambiguous; (ii) the expression “the Picard group of the Picard stack/scheme” seems a bit cacophonic.

In Section 3, we describe the irreducible components of the boundary divisor \overline{J ac}_{d,g} \ J ac_{d,g}. Clearly, the boundary of \overline{J ac}_{d,g} is the pull-back via the morphism Φ̃_{d} : \overline{J ac}_{d,g} → \overline{M}_{g} of the boundary of \overline{M}_{g}. Recall that \overline{M}_{g} \ M_{g} =

⋃_{i=0}^{[g/2]} δ_{i}, where δ_{0} is the irreducible divisor whose generic point is an irreducible

curve with one node and, for i = 1, …, [g/2], δ_{i} is the irreducible divisor whose generic point is the stable curve with two irreducible components of genera i and g − i meeting in one point. In Theorem 3.2, we prove that δ̃_{i} := Φ̃_{d}^{−1}(δ_{i}) is irreducible if either i = 0 or i = g/2 or the number \frac{2g−2}{(2g−2,d+g−1)} does not divide

(2i − 1) and, otherwise, that Φ̃_{d}^{−1}(δ_{i}) is the union of two irreducible divisors, that we call δ̃_{i}^{1} and δ̃_{i}^{2} (see Section 3 for the precise description of these two divisors). Since \overline{J ac}_{d,g} is a smooth stack, the boundary divisors \{δ̃_{i}, δ̃_{i}^{1}, δ̃_{i}^{2}\} are Cartier divisors and therefore they give rise to line bundles on \overline{J ac}_{d,g} that we denote by {O(δ̃_{i}), O(δ̃_{i}^{1}), O(δ̃_{i}^{2})} and we call the boundary line bundles of \overline{J ac}_{d,g}. Note that the irreducible components of the boundary of \overline{J}_{d,g} are the divisors \overline{δ}_{i} := ν_{d}(δ̃_{i}), \overline{δ}_{i}^{1} := ν_{d}(δ̃_{i}^{1}) and \overline{δ}_{i}^{2} := ν_{d}(δ̃_{i}^{2}). The associated line bundles \{O(\overline{δ}_{i}), O(\overline{δ}_{i}^{1}), O(\overline{δ}_{i}^{2})\} are called boundary line bundles of \overline{J}_{d,g} and clearly we have that ν_{d}^{∗}O(\overline{δ}_{i}) = O(δ̃_{i}), ν_{d}^{∗}O(\overline{δ}_{i}^{1}) = O(δ̃_{i}^{1}) and ν_{d}^{∗}O(\overline{δ}_{i}^{2}) = O(δ̃_{i}^{2}) (see Corollary 3.3). In Section 5, we introduce the line bundles K_{1,0}, K_{0,1}, K_{−1,2} and Λ(m, n) (for n, m ∈ ℤ) on \overline{J ac}_{d,g}, which we call tautological line bundles. The tautological line bundles are defined in terms of the determinant of cohomology d_{π}(−) and of the Deligne pairing ⟨−, −⟩_{π} applied to the universal family π : \overline{J ac}_{d,g,1} → \overline{J ac}_{d,g} (see §2.6 for the definition and basic properties of the determinant of cohomology and of the Deligne pairing). More precisely, we define

K_{1,0} := ⟨ω_{π}, ω_{π}⟩_{π},
K_{0,1} := ⟨ω_{π}, L_{d}⟩_{π},
K_{−1,2} := ⟨L_{d}, L_{d}⟩_{π},
Λ(n, m) = d_{π}(ω_{π}^{n} ⊗ L_{d}^{m}),

where ω_{π} is the relative dualizing sheaf for π and L_{d} is the universal line bundle on \overline{J ac}_{d,g,1}. Following a strategy due to Mumford [Mum83], we apply the Grothendieck-Riemann-Roch theorem to the morphism π : \overline{J ac}_{d,g,1} → \overline{J ac}_{d,g} in order to produce relations among the tautological line bundles, at least in the rational Picard group. In particular, we prove in Theorem 5.2 that all the tautological line bundles can be expressed in Pic(\overline{J ac}_{d,g}) ⊗ ℚ in terms of Λ(1, 0), Λ(0, 1) and Λ(1, 1). Therefore, we define the tautological subgroup Pic^{taut}(\overline{J ac}_{d,g}) ⊆ Pic(\overline{J ac}_{d,g}) as the subgroup generated by the line bundles Λ(1, 0), Λ(0, 1), Λ(1, 1) together with the boundary line bundles of \overline{J ac}_{d,g}. Similarly, we consider the subgroup Pic^{taut}(J ac_{d,g}) ⊆ Pic(J ac_{d,g}) generated by the restriction of Λ(1, 0), Λ(0, 1), Λ(1, 1) to J ac_{d,g}. Moreover, using the pull-back morphism ν_{d}^{∗} (see diagram (1.1)), we can define the tautological subgroups Pic^{taut}(\overline{J}_{d,g}) := (ν_{d}^{∗})^{−1}(Pic^{taut}(\overline{J}_{d,g})) ⊆ Pic(\overline{J}_{d,g}) and Pic^{taut}(J_{d,g}) := (ν_{d}^{∗})^{−1}(Pic^{taut}(J_{d,g})) ⊆ Pic(J_{d,g}). After these preliminaries, we can now state the main results of this paper, concerning the Picard groups of J ac_{d,g} and J_{d,g} and of their compactifications \overline{J ac}_{d,g} and \overline{J}_{d,g}. We prove that all the Picard groups in question are free and generated by tautological line bundles and boundary line bundles (if any). More precisely, we have the following.

Theorem A. Assume that g ≥ 3.

(i) The Picard group of J ac_{d,g} is freely generated by Λ(1, 0), Λ(0, 1) and Λ(1, 1).

(ii) The Picard group of \overline{J ac}_{d,g} is freely generated by the boundary line bundles and the tautological line bundles Λ(1, 0), Λ(0, 1) and Λ(1, 1).

Theorem B. Assume that g ≥ 3.

(i) The Picard group of J_{d,g} is freely generated by the tautological line bundles Λ(1, 0) and

(1.2)  Ξ := Λ(0, 1)^{\frac{d+g−1}{(d+g−1,d−g+1)}} ⊗ Λ(1, 1)^{− \frac{d−g+1}{(d+g−1,d−g+1)}} .

(ii) The Picard group of \overline{J}_{d,g} is freely generated by the boundary line bundles and the tautological line bundles Λ(1, 0) and Ξ.

Let us now sketch the strategy that we use to prove Theorems A and B. Since the stack \overline{J ac}_{d,g} is smooth we have a natural exact sequence (1.3)

⨁_{or i=g/2 or i=0^{k_{d,g} ∤ (2i−1)}} ⟨O(δ̃_{i})⟩ ⨁_{and i≠0,g/2^{k_{d,g} |(2i−1)}} ⟨O(δ̃_{i}^{1}), O(δ̃_{i}^{2})⟩ → Pic(\overline{J ac}_{d,g}) → Pic(J ac_{d,g}) → 0.

In Theorem 4.1, we prove that the above exact sequence is also exact on the left, or in other words that the boundary line bundles are linearly independent in the Picard group of \overline{J ac}_{d,g}. In order to prove this, we use the same strategy used by Arbarello-Cornalba in [AC87] to prove the analogous statement for the boundary line bundles of \overline{M}_{g}: we construct some test curves F̃_{j} → \overline{J ac}_{d,g}, in number equal to the number of boundary line bundles, and prove that the intersection matrix between these test curves F̃_{j} and the boundary line bundles of \overline{J ac}_{d,g} is non-degenerate. This reduces the proof of Theorem A(ii) to the proof of Theorem A(i). Moreover, using the fact that the pull-back morphism ν_{d}^{∗} : Pic(\overline{J}_{d,g}) → Pic(\overline{J ac}_{d,g}) is injective and it sends the boundary line bundles of \overline{J}_{d,g} into the boundary line bundles of \overline{J ac}_{d,g}, we get that also the boundary line bundles of \overline{J}_{d,g} are linearly independent (see Corollary 4.6), or in other words that we have an exact sequence: (1.4)

0 → ⨁_{or i=g/2 or i=0^{k_{d,g} ∤ (2i−1)}} ⟨O(\overline{δ}_{i})⟩ ⨁_{and i≠0,g/2^{k_{d,g} |(2i−1)}}⟨O(\overline{δ}_{i}^{1}), O(\overline{δ}_{i}^{2})⟩ → Pic(\overline{J}_{d,g}) → Pic(J_{d,g}) → 0.

This reduces the proof of Theorem B(ii) to the proof of Theorem B(i). The Picard groups of J ac_{d,g} and of J_{d,g} are related via the following exact sequence coming from the Leray spectral sequence for the étale sheaf 𝔾_{m} with respect to the rigidification map ν_{d} : J ac_{d,g} → J_{d,g} (see (6.1)):

0 → Pic(J_{d,g}) ⟶^{ν_{d}^{∗}} Pic(J ac_{d,g}) ⟶^{res} Pic B𝔾_{m} = Hom(𝔾_{m}, 𝔾_{m}) ≅ ℤ ⟶^{obs} Br(J_{d,g}).

The map res is the restriction to the fibers of ν_{d} (which are isomorphic to the classifying stack B𝔾_{m} of the multiplicative group 𝔾_{m}) and obs sends 1 ∈ ℤ into the class [ν_{d}] of the 𝔾_{m}-gerbe ν_{d} : J ac_{d,g} → J_{d,g} in the cohomological Brauer group Br(J_{d,g}) := H_{ét}^{2}(J_{d,g}, 𝔾_{m}) of J_{d,g}. In Theorem 6.4, we prove that the order of [ν_{d}] is the greatest common divisor (d + 1 − g, 2g − 2). In proving this, we interpret in Proposition 6.6 the order of [ν_{d}] as the smallest natural number m for which there exists an m-Poincaré line bundle (in the sense of Definition 6.5) on the universal family J ac_{d,g,1} over J_{d,g}. Using Proposition 6.6, Theorem 6.4 follows then from a result of Kouvidakis (see [Kou93, p. 514]). Note also that by combining Theorem 6.4 and Proposition 6.6, we recover the well-known result of Mestrano-Ramanan ([MR85, Cor. 2.9]): there exists a Poincaré line bundle on J ac_{d,g,1} if and only if (d + 1 − g, 2g − 2) = 1. We conjecture that the cohomological Brauer group Br(J_{d,g}) is generated by [ν_{d}] (see Conjecture 6.9 and the discussion following it). From the computation of the order of [ν_{d}] and the above exact sequence, we get that res(Pic(J ac_{d,g})) = (2g −2, d+1−g)·ℤ. Moreover, we compute the values of the map res on the generators of the tautological subgroup Pic^{taut}(J ac_{d,g}) ⊆ Pic(J ac_{d,g}) in Lemma 6.2 and deduce that res(Pic^{taut}(J ac_{d,g})) = (2g − 2, d + 1−g)·ℤ. This easily reduces the proof of Theorem A(i) to the proof of Theorem B(i). Furthermore, it shows that Pic^{taut}(J_{d,g}) is generated by Λ(1, 0) and the line bundle Ξ of (1.2). The Picard group of J_{d,g} can be determined with the help of the following exact sequence

(1.5)  0 → Pic(M_{g}) ⟶^{Φ_{d}^{∗}} Pic(J_{d,g}) ⟶^{χ_{d}} ℤ,

where the map χ_{d} sends a line bundle L ∈ Pic(J_{d,g}) to the integer m ∈ ℤ such that the class of the restriction of L to the fiber Φ_{d}^{−1}(C) = J^{d}(C) in the Néron-Severi group NS(J^{d}(C)) is isomorphic to m times the class θ_{C} of the theta divisor (see Section 7 for more details). A well-known result of Harer and Arbarello-Cornalba says that Pic(M_{g}) is freely generated by the Hodge line bundle Λ if g ≥ 3 (see Theorem 2.12) and we prove in Lemma 5.1 that Φ_{d}^{∗}(Λ) = Λ(1, 0). On the other hand, a result of Kouvidakis in [Kou91] implies

that Im(χ_{d}) ⊆ \frac{2g − 2}{(2g − 2, d + 1 − g)} · ℤ. In Theorem 7.2, we compute the values of χ_{d} on the generators of the tautological subgroup Pic^{taut}(J_{d,g}) ⊆ Pic(J_{d,g})

Documenta Mathematica 19 (2014) 457–507

and we deduce that χ_{d}(Pic^{taut}(J_{d,g})) = \frac{2g − 2}{(2g − 2, d + 1 − g)} · ℤ. From the exact

sequence (1.5), we deduce now that Pic^{taut}(J_{d,g}) = Pic(J_{d,g}) is free of rank two; Theorem B(i) now follows. In the last Section of the paper, we relate the Picard group of the moduli stack \overline{J}_{d,g} with the divisor class group Cl(\overline{J}_{d,g}) of its moduli scheme \overline{J}_{d,g}, which was computed by Fontanari [Fon05] based upon the work of Kouvidakis [Kou91] on the Picard group of the open subscheme J_{d,g}^{0} ⊂ J_{d,g} consisting of pairs (C, L) such that C does not have non-trivial automorphisms. Fontanari proved in [Fon05] that the boundary of \overline{J}_{d,g} is the union of the irreducible divisors Δ̃_{i} := ϕ_{d}^{−1}(Δ_{i}) for i = 1, …, [g/2], where ϕ_{d} : \overline{J}_{d,g} → \overline{M}_{g} is the natural map towards the moduli scheme of stable curves of genus g and Δ_{i} ⊆ \overline{M}_{g} is, as usual, the irreducible divisor of \overline{M}_{g} whose generic point is an irreducible curve with one node if i = 0 or, for i > 0, the union of two irreducible components of genera i and g − i meeting in one point. Moreover, Fontanari proved that there is an exact sequence

(1.6)  0 → ⨁_{i=0}^{[g/2]} ℤ · Δ̃_{i} → Cl(\overline{J}_{d,g}) → Cl(J_{d,g}) → 0,

where the last map is the restriction map and the first map sends each Δ̃_{i} into its class in Cl(\overline{J}_{d,g}). The Picard group of \overline{J}_{d,g} and the divisor class group of \overline{J}_{d,g} are related by the pull-back via the natural map Ψ_{d} : \overline{J}_{d,g} → \overline{J}_{d,g}, which induces a map from the exact sequence (1.4) into the exact sequence (1.6). In Section 8 we prove the following result.

Theorem C. The pull-back map Ψ_{d}^{∗} : Cl(\overline{J}_{d,g}) → Pic(\overline{J}_{d,g}) induced by the natural map Ψ_{d} : \overline{J}_{d,g} → \overline{J}_{d,g} fits into a commutative diagram with exact rows

\xymatrix{0 \ar[r] & ⨁_{i=0}^{[g/2]} ℤ · Δ̃_{i} \ar[r] \ar[d]^{α_{d}} & Cl(\overline{J}_{d,g}) \ar[r] \ar[d]^{Ψ_{d}^{∗}} & Cl(J_{d,g}) \ar[r] \ar[d]^{β_{d}} & 0 \\ 0 \ar[r] & ⨁_{or i=g/2^{k_{d,g} ∤ 2i−1}} ⟨O(\overline{δ}_{i})⟩ ⨁_{and i≠g/2^{k_{d,g} |2i−1}} ⟨O(\overline{δ}_{i}^{1}), O(\overline{δ}_{i}^{2})⟩ \ar[r] & Pic(\overline{J}_{d,g}) \ar[r] & Pic(J_{d,g}) \ar[r] & 0,}

such that:

(i) the map β_{d} is an isomorphism; (ii) the map α_{d} satisfies

α_{d}(Δ̃_{i}) = \begin{cases}O(\overline{δ}_{i}) & if k_{d,g} ∤ (2i − 1), \\ O(\overline{δ}_{i}^{1}) + O(\overline{δ}_{i}^{2}) & if k_{d,g} | (2i − 1) and i ≠ g/2, \\ O(2\overline{δ}_{i}) & if k_{d,g} | (2i − 1) and i = g/2. \end{cases}

It is likely that the same techniques used in this paper could lead to the computation of the Picard group of the degree-d compactified universal Jacobian stack \overline{J ac}_{d,g,n} over the stack \overline{M}_{g,n} of n-pointed stable curves of genus g constructed in [Mel10] and of the universal vector bundle over \overline{M}_{g} constructed in [Pan96]. We plan to come back to these problems in a near future. The paper is organized as follows. In Section 2, we summarize the known properties of the stacks \overline{J ac}_{d,g} and \overline{J}_{d,g} as well as the properties of their moduli scheme \overline{J}_{d,g} (see 2.1). Moreover, we recall some basic facts about the Picard group of a stack and how to construct natural line bundles on moduli stacks by using the determinant of cohomology and the Deligne pairing (see 2.6). Finally, we recall the computation of the Picard group of the stack \overline{M}_{g} of stable curves of genus g by Harer and Arbarello-Cornalba (see 2.11). In Section 3, we describe the boundary divisors of \overline{J ac}_{d,g} and we explain how they are related to the pull-back of the boundary divisors of \overline{M}_{g}. In Section 4, we show that the line bundles on \overline{J ac}_{d,g} associated to the boundary divisors are linearly independent. In Section 5, we introduce the tautological line bundles on \overline{J ac}_{d,g} and we study the relations among them. In Section 6, we compare the Picard groups of J ac_{d,g} and of J_{d,g} using the Leray’s spectral sequence associated to the rigidification map ν_{d} : J ac_{d,g} → J_{d,g}. Moreover, we compute the order of the 𝔾_{m}-gerbe ν_{d} in the Brauer group of J_{d,g}. In Section 7, we compute the Picard group of J_{d,g} using the fibration Φ_{d} : J_{d,g} → M_{g}. Moreover, we investigate the relation between the line bundle Ξ and the universal theta divisor (see 7.1) and we prove that the pull-back via the Abel-Jacobi map provides an isomorphism between the Picard groups of J ac_{d,g} and of the d-th symmetric product of the universal curve M_{g,1} → M_{g}, when d > 2g − 2 (see 7.2). In Section 8, we compare the Picard group of \overline{J}_{d,g} with the divisor class group of its moduli scheme \overline{J}_{d,g}.

1.1. Relation to algebraic topology. After a preliminary version of this manuscript has been posted on arXiv, J. Ebert and O. Randal-Williams posted on arXiv a preliminary version of the paper [ERW12], which contains, among other things, some results that are closely related to Theorem A(i) and Theorem B(i) in the case when our base field k is the field of complex numbers. We now explain the relation between our results and the results of [ERW12]. In [ERW12], the authors introduce two holomorphic stacks Hol_{g}^{d} and Pic_{g}^{k}, defined as follows (see [ERW12, Sec. 4.1] for details): Hol_{g}^{d} is the holomorphic stack whose fibers over a topological space B consists of families of Riemann surfaces π : E → B of genus g equipped with a fiberwise holomorphic line bundle L → E of relative degree d; Pic_{g}^{d} is the holomorphic stack parametrizing families of Riemann surfaces of genus g equipped with a section of the associated bundle of Jacobian varieties of degree d. There is a morphism ϕ_{g}^{d} : Hol_{g}^{d} → Pic_{g}^{d} defined by sending a fiberwise holomorphic line bundle to its isomorphism class. It turns out that ϕ_{g}^{d} is a gerbe with band ℂ^{∗} (see [ERW12, Thm. 4.5]).

The relation with our algebraic stacks J ac_{d,g} and J_{d,g} (over the complex numbers) is provided by a commutative diagram

(1.7)  \xymatrix{(J ac_{d,g})^{an} \ar[r] \ar[d]^{ν_{d}} & Hol_{g}^{d} \ar[d]^{ϕ_{g}^{d}} \\ (J_{d,g})^{an} \ar[r] & _{Picd g}}

where (J ac_{d,g})^{an} and (J_{d,g})^{an} are the analytifications of the complex algebraic stacks J ac_{d,g} and J_{d,g}. The horizontal maps are most likely isomorphisms although we have not checked this in detail. The authors of loc. cit. consider tautological classes κ_{i,j} ∈ H^{2i+2j}(Hol_{g}^{d}, ℤ) for i ≥ −1 and j ≥ 0 defined by associating to every element (π : E → B, L → E) ∈ Hol_{g}^{d}(B) the cohomology class

(1.8) κ_{i,j}(π : E → B, L → E) := π_{!}(c_{1}(T^{v}E)^{i+1} · c_{1}(L)^{j}) ∈ H^{2i+2j}(B, ℤ),

where T^{v}E is the relative tangent line bundle of the family π : E → B of Riemann surfaces, which is of course dual to the sheaf ω_{π} of relative differentials of π. In particular, the classes κ_{i,0} are the pull-back to Hol_{g}^{d} of the Mumford-Morita-Miller classes κ_{i} on M_{g}. Moreover, one denotes by λ the pull-back to Hol_{g}^{d} of the Hodge class on M_{g}. Among other beautiful results, Ebert and Randal-Williams compute the analytic Néron-Severi group NS, the topological Picard group Pic_{top} and the second cohomology group with integer values H^{2}(−, ℤ) of the above two stacks (see [ERW12, Thm. C, Thm. E]), under the assumption that g ≥ 6.

Theorem 1.1 (Ebert, Randal-Williams). Assume that g ≥ 6. Then

(i) NS(Hol_{g}^{d}) = Pic_{top}(Hol_{g}^{d}) = H^{2}(Hol_{g}^{d}, ℤ) is freely generated by λ, κ_{−1,2},

and ζ := \frac{κ_{0,1} − κ_{−1,2}}{2} .

(ii) NS(Pic_{g}^{d}) = Pic_{top}(Hol_{g}^{d}) = H^{2}(Pic_{g}^{d}, ℤ) is the subgroup of H^{2}(Hol_{g}^{d}, ℤ) generated by λ and

η := \frac{d κ_{0,1} + (g − 1)κ_{−1,2}}{(2g − 2, g + d − 1)} .

The diagram (1.7) gives two natural homomorphisms

(1.9)

c_{1} : Pic(J ac_{d,g}) → H^{2}(Hol_{g}^{d}, ℤ),
c_{1} : Pic(J_{d,g}) → H^{2}(Pic_{g}^{d}, ℤ).

The next result is obtained by comparing Theorems A(i) and B(i) with Theorem 1.1.

Corollary 1.2. Assume that g ≥ 6. The homomorphisms of (1.9) are isomorphisms.

Proof. The fact that the first map in (1.9) is an isomorphism follows by comparing Theorem A(i) and Theorem 1.1(i) by mean of the formulas

(*)  \{ \begin{matrix}c_{1}(Λ(1, 0)) = λ, \\ c_{1}(Λ(1, 1)) = \frac{κ_{−1,2} − κ_{0,1}}{2} = −ζ, \\ c_{1}(Λ(0, 1)) = \frac{κ_{−1,2} + κ_{0,1}}{2} + λ = ζ + κ_{−1,2} + λ, \end{matrix}

where the first formula follows from Lemma 5.1 and the last two formulas follow from Theorem 5.2 together with the facts that c_{1}(K_{−1,2}) = κ_{−1,2} and c_{1}(K(0, 1)) = −κ_{0,1}. Note that the minus sign appearing in this last equality is due to the fact that in defining the classes κ_{i,j} ∈ H^{2}(Hol_{g}^{d}, ℤ) (see (1.8)), Ebert and Randal-Williams use the relative tangent sheaf while our definition (5.1) of the tautological line bundles K_{i,j} ∈ Pic(J ac_{d,g}) uses its dual sheaf, namely the sheaf of relative differentials. The fact that the second map in (1.9) is an isomorphism follows by comparing Theorem B(i) and Theorem 1.1(ii) using the formula

c_{1}(Ξ) = \frac{(d + g − 1)c_{1}(Λ(0, 1)) − (d − g + 1)c_{1}(Λ(1, 1))}{(d + g − 1, d − g + 1)} =

= η + \frac{d + g − 1}{(d + g − 1, d − g + 1)}λ.

■

Acknowledgements. The first author would like to thank Lucia Caporaso for suggesting this problem to her while a PhD student of hers. We thank Alexis Kouvidakis for pointing out to us the reference [Kou93, p. 514] which simplified the proof of Theorem 6.4. We are grateful to Oscar Randal-Williams for some enlightening discussions on the paper [ERW12] and for pointing out to us a mistake in a previous version of Lemma 7.4. We thank F. Poma, M. Talpo and F. Tonini for pointing out that the proof of Theorem 5.2 gives a priori only relations in the rational Picard group. We are grateful to the referee for suggesting an interesting connection between the Picard groups of the universal Jacobian and of the universal symmetric product, which is now included in subsection 7.2. We have benefited from useful conversations with Gilberto Bini, Silvia Brannetti and Claudio Fontanari.

Notations.

1.3. We fix two integers g ≥ 2 and d: g will always denote the genus of the curves and d the degree of the Jacobian varieties. Given two integers m and n, we set (n, m) for the greatest common divisor of n and m. In particular the greatest common divisor

(2g − 2, d + 1 − g) = (2g − 2, d − 1 + g) = (d + 1 − g, d − 1 + g)

will appear often in what follows. Similarly the number

(1.10)  k_{d,g} := \frac{2g − 2}{(2g − 2, d + g − 1)}

will appear repeatedly throughout the paper and hence it deserves a special notation.

1.4. We work over an algebraically closed field k of characteristic 0. All the schemes and stacks we will deal with are of finite type over k. There are two places in our work where the assumption on the characteristic of k is used. The first one is the explicit computation of the Picard group of \overline{M}_{g} by Harer and Arbarello-Cornalba (see Theorem 2.12 for the precise statement), which is known to be true only in characteristic zero (in positive characteristic, the same statement remains true for the rational Picard group of \overline{M}_{g} by the work of Moriwaki in [Mor01]). The second one is a result of Kouvidakis [Kou91] (see Theorem 7.1), whose proof over the complex numbers does not immediately extend to a base field k of positive characteristics^{2}.

1.5. We will often assume, for simplicity, that g ≥ 3. This is the case for two of the main results of this paper, namely Theorems A and B. The reason for this assumption is that the Picard group of \overline{M}_{g} is freely generated by the Hodge line bundle Λ and the boundary line bundles \{O(δ_{0}), …, O(δ_{[g/2]})\} if g ≥ 3 (see Theorem 2.12) while if g = 2 then Pic(\overline{M}_{g}) is still generated by Λ and the boundary line bundles but with the relation Λ^{10} ⊗ O(−δ_{0} − 2δ_{1}) = 0 (see 2.11). Indeed, all the above mentioned results continue to hold for g = 2 if we add the relation pull-backed from the relation Λ^{10} ⊗ O(−δ_{0} − 2δ_{1}) = 0 in Pic(\overline{M}_{2}) or its image Λ^{10} = 0 in Pic(M_{2}).

2. Preliminaries

2.1. The stacks \overline{J ac}_{d,g} and \overline{J}_{d,g} and their moduli space \overline{J}_{d,g} Let J ac_{d,g} be the universal Jacobian stack over the moduli stack M_{g} of smooth curves of genus g. The fiber of J ac_{d,g} over a scheme S is the groupoid whose objects are families of smooth curves C → S endowed with a line bundle L over C of relative degree d over S and whose arrows are the obvious isomorphisms. J ac_{d,g} is a smooth irreducible (Artin) algebraic stack of dimension 4g − 4 endowed with a natural forgetful morphism Φ̃_{d} : J ac_{d,g} → M_{g}. The multiplicative group 𝔾_{m} naturally injects into the automorphism group of every object (C → S, L) ∈ J ac_{d,g}(S) as multiplication by scalars on L, endowing J ac_{d,g} with the structure of a 𝔾_{m}-stack in the sense of [Hof07, Def. 3.1] or, equivalently, with a 𝔾_{m}-2-structure in the sense of [AGV09, Appendix C.1]. There is a canonical procedure to remove such automorphisms, called 𝔾_{m}- rigidification (see [ACV03, Sec. 5], [Rom05, Sec. 5] and [AGV09, Appendix C]). The outcome is a new stack J_{d,g} := J ac_{d,g} 𝔾_{m} together with a smooth and surjective map ν_{d} : J ac_{d,g} → J_{d,g}. Indeed, the map ν_{d} makes J ac_{d,g} into a gerbe banded by 𝔾_{m} (or a 𝔾_{m}-gerbe in short) over J_{d,g} (we refer to [Gir71] for the theory of gerbes). The forgetful map Φ̃_{d} factors via ν_{d} and we get a commutative diagram

2We thank F. Poma, M. Talpo and F. Tonini for pointing out this to us.

\xymatrix{J ac_{d,g} \ar[dr]_{Φ̃_{d}} \ar[rr]^{ν_{d}} &  & J_{d,g} \ar[dl]^{Φ_{d}} \\  & M_{g}}

The new stack J_{d,g} is a smooth, irreducible and separated Deligne-Mumford stack of dimension 4g − 3 and the map Φ_{d} is representable. A modular compactification of the stacks J ac_{d,g} and J_{d,g} was described by Caporaso in [Cap05] for some degrees and later by Melo in [Mel09] for the general case, based upon previous work of Caporaso in [Cap94]. Let us review this compactification.

Definition 2.2. [Cap94, Sec. 3.3] A connected, projective nodal curve X is said to be quasistable if it is (Deligne-Mumford) semistable and if the exceptional components of X do not meet.

Definition 2.3. [BFMV, Def. 3.5] Let X be a quasistable curve of genus g ≥ 2. A line bundle L of degree d on X (or its multidegree) is said to be properly balanced if

• for every subcurve Z of X the following (“Basic Inequality”) holds

(2.1)  m_{Z}(d) := \frac{dw_{Z}}{2g − 2} − \frac{k_{Z}}{2} ≤ deg_{Z} L ≤ \frac{dw_{Z}}{2g − 2} + \frac{k_{Z}}{2} := M_{Z}(d),

where w_{Z} := deg_{Z}(ω_{X}) and k_{Z} := ♯(Z ∩ \overline{X \ Z}).

• deg_{E} L = 1 for every exceptional component E of X.

Remark 2.4. In order to check that a line bundle is properly balanced, it is enough to check the basic inequality (2.1) for all subcurves Z such that Z and Z^{c} are connected (see [BFMV, Rmk. 3.8]).

Let \overline{J ac}_{d,g} be the category fibered in groupoids whose fiber over a scheme S consists of the groupoid whose objects are families of quasistable curves C → S endowed with a line bundle L of relative degree d, whose restriction to each geometric fiber is properly balanced (we say that L is properly balanced), and whose arrows are the obvious isomorphisms. The multiplicative group 𝔾_{m} injects into the automorphism group of every object (C → S, L) ∈ \overline{J ac}_{d,g}(S) as multiplication by scalars on L. As in the smooth case, the rigidification morphism ν_{d} : \overline{J ac}_{d,g} → \overline{J}_{d,g} := \overline{J ac}_{d,g} 𝔾_{m} endows \overline{J ac}_{d,g} with the structure of a 𝔾_{m}-gerbe over \overline{J}_{d,g}. There is a natural morphism of category fibered in groupoids Φ̃_{d} : \overline{J ac}_{d,g} → \overline{M}_{g} obtained by sending (C → S, L) ∈ \overline{J ac}_{d,g}(S) into the stabilization C^{st} → S ∈ \overline{M}_{g}(S) of the family of quasi-stable curves C → S. Clearly, the morphism Φ̃_{d} factors through a morphism Φ_{d} : \overline{J}_{d,g} → \overline{M}_{g}.

The following theorem summarizes the known properties of \overline{J ac}_{d,g} and of \overline{J}_{d,g}, proved in [Cap05] under the assumption that (d + g − 1, 2g − 2) = 1 and in [Mel09] for arbitrary d, and of their moduli space \overline{J}_{d,g} constructed in [Cap94].

Theorem 2.5 (Caporaso, Melo).

(1) \overline{J ac}_{d,g} (resp. \overline{J}_{d,g}) is an irreducible and smooth quotient stack of finite type over k and of dimension 4g − 4 (resp. 4g − 3). It contains the stack J ac_{d,g} (resp. J_{d,g}) as a dense open substack.

(2) The morphism Φ̃_{d} : \overline{J ac}_{d,g} → \overline{M}_{g} (resp. Φ_{d} : \overline{J}_{d,g} → \overline{M}_{g}) is surjective and universally closed.

(3) There exists a projective irreducible normal variety \overline{J}_{d,g}, endowed with a surjective morphism ϕ_{d} : \overline{J}_{d,g} → \overline{M}_{g}, which is an adequate moduli space in the sense of [Alp2] (and even a good moduli space in the sense of [Alp1] if char(k) = 0) for \overline{J ac}_{d,g} and \overline{J}_{d,g}.

Indeed, if (and only if) (d + 1 − g, 2g − 2) = 1 then \overline{J}_{d,g} is a Deligne-Mumford stack, the morphism Φ_{d} is proper and \overline{J}_{d,g} is a coarse moduli space for \overline{J}_{d,g}. For later use, we record the morphisms introduced in this subsection into the following commutative diagram:

(2.2)  \xymatrix{\overline{J ac}_{d,g} \ar[d]^{Φ̃_{d}} \ar[r]^{ν_{d}} & \overline{J}_{d,g} \ar[dl]^{Φ_{d}} \ar[r]^{Ψ_{d}} & \overline{J}_{d,g} \ar[d]_{ϕ_{d}} \\ _{\overline{M}_{g}} \ar[rr] &  & _{\overline{M}_{g}}}

2.6. The Picard and the Chow groups of a stack In this subsection, we are going to briefly recall the definition and the main properties of the Picard group and of the Chow group of an algebraic stack that we are going to use later. We refer to [Edi12] for a nice survey on the subject. Let X be an Artin stack of finite type over k. The definition of the (functorial) Picard group of X was introduced by Mumford (see [Mum65, p. 64]).

Definition 2.7 (Mumford). A line bundle L on X is the data consisting of a line bundle L(f) ∈ Pic(S) for every morphism f : S → X from a scheme S and, for every composition of morphisms T →^{g} S →^{f} X, an isomorphism L(f ∘ g) ≅ g^{∗}L(f), with the obvious compatibility requirements. The tensor product of two line bundles L and M on X is the new line bundle L⊗ M on X defined by (L⊗M)(f) := L(f)⊗M(f) together with the isomorphisms (L ⊗ M)(f ∘ g) ≅ g^{∗}(L ⊗ M)(f) induced by those of L and M. The abelian group consisting of all the line bundles on X together with the operation of tensor product is called the Picard group of X and is denoted by Pic(X).

If X is isomorphic to a quotient stack [X/G], where X is a scheme of finite type over k and G is a group scheme of finite type over k, then Pic(X) is isomorphic to the group Pic^{G}(X) of G-linearized line bundles on X in the sense of [GIT65, I.3] (see e.g. [EG98, Prop. 18]). The (operational) Chow groups of an Artin stack X were introduced by Edidin-Graham in [EG98, Sec. 5.3] (see also [Edi12, Def. 3.5]), generalizing the definition of the operational (or bivariant) Chow groups of a scheme (see [Ful98, Chap. 17]).

Definition 2.8 (Edidin-Graham). An i-th Chow cohomology class c on X is the data consisting of an element c(f) belonging to the i-th operational Chow group A^{i}(S) for every morphism f : S → X from a scheme S and, for every composition of morphisms T →^{g} S →^{f} X, an isomorphism c(f ∘ g) ≅ g^{∗}c(f), with the obvious compatibility requirements. The sum of two i-th Chow cohomology classes c and d on X is the new i-th Chow cohomology class c⊕d on X defined by (c⊕d)(f) := c(f)⊕d(f) together with the isomorphisms (c ⊕ d)(f ∘ g) ≅ g^{∗}(c ⊕ d)(f) induced by those of c and d. The abelian group consisting of all the i-th Chow cohomology classes on X together with the operation of sum is called the i-th Chow group of X and is denoted by A^{i}(X).

If X is isomorphic to a quotient stack [X/G], where X is a scheme of finite type over k and G is a group scheme of finite type over k, then A^{i}(X) is isomorphic to the i-th (operational) equivariant Chow group A_{G}^{i}(X) defined by Edidin-Graham in [EG98, Sec. 2.6] (see [EG98, Prop. 19]). The first Chern class gives an homomorphism

(2.3)  \begin{matrix}c_{1} : Pic(X) & ⟶ A^{1}(X) \\ L & ↦ c_{1}(L) \end{matrix}

where c_{1}(L) ∈ A^{1}(X) is defined by setting c_{1}(L)(f) := c_{1}(L(f)) for every morphism f : S → X from a scheme S. In the sequel, we will use the following results concerning the Picard group of a smooth quotient stack.

Fact 2.9 (Edidin-Graham). Let X be a smooth quotient stack, i.e. X = [X/G] where X is a smooth variety and G is an algebraic group acting on X.

(i) The first Chern class map c_{1} : Pic(X) → A^{1}(X) is an isomorphism. In particular, every Weil divisor D on X is a Cartier divisor and hence it gives rise to a line bundle O_{X} (D) on X.

(ii) Given a Weil divisor D of X with irreducible components D_{i}, there is an exact sequence

⨁_{i} ℤ · ⟨O_{X} (D_{i})⟩ → Pic(X) → Pic(X \ D) → 0.

(iii) If Y is a closed substack of X of codimension greater than 1 then there is

an isomorphism

Pic(X) →^{≅} Pic(X \ Y).

Proof. Part (i) follows from [EG98, Cor. 1]. Part (ii) follows from [EG98, Prop. 5]. Part (iii) follows from [EG98, Lemma 2(a)]. ■

By Theorems 2.5, all the properties stated in Fact 2.9 hold for the stacks we will deal with, namely J ac_{d,g}, J_{d,g}, \overline{J ac}_{d,g} and \overline{J}_{d,g}. Moreover, it is well-known that the same properties hold true for \overline{M}_{g} and M_{g}. There are two standard methods to produce line bundles on a stack parametrizing nodal curves with some extra-structure (as \overline{J ac}_{d,g}), namely the determinant of cohomology (introduced in [KM76]) and the Deligne pairing (introduced in [Del87]). Let us review briefly the definition and main properties of these two constructions, following the presentation given in [ACG11, Chap. 13, Sec. 4 and 5]. Let π : X → S be a family of nodal curves, i.e. a proper and flat morphism whose geometric fibers are nodal curves. Given a coherent sheaf F on X flat over S (e.g. a line bundle on X), the determinant of cohomology of F is a line bundle d_{π}(F) ∈ Pic(S) defined as it follows: we choose a complex of locally free sheaves f : K^{0} → K^{1} on S such that ker f = π_{∗}(F) and coker f = R^{1}π_{∗}(F) (this is always possible) and we set

d_{π}(F) := det K^{0} ⊗ (det K^{1})^{−1}.

The determinant of cohomology is functorial, multiplicative for short exact sequence and its first Chern class is equal to

(2.4)  c_{1}(d_{π}(F)) = c_{1}(π_{!}(F)) := c_{1}(π_{∗}(F)) − c_{1}(R^{1}π_{∗}(F)).

For more details, the reader is referred to [ACG11, Chap. 13, Sec. 4]. Given two line bundles M and L on the total space of a family of nodal curves π : X → S, the Deligne pairing of M and L is a line bundle ⟨M, L⟩_{π} ∈ Pic(S) which can be defined as

(2.5)  ⟨M, L⟩_{π} := d_{π}(M ⊗ L) ⊗ d_{π}(M)^{−1} ⊗ d_{π}(L)^{−1} ⊗ d_{π}(O_{X}).

The Deligne pairing is functorial, symmetric and bilinear in each factor, and its first Chern class satisfies

(2.6)  c_{1}(⟨M, L⟩_{π}) = π_{∗}(c_{1}(M) · c_{1}(L)).

For more details, the reader is referred to [ACG11, Chap. 13, Sec. 5].

Remark 2.10. Since the determinant of cohomology and the Deligne pairing are functorial, we can extend their definition to the case when π : Y → X is a representable, proper and flat morphism of Artin stacks whose geometric fibers are nodal curves.

2.11. The Picard group of \overline{M}_{g} In this subsection, in order to fix the notation, we recall the description of the Picard group Pic(\overline{M}_{g}). The universal family \overline{π} : \overline{M}_{g,1} → \overline{M}_{g} is a representable, proper and flat morphism whose geometric fibers are nodal curves. Applying the determinant of cohomology to the relative dualizing sheaf ω\overline{π} (see 2.6), we define the Hodge line bundle

(2.7)  Λ := d\overline{π}(ω\overline{π}) ∈ Pic(\overline{M}_{g}).

Using the functoriality of the determinant of cohomology, it is easily checked that Λ associates to a family of stable curves \{f : C → S\} ∈ \overline{M}_{g}(S) the line bundle

Λ(f) = det f_{∗}(ω_{C/S}) ⊗ det(R^{1}f_{∗}(ω_{C/S}))^{−1} = ⋀^{g} f_{∗}(ω_{C/S}) ∈ Pic(S).

We will abuse the notation and denote also with Λ the restriction of Λ to M_{g} is also denoted by Λ. Recall that the boundary \overline{M}_{g} \ M_{g} decomposes as the union of irreducible divisors δ_{i} for i = 0, …, [g/2] which are defined as follows: δ_{0} is the boundary divisor of \overline{M}_{g} whose generic point is an irreducible nodal curve of genus g with one node while, for any 1 ≤ i ≤ [g/2], δ_{i} is the boundary divisor of \overline{M}_{g} whose generic point is a stable curve formed by two irreducible components of genera i and g − i meeting in one point. We will denote by Δ_{i} ⊂ \overline{M}_{g} the image of δ_{i} ⊂ \overline{M}_{g} via the natural map \overline{M}_{g} → \overline{M}_{g}. We set δ := ∑_{i} δ_{i} and denote by O(δ) the associated line bundle on \overline{M}_{g} (see Fact 2.9(i)). Similarly for O(δ_{i}) ∈ Pic(\overline{M}_{g}). The Picard groups of \overline{M}_{g} and of M_{g} are described by the following theorem proved by Arbarello-Cornalba in [AC87, Thm. 1], based upon a result of Harer [Har83].

Theorem 2.12 (Harer, Arbarello-Cornalba). Assume that g ≥ 3. Then

(i) Pic(M_{g}) is freely generated by Λ. (ii) Pic(\overline{M}_{g}) is freely generated by Λ, O(δ_{0}), ⋯, O(δ_{[g/2]}).

If g = 2, then Pic(M_{g}) (resp. Pic(\overline{M}_{g})) is still generated by Λ (resp. by Λ, O(δ_{0}), O(δ_{1})) but with the extra relation Λ^{10} = 0 (resp. Λ^{10}⊗O(−δ_{0}−2δ_{1}) = 0), see respectively [Vis98] and [Cor07].

3. Boundary divisors of \overline{J ac}_{d,g}

The aim of this Section is to describe the irreducible components of the boundary divisor \overline{J ac}_{d,g} and their relationship with the boundary divisors of \overline{M}_{g}. Consider the following divisors in the boundary of \overline{J ac}_{d,g}:

(A) δ̃_{0} is the divisor whose generic point is a pair (C, L) where C is an irreducible curve of genus g with one node and L is a degree d line bundle on it.

(B) For 1 ≤ i ≤ g/2 and k_{d,g} ∤ (2i − 1), δ̃_{i} is the divisor whose generic point is a pair (C, L), where C is formed by two smooth irreducible curves C_{1} and C_{2} of genera respectively i and g − i meeting in one point, and L is a line bundle of multidegree

(deg_{C_{1}} L, deg_{C_{2}} L) = ([d\frac{2i − 1}{2g − 2} + \frac{1}{2}] , [d\frac{2(g − i) − 1}{2g − 2} + \frac{1}{2}]) .

(C) For 1 ≤ i < g/2 and k_{d,g} | (2i − 1), δ̃_{i}^{1} (resp. δ̃_{i}^{2}) is the divisor whose generic point is a pair (C, L_{1}) (resp. (C, L_{2})), where C consists of two smooth irreducible curves C_{1} and C_{2} of genera respectively i and g − i meeting in one point, and L_{1} and L_{2} are line bundles of multidegree

(deg_{C_{1}} L_{1}, deg_{C_{2}} L_{1}) = (d\frac{2i − 1}{2g − 2} − \frac{1}{2}, d\frac{2(g − i) − 1}{2g − 2} + \frac{1}{2}) .

(deg_{C_{1}} L_{2}, deg_{C_{2}} L_{2}) = (d\frac{2i − 1}{2g − 2} + \frac{1}{2}, d\frac{2(g − i) − 1}{2g − 2} − \frac{1}{2}) .

(D) If g is even and k_{d,g} | (g−1) (i.e. d is odd), δ̃_{g/2} is the divisor whose generic point is a pair (C, L), where C is formed by two smooth irreducible curves C_{1} and C_{2} both of genera g/2 meeting in one point, and L is a line bundle of multidegree

(deg_{C_{1}} L, deg_{C_{2}} L) = (\frac{d − 1}{2}, \frac{d + 1}{2}) .

Note that in the above cases (C) and (D), the divisibility condition k_{d,g} | (2i−1) is equivalent to the condition that M_{C_{i}}(d) and m_{C_{i}}(d) are integers (see Definition 2.3). Moreover, the case (D) is different from the case (C) since in the case (D) the two components C_{1} and C_{2} have the same genus and hence it is not possible to distinguish “numerically” a line bundle of multidegree (deg_{C_{1}} L, deg_{C_{2}} L) = (\frac{d−1}{2}, \frac{d+1}{2}) from one of multidegree (deg_{C_{1}} L, deg_{C_{2}} L) = (\frac{d+1}{2}, \frac{d−1}{2}).

3.1. Notation: Sometimes it is convenient to unify the notation for the cases (A) and (B) and for the cases (C) and (D). For this reason, we always assume that k_{d,g} ∤ (2 · 0 − 1) = −1 (even when k_{d,g} = 1) and we set δ̃_{g/2}^{1} = δ̃_{g/2}^{2} = δ̃_{g/2} if g is even and k_{d,g} | (g − 1) (i.e. if g is even and d is odd).

As usual, we denote by O(δ̃_{i}) the line bundle on \overline{J ac}_{d,g} associated to δ_{i} and similarly for O(δ̃_{i}^{1}) and O(δ̃_{i}^{2}). Using the above Notation 3.1, we also set

(3.1)  δ̃:= ∑_{k_{d,g}∤(2i−1)} δ̃_{i} + ∑_{k_{d,g}|(2i−1)}(δ̃_{i}^{1} + δ̃_{i}^{2}),

and we denote by O(δ̃) ∈ Pic(\overline{J ac}_{d,g}) its associated line bundle. Note that, according to Notation 3.1, if g is even and d is odd then δ̃_{g/2} = δ̃_{g/2}^{1} = δ̃_{g/2}^{2}

appears with coefficient two in δ̃. Via the natural forgetful map Φ̃_{d} : \overline{J ac}_{d,g} → \overline{M}_{g}, we can relate the boundary divisors of \overline{J ac}_{d,g} with those of \overline{M}_{g} as follows.

Theorem 3.2.

(i) The boundary \overline{J ac}_{d,g} \J ac_{d,g} of \overline{J ac}_{d,g} consists of the irreducible divisors {δ̃_{i} : k_{d,g} ∤ (2i − 1) or i = g/2} and {δ̃_{i}^{1}, δ̃_{i}^{2} : k_{d,g} | (2i − 1) and i < g/2}.

(ii) For any 0 ≤ i ≤ g/2, we have

Φ̃_{d}^{∗} O(δ_{i}) = \{ \begin{matrix}O(δ̃_{i}) & if k_{d,g} ∤ (2i − 1), \\ O(δ̃_{i}^{1} + δ̃_{i}^{2}) & if k_{d,g} | (2i − 1). \end{matrix}

In particular, Φ̃_{d}^{∗} O(δ) = O(δ̃).

Proof. By construction we have that \overline{J ac}_{d,g} \ J ac_{d,g} = Φ̃_{d}^{−1}(\overline{M}_{g} \ M_{g}) (see 2.1) and moreover \overline{M}_{g} \M_{g} = ⋃_{i} δ_{i} (see 2.11). By the Definition 2.3, it is easy to check that we have a set-theoretical equality

(3.2)  Φ̃_{d}^{−1}(δ_{i}) = \{ \begin{matrix}δ̃_{i} & if k_{d,g} ∤ (2i − 1), \\ δ̃_{i}^{1} ∪ δ̃_{i}^{2} & if k_{d,g} | (2i − 1). \end{matrix}

Finally, by looking at their definition, it is easy to see that the divisors δ̃_{i}, δ̃_{i}^{1}, δ̃_{i}^{2} are irreducible. This completes the proof of part (i). Part (ii) is equivalent to proving that we have a scheme-theoretic equality in (3.2). To achieve that, we need a local description of the morphism Φ̃_{d} : \overline{J ac}_{d,g} → \overline{M}_{g} at a general point (C, L) of δ̃_{i} or of δ̃_{i}^{1} ∩ δ̃_{i}^{2}. Recall that locally at (C, L), the morphism Φ̃_{d} looks like

q : [Def_{(C,L)} / Aut(C, L)] → [Def_{C^{st}} / Aut(C^{st})],

where Def_{C^{st}}(resp. Def_{(C,L)}) is the miniversal deformation space of the stabilization C^{st} of C (resp. of the pair (C, L)) and Aut(C^{st}) (resp. Aut(C, L)) is the automorphism group of C^{st} (resp. the automorphism group of the pair (C, L)). Using the results on the local structure of \overline{J ac}_{d,g} given in [BFV12, Sec. 2.15], we can describe explicitly the above morphism q at a general point of δ̃_{i} or of δ̃_{i}^{1}∩δ̃_{i}^{2} in the boundary of \overline{J ac}_{d,g}. To this aim, we need to distinguish between the case k_{d,g} ∤ (2i − 1) (cases (A) and (B)) and the case k_{d,g} | (2i − 1) (cases (C) and (D)). Suppose first that k_{d,g} ∤ (2i − 1). Consider a general point (C, L) of δ̃_{i}. Since C = C^{st} is a general element of δ_{i}, it is well-known that Def_{C} = Spf k[[x_{1}, ⋯, x_{3g−3}]] and

(3.3)  Aut(C) = \begin{cases}\{1\} & if i ≠ 1, \\ ℤ/2ℤ & if i = 1, \end{cases}

where, in the second case, the unique non-trivial automorphism is the elliptic involution on the elliptic tail of C. On the other hand, we have that Def_{(C,L)} = Spf k[[x_{1}, ⋯, x_{3g−3}, t_{1}, ⋯, t_{g}]] and Aut(C, L) = 𝔾_{m} acts trivially on it (see [BFV12, Proof of Thm. 1.5, Cases (1) and (2)]), where the coordinates x_{i}’s correspond to the deformation of the curve C and the coordinates t_{j}’s correspond to the deformation of the line bundle L. The morphism q is given by the natural equivariant projection Def_{(C,L)} ↠ Def_{C}. Moreover, we can choose local coordinates x_{1}, ⋯, x_{3g−3} for Def_{C} in such a way that the first coordinate x_{1} corresponds to the smoothing of the unique node of C and, if i = 1, the action of the generator of Aut(C) = ℤ/2ℤ sends x_{1} into −x_{1} and fixes the other coordinates. For such a choice of the coordinates, we have that the equation of δ_{i} inside Def_{C} is given by (x_{1} = 0) and the equation of δ̃_{i} inside Def_{(C,L)} is given by (x_{1} = 0). Since q^{∗}(x_{1}) = (x_{1}), we conclude in this case. Suppose now that k_{d,g} | (2i − 1) (hence that i > 0 by Notation 3.1). If i < g/2 then a general point (C, L) of δ̃_{i}^{1} ∩ δ̃_{i}^{2} consists of the two general curves C_{1} and C_{2} of genera respectively i and g − i joined by a rational curve R ≅ ℙ^{1}. By convention, in the case i = g/2 and k_{d,g} | (g − 1), we set δ̃_{g/2}^{1} ∩

δ̃_{g/2}^{2} to be the closure of the locus of curves consisting of two smooth curves of genera g/2 joined by a rational curve R ≅ ℙ^{1}. The stabilization C^{st} is obtaining by contracting the rational curve R to a node n and it will be a general point of δ_{i}. As before, we have that Def_{C^{st}} = Spf k[[x_{1}, ⋯, x_{3g−3}]], where x_{1} can be chosen as the coordinate corresponding to the smoothing of the node n, and Aut(C^{st}) is as in (3.3). On the other hand, by [BFV12, Proof of Theorem 1.5, Case (3)], we have that Aut (C, L) = 𝔾_{m}^{2}, Def_{(C,L)} = Spf k[[u_{1}, v_{1}, x_{2}, ⋯, x_{3g−3}, t_{1}, ⋯, t_{g}]] where u_{1} corresponds to the node C_{1}∩R and v_{1} corresponds to the node C_{2} ∩R. Moreover, the action of 𝔾_{m}^{2} on Def_{(C,L)} is given by (λ, μ) · (u_{1}, v_{1}) = (λμ^{−1}u_{1}, λ^{−1}μv_{1}) while it is the identity on the other coordinates. The morphism q is induced by the equivariant morphism Def_{(C,L)} → Def_{C^{st}} that, at the level of rings, sends x_{1} into u_{1} · v_{1} and x_{i} into x_{i} for i > 1. The equation of δ_{i} inside Def_{C^{st}} is given by (x_{1} = 0) while the equations of δ̃_{i}^{1} and δ̃_{i}^{2} inside Def_{(C,L)} are given by (u_{1} = 0) and (v_{1} = 0) (note that in the special case i = g/2 and k_{d,g} | (g − 1), the divisor δ̃_{g/2}, even though irreducible, has two branches locally at (C, L), which we call δ̃_{g/2}^{1} and

δ̃_{g/2}^{2}, whose equations are (u_{1} = 0) and (v_{1} = 0)). Since q^{∗}(x_{1}) = (u_{1} · v_{1}), we conclude also in this case.

■

As a Corollary of the above Theorem 3.2, we can determine also the irreducible components of the boundary of \overline{J}_{d,g}. We set \overline{δ}_{i} := ν_{d}(δ̃_{i}), \overline{δ}_{i}^{1} = ν_{d}(δ̃_{i}^{1}) and \overline{δ}_{i}^{2} := ν_{d}(δ̃_{i}^{2}) according to the above Cases (A)–(B), where as usual ν_{d} : \overline{J ac}_{d,g} → \overline{J}_{d,g} is the rigidification map.

Corollary 3.3.

(i) The boundary \overline{J}_{d,g} \ J_{d,g} of \overline{J}_{d,g} consists of the irreducible divisors \{\overline{δ}_{i} : k_{d,g} ∤ (2i − 1) or i = g/2\} and \{\overline{δ}_{i}^{1}, \overline{δ}_{i}^{2} : k_{d,g} | (2i − 1) and i < g/2\}.

(ii) For any 0 ≤ i ≤ g/2, we have

\{ \begin{matrix}ν_{d}^{∗} O(\overline{δ}_{i}) = O(δ̃_{i}) & if k_{d,g} ∤ (2i − 1), \\ ν_{d}^{∗} O(\overline{δ}_{i}^{j}) = O(δ̃_{i}^{j}) & if k_{d,g} | (2i − 1) and j = 1, 2. \end{matrix}

Proof. The Corollary follows straightforwardly from Theorem 3.2 and the fact that ν_{d} : \overline{J ac}_{d,g} → \overline{J}_{d,g} is a 𝔾_{m}-gerbe such that ν_{d}^{−1}(J_{d,g}) = J ac_{d,g}. ■

4. Independence of the boundary divisors

The aim of this Section is to prove that the line bundles corresponding to the irreducible components of the boundary of \overline{J ac}_{d,g} are linearly independent in Pic(\overline{J ac}_{d,g}). More precisely, we will prove the following result.

Theorem 4.1. We have an exact sequence (4.1)

0 → ⨁_{or i=g/2^{k_{d,g} ∤ 2i−1}} ⟨O(δ̃_{i})⟩ ⨁_{and i≠g/2^{k_{d,g} |2i−1}} ⟨O(δ̃_{i}^{1})⟩ ⊕ ⟨O(δ̃_{i}^{2})⟩ → Pic(\overline{J ac}_{d,g}) → Pic(J ac_{d,g}) → 0,

where the right map is the natural restriction morphism and the left map is the natural inclusion.

Using Theorem 3.2(i) and Fact 2.9(ii), we have that the exact sequence (4.1) is exact except perhaps to the left. It remains to prove that the map on the left is injective, or in other words that the line bundles associated to the boundary divisors of \overline{J ac}_{d,g} are linearly independent in Pic(\overline{J ac}_{d,g}). The strategy that we will use to prove this is the same as the one used by Arbarello-Cornalba in [AC87]: we shall construct maps B → \overline{J ac}_{d,g} from irreducible smooth projective curves B (i.e. families of quasistable curves of genus g parametrized by B, endowed with a properly balanced line bundle of relative degree d) and compute the degree of the pullbacks of the boundary divisors of Pic(\overline{J ac}_{d,g}) to B. Actually, we will construct liftings of the families F_{h} (for 1 ≤ h ≤ (g − 2)/2), F and F′ used by Arbarello-Cornalba in [AC87, p. 156-159]. For that reason, we will be using their notations. Note that, for every n ∈ ℤ, there are isomorphisms

(4.2)  \begin{matrix}ϕ̃_{d}^{n} : \overline{J ac}_{d,g} & ⟶^{≅} \overline{J ac}_{d+n(2g−2),g} \\ (C → S, L) & ↦ (C → S, L ⊗ ω_{C/S}^{⊗n}). \end{matrix}

Clearly, ϕ̃_{d}^{n} is an isomorphism of 𝔾_{m}-stacks and therefore, by passing to the 𝔾_{m}-rigidification, it induces an isomorphism ϕ_{d}^{n} : \overline{J}_{d,g} →^{≅} \overline{J}_{d+n(2g−2),g}. Since \overline{J ac}_{d,g} ≅ \overline{J ac}_{d′,g} if d ≡ d′ mod (2g−2) (see 4.2), throughout this section we can make the following

Assumption 4.2. The degree d satisfies 0 ≤ d < 2g − 2.

The Family F̃ Start from a general pencil of conics in ℙ^{2}. Blowing up the four base points of the pencil, we get a conic bundle ϕ : X → ℙ^{1}. The four exceptional divisors E_{1}, E_{2}, E_{3}, E_{4} ⊂ X of the blow-up of ℙ^{2} are sections of ϕ through the smooth locus of ϕ. Note that ϕ will have three singular fibers consisting of two incident lines. Let C be a fixed irreducible, smooth and projective curve of genus g − 3 and p_{1}, p_{2}, p_{3}, p_{4} four points of C. We construct a surface Y by setting

Y = (X ∐(C × ℙ^{1})) /(E_{i} ∼ {p_{i}} × ℙ^{1} : i = 1, ⋯, 4).

We get a family f : Y → ℙ^{1} of stable curves of genus g: the general fiber of f consists of C and a smooth conic Q meeting in 4 points (see Figure 1 below), while the three special fibers consist of C and two lines R_{1} and R_{2} such that |R_{1} ∩ R_{2}| = 1, |R_{1} ∩ C| = |R_{2} ∩ C| = 2 (see Figure 2 below).

C  Q g − 3

Figure 1. The general fiber of f : Y → ℙ^{1}

R_{1}  R_{2} C

Figure 2. The three special fibers of f : Y → ℙ^{1}

Choose a line bundle L of degree d on C, pull it back to C × ℙ^{1} and call it again L. Since L is trivial when restricted to {p_{i}} × ℙ^{1}, we can glue it with the trivial line bundle on X and, thus, we obtain a line bundle L on the family Y → ℙ^{1} of relative degree d.

Lemma 4.3. The line bundle L is properly balanced.

Proof. Since the property of being properly balanced is an open condition, it is enough to check that L is properly balanced on the three special fibers of f : Y → ℙ^{1}. According to Remark 2.4, it is enough to check the basic inequality for the three subcurves R_{1}∪R_{2}, R_{1} and R_{2}. The balancing condition for R_{1}∪R_{2}

|deg_{R_{1}∪R_{2}}(L) − {d · 2 \atop 2g − 2}| ≤ {4 \atop 2},

is true because deg_{R_{1}∪R_{2}}(L) = 0 and 0 ≤ d < 2g − 2. The balancing condition for each of the subcurves R_{i} (i = 1, 2) is

|deg_{R_{i}}(L) − \frac{d · 1}{2g − 2}| ≤ \frac{3}{2},

which is satisfied because deg_{R_{i}}(L) = 0 and 0 ≤ d < 2g − 2. ■

We call F̃ the family f : Y → ℙ^{1} endowed with the line bundle L. Forgetting the line bundle L, we are left with the family F of [AC87, p. 158]. We can compute the degree of the pull-backs of the boundary classes in Pic(\overline{J ac}_{d,g}) to the curve F̃: (4.3)

degF̃ O(δ̃_{0}) = −1,
degF̃ O(δ̃_{i}) = 0  if 1 ≤ i and k_{d,g} ∤ (2i − 1) or i = g/2,
degF̃ O(δ̃_{i}^{1}) = degF̃ O(δ̃_{i}^{2}) = 0  if 1 ≤ i < g/2 and k_{d,g} | (2i − 1).

The first relation follows from the fact that degF̃ O(δ̃_{0}) = deg_{F} O(δ_{0}) (by using the projection formula) and the relation deg_{F} O(δ_{0}) = −1 proved in [AC87, p. 158]. The last two relations follow by the obvious fact that F̃ does not meet the divisors δ̃_{i} or δ̃_{i}^{1} and δ̃_{i}^{2} for i ≥ 1. The Families F̃′_{1} and F̃′_{2} We start with the same family of conics ϕ : X → ℙ^{1} that we considered in the construction of the family F̃. Let C be a fixed irreducible, smooth and projective curve of genus g − 3, E be a fixed irreducible, smooth and projective elliptic curve and take points p_{1} ∈ E and p_{2}, p_{3}, p_{4} ∈ C. We construct a surface Z by setting

Z = (X ∐(C × ℙ^{1}) ∐(E × ℙ^{1})) /(E_{i} ∼ {p_{i}} × ℙ^{1} : i = 1, ⋯, 4).

We get a family g : Z → ℙ^{1} of stable curves of genus g: the general fiber of g consists of C, E and a smooth conic Q intersecting as in Figure 3. The three special fibers consist of C, E and two lines R_{1} and R_{2}, intersecting as shown in Figure 4. We choose two line bundles of degree d and d − 3 on C, we pull them back to C × ℙ^{1} and call them, respectively, L_{1} and L_{2}. Similarly, we choose two line bundles of degree 0 and 1 on E, we pull them back to E × ℙ^{1} and call them, respectively, M_{1} and M_{2}. We glue the line bundle L_{1} (resp. L_{2}) on C × ℙ^{1}, the line bundle M_{1} (resp. M_{2}) on E ×ℙ^{1} and the line bundle O_{X} (resp. ω_{X/ℙ^{1}}^{−1}, the relative anti-canonical bundle of ϕ : X → ℙ^{1}) on X, obtaining a line bundle M_{1} (resp. M_{2}) on Z of relative degree d.

Lemma 4.4. The line bundle M_{1} is properly balanced if 0 ≤ d ≤ g − 1. The line bundle M_{2} is properly balanced if g − 1 ≤ d < 2g − 2.

Proof. The proof is straightforward and similar to the one of Lemma 4.3: we leave it to the reader.

E Q C

Figure 3. The general fibers of g : Z → ℙ^{1}.

E C R_{1}  R_{2}

Figure 4. The three special fibers of g : Z → ℙ^{1}.

■

If 0 ≤ d ≤ g −1, we call F̃′_{1} the family g : Z → ℙ^{1} endowed with the line bundle M_{1}; if g − 1 ≤ d < 2g − 2, we call F̃′_{2} the family g : Z → ℙ^{1} endowed with the line bundle M_{2}. Both families F̃′_{1} and F̃′_{2}, when defined, are liftings of the family F′ of [AC87, p. 158]. We can compute the degree of the pull-backs of some of the boundary classes in Pic(J ac_{d,g}) to the curves F̃′_{1} and F̃′_{2}, in the ranges of degrees where they are defined (note that Φ̃_{d}^{−1}(δ_{1}) is the union of two irreducible divisors if and only if k_{d,g} = 1, i.e. iff d = g − 1): (4.4)

degF̃′_{1} O(δ̃_{1}) = degF̃′_{2} O(δ̃_{1}) = −1 if d ≠ g − 1,
degF̃′_{1} O(δ̃_{1}^{1}) = degF̃′_{2} O(δ̃_{1}^{2}) = −1
and degF̃′_{1} O(δ̃_{1}^{2}) = degF̃′_{2} O(δ̃_{1}^{1}) = 0 if d = g − 1,

degF̃′_{1} O(δ̃_{i}) = degF̃′_{2} O(δ̃_{i}) = 0 if 1 < i and k_{d,g} ∤ (2i − 1) or i = g/2,

degF̃′_{1} O(δ̃_{i}^{j}) = degF̃′_{2} O(δ̃_{i}^{j}) = 0 if 1 < i < g/2
and k_{d,g} | (2i − 1), for j = 1, 2.

The first relation follow, by using the projection formula, from the relation deg_{F′} O(δ_{1}) = −1 proved in [AC87, p. 159]. The second relation is deduced in a similar way using the projection formula and the (easily checked) fact that F̃′_{1} does not meet δ̃_{1}^{2} and that F̃′_{2} does not meet δ̃_{1}^{1}. The last two relations follow from the fact that F̃′_{1} and F̃′_{2} do not meet the divisors δ̃_{i} or δ̃_{i}^{1} and δ̃_{i}^{2} for i > 1. The Families F̃_{h,1} and F̃_{h,2} (for 1 ≤ h ≤ \frac{g−2}{2}) Fix irreducible, smooth and projective curves C_{1}, C_{2} and Γ of genera h, g−h−1 and 1, and points x_{1} ∈ C_{1}, x_{2} ∈ C_{2} and γ ∈ Γ. Consider the surfaces Y_{1} = C_{1} × Γ, Y_{3} = C_{2} × Γ and Y_{2} given by the blow-up of Γ × Γ at (γ, γ). Let us denote by p_{2} : Y_{2} → Γ the map given by composing the blow-down Y_{2} → Γ × Γ with the second projection, and by π_{1} : Y_{1} → Γ and π_{3} : Y_{3} → Γ the projections along the second factor. As in [AC87, p. 156], we set (see also Figure 5):

A = {x_{1}} × Γ,

B = {x_{2}} × Γ,

E = exceptional divisor of the blow-up of Γ × Γ at (γ, γ), Δ = proper transform of the diagonal in Y_{2},

S = proper transform of {γ} × Γ in Y_{2},

T = proper transform of Γ × {γ} in Y_{2}.

We construct a surface X by identifying S with A and Δ with B. The surface X comes equipped with a projection f : X → Γ. The fibers over all the points γ′ ≠ γ are shown in Figure 6, while the fiber over the point γ is shown in Figure 7. We will first construct several line bundles over the three surfaces Y_{1}, Y_{2} and Y_{3}, and then we will glue them in a suitable way. Consider the line bundles M_{i} (i = 1, ⋯, 4) on Y_{2} given by

M_{1} := O_{Y_{2}}, M_{2} := O_{Y_{2}}(Δ), M_{3} := O_{Y_{2}}(Δ + E), M_{4} := O_{Y_{2}}(2Δ + E).

[figure: A; Γ; Γ; Γ; π_{1}; π_{3}; C_{1}; B; S; Γ; Δ; T; E; C_{2}]

Figure 5. Constructing f : X → Γ.

C_{1} h  Γ 1  C_{2} g − h − 1

Figure 6. The general fiber of f : X → Γ.

E C_{1}  C_{2}  Γ

Figure 7. The special fiber of f : X → Γ.

Using that deg_{E} O(E) = −1, we get that the restrictions of M_{i} to E and T have degrees:

(deg_{E} M_{i}, deg_{T} M_{i}) = \begin{cases}(0, 0) & if i = 1, \\ (1, 0) & if i = 2, \\ (0, 1) & if i = 3, \\ (1, 1) & if i = 4. \end{cases}

Documenta Mathematica 19 (2014) 457–507

Notice that the diagonal \overline{Δ} of Γ×Γ is such that O_{Γ×Γ}(\overline{Δ})_{|\overline{Δ}} = O\overline{Δ} since Γ is an elliptic curve. By applying the projection formula to the blow-up Y_{2} → Γ × Γ, we get that O_{Y_{2}}(Δ)_{|Δ} = O_{Δ}(−γ). Using this, we can easily compute the restrictions of M_{i} to S and Δ (which are canonically isomorphic to Γ):

(4.5) (M_{i})_{|Δ} = \begin{cases}O_{Γ} & if i = 1, 3 \\ O_{Γ}(−γ) & if i = 2, 4 \end{cases} and (M_{i})_{|S} = \begin{cases}O_{Γ} & if i = 1, 2 \\ O_{Γ}(γ) & if i = 3, 4. \end{cases}

Consider now the integers α_{1}, α_{2} defined by:

α_{1} := ⌊\frac{d(2g − 2h − 3)}{2g − 2}⌋ , α_{2} := ⌈\frac{d(2g − 2h − 3)}{2g − 2}⌉ ,

if \frac{d(2g − 2h − 3)}{2g − 2} ≡ \frac{1}{2} mod ℤ

α_{1} = α_{2} := the unique integer which is closest to \frac{d(2g − 2h − 3)}{2g − 2},

otherwise.

Take two line bundles on C_{2} of degrees α_{1} and α_{2}, and call, respectively, L_{1} and L_{2} their pull-backs to Y_{3} = C_{2} × Γ. We may assume that L_{1} = L_{2} if α_{1} = α_{2}. Analogously, consider the integers β_{1}, β_{2} defined by:

β_{1} := ⌊\frac{d(2h − 1)}{2g − 2}⌋ , β_{2} := ⌈\frac{d(2h − 1)}{2g − 2}⌉, if \frac{d(2h − 1)}{2g − 2} ≡ \frac{1}{2} mod ℤ

β_{1} = β_{2} := the unique integer which is closest to \frac{d(2h − 1)}{2g − 2}, otherwise.

Consider two line bundles on C_{1} of degrees β_{1} and β_{2}, and call, respectively, N_{1} and N_{2} their pull-back to Y_{1} = C_{1} × Γ. We may assume that N_{1} = N_{2} if β_{1} = β_{2}. We now want to define two (possibly equal) line bundles I_{1} and I_{2} on X, by gluing in a suitable way some of the line bundles on Y_{1}, Y_{2} and Y_{3}, we have just defined. We shall distinguish between several cases: \underline{CASE A:} \frac{d(2g−2h−3)}{2g−2} ≢ \frac{1}{2} mod ℤ (i.e. α_{1} = α_{2}). In this case, we have that

(4.6) α_{1} − \frac{1}{2} < \frac{d(2g − 2h − 3)}{2g − 2} < α_{1} + \frac{1}{2} and β_{1} − \frac{1}{2} < \frac{d(2h − 1)}{2g − 2} ≤ β_{1} + \frac{1}{2}.

\underline{Subcase A1:} 0 ≤ d ≤ g − 1. Using the inequalities (4.6), we get that

−1 ≤ −1 + \frac{d}{g − 1} = −1 + d − \frac{d(2g − 2h − 3)}{2g − 2} − \frac{d(2h − 1)}{2g − 2} < d − α_{1} − β_{1} <

(4.7)  < 1 + d − \frac{d(2g − 2h − 3)}{2g − 2} − \frac{d(2h − 1)}{2g − 2} = 1 + \frac{d}{g − 1} < 2.

If d − α_{1} − β_{1} = 0 then we define I_{1} = I_{2} to be equal to the line bundle on X obtained by gluing N_{1}, M_{1} and L_{1} = L_{2}, which is possible since, by (4.5), we have that (N_{1})_{|A} = O_{Γ} = (M_{1})_{|S} and (L_{1})_{|B} = O_{Γ} = (M_{1})_{|Δ}.

Otherwise, if d − α_{1} − β_{1} = 1, then we define I_{1} = I_{2} to be equal to the line bundle on X obtained by gluing the sheaves N_{1}, M_{2} and L_{1} ⊗ π_{3}^{∗}O_{Γ}(−γ), which is possible since, by (4.5), we have that (N_{1})_{|A} = O_{Γ} = (M_{2})_{|S} and (L_{1} ⊗ π_{3}^{∗}O_{Γ}(−γ))_{|B} = O_{Γ}(−γ) = (M_{2})_{|Δ}. \underline{Subcase A2:} g − 1 < d < 2g − 2. Arguing similarly to the above inequality (4.7), we get that d − α_{1} − β_{1} = 1, 2. If d − α_{1} − β_{1} = 1, then we define I_{1} = I_{2} to be equal to the line bundle on X obtained by gluing N_{1} ⊗π_{1}^{∗}O_{Γ}(γ), M_{3} and L_{1}, which is possible since, by (4.5), we have that (N_{1} ⊗π_{1}^{∗}O_{Γ}(γ))_{|A} = O_{Γ}(γ) = (M_{3})_{|S} and (L_{1})_{|B} = O_{Γ} = (M_{3})_{|Δ}. If d − α_{1} − β_{1} = 2, then we define I_{1} = I_{2} to be equal to the line bundle on X obtained by gluing N_{1} ⊗ π_{1}^{∗}O_{Γ}(γ), M_{4} and L_{1} ⊗ π_{3}^{∗}O_{Γ}(−γ), which is possible since, by (4.5), we have that (N_{1} ⊗π_{1}^{∗}O_{Γ}(γ))_{|A} = O_{Γ}(γ) = (M_{4})_{|S} and (L_{1} ⊗ π_{3}^{∗}O_{Γ}(−γ))_{|B} = O_{Γ}(−γ) = (M_{4})_{|Δ}. \underline{CASE B:} \frac{d(2g−2h−3)}{2g−2} ≡ \frac{1}{2} mod ℤ (i.e. α_{1} = α_{2} − 1).

In this case, we have that α_{1} + \frac{1}{2} \frac{d(2g−2h−3)}{2g−2} = α_{2} − \frac{1}{2}, β_{1} − \frac{1}{2} < \frac{d(2g−2h−3)}{2g−2} ≤

β_{1} + \frac{1}{2}, and that β_{2} − \frac{1}{2} ≤ \frac{(2h−1)}{2g−2} < β_{2} + \frac{1}{2}. So, arguing similarly to the above inequality (4.7), we get that

d − α_{1} − β_{2} = \begin{cases}1 & if 0 ≤ d ≤ g − 1, \\ 2 & if g − 1 < d < 2g − 2. \end{cases}

If 0 ≤ d ≤ g − 1, we define I_{1} to be equal to the line bundle on X obtained by gluing the sheaves N_{2}, M_{2} and L_{1} ⊗ π_{3}^{∗}O_{Γ}(−γ), which is possible since, by (4.5), we have that (N_{2})_{|A} = O_{Γ} = (M_{2})_{|S} and (L_{1} ⊗ π_{3}^{∗}O_{Γ}(−γ))_{|B} = O_{Γ}(−γ) = (M_{2})_{|Δ}. If g−1 < d < 2g−2, we define I_{1} to be equal to the line bundle on X obtained by gluing N_{2} ⊗π_{1}^{∗}O_{Γ}(γ), M_{4} and L_{1} ⊗π_{3}^{∗}O_{Γ}(−γ). which is possible since, by (4.5), we have that (N_{2} ⊗ π_{1}^{∗}O_{Γ}(γ))_{|A} = O_{Γ}(γ) = (M_{4})_{|S} and (L_{1} ⊗ π_{3}^{∗}O_{Γ}(−γ))_{|B} = O_{Γ}(−γ) = (M_{4})_{|Δ}. Similarly, we get that

d − α_{2} − β_{1} = \begin{cases}0 & if 0 ≤ d < g − 1, \\ 1 & if g − 1 ≤ d < 2g − 2. \end{cases}

If 0 ≤ d < g − 1, we define I_{2} to be equal to the line bundle on X obtained by gluing N_{1}, M_{1} and L_{2}, which is possible since, by (4.5), we have that (N_{1})_{|A} = O_{Γ} = (M_{1})_{|S} and (L_{1})_{|B} = O_{Γ} = (M_{1})_{|Δ}. If g −1 ≤ d < 2g −2, we define I_{2} to be equal to the line bundle on X obtained by gluing N_{1} ⊗ π_{1}^{∗}O_{Γ}(γ), M_{3} and L_{2}, which is possible since, by (4.5), we have that (N_{1} ⊗ π_{1}^{∗}O_{Γ}(γ))_{|A} = O_{Γ}(γ) = (M_{3})_{|S} and (L_{2})_{|B} = O_{Γ} = (M_{3})_{|Δ}.

Lemma 4.5. The line bundles I_{1} and I_{2} on X are properly balanced of relative degree d.

Proof. The proof is straightforward and similar to the one of Lemma 4.3: we leave it to the reader.

■

We call F̃_{h,1} the family f : X → Γ endowed with the line bundle I_{1} and F̃_{h,2} the family f : X → Γ endowed with the line bundle I_{2}. Note that F̃_{h,1} = F̃_{h,2} if and only if we are in case A, which happens exactly when k_{d,g} ∤ 2h + 1. Both families F̃_{h,1} and F̃_{h,2} are liftings of the family F_{h} of [AC87, p. 156]. We can compute the degrees of the pull-backs of some of the boundary classes in Pic(\overline{J ac}_{d,g}) to the curves F̃_{h,1} and F̃_{h,2}:

(4.8)

degF̃_{h,1} O(δ̃_{h+1}) = −1 if k_{d,g} ∤ 2h + 1 or h + 1 = g/2,

degF̃_{h,1} O(δ̃_{h+1}^{1}) = degF̃_{h,2} O(δ̃_{h+1}^{2}) = −1 if k_{d,g} | 2h + 1 and h + 1 ≠ g/2,

degF̃_{h,1} O(δ̃_{h+1}^{2}) = degF̃_{h,2} O(δ̃_{h+1}^{1}) = 0 if k_{d,g} | 2h + 1 and h + 1 ≠ g/2,

degF̃_{h,1} O(δ̃_{i}) = 0 if h + 1 < i and k_{d,g} ∤ (2i − 1) or i = g/2,

degF̃_{h,1} O(δ̃_{i}^{j}) = degF̃_{h,2} O(δ̃_{i}^{j}) = 0 if h + 1 < i < g/2 and k_{d,g} | (2i − 1),

for j = 1, 2.

The first relations follow, by using the projection formula, from the relation deg_{F_{h}} O(δ_{h+1}) = −1 proved in [AC87, p. 157]. The second and third relations are deduced in a similar way using the projection formula and the (easily checked) fact that F̃_{h,1} does not meet δ̃_{h+1}^{2} and F̃_{h,2} does not meet δ̃_{h+1}^{1}. The last two relations follow from the fact that F̃_{h,1} and F̃_{h,2} do not meet the divisors δ̃_{i} or δ̃_{i}^{1} and δ̃_{i}^{2} for i > h + 1. With the help of the above families, we can finally conclude the proof of our main theorem.

Proof of Theorem 4.1. As observed before, it is enough to prove that the line bundles associated to the boundary divisors {δ̃_{i} : k_{d,g} ∤ 2i − 1 or i = g/2}, {δ̃_{i}^{1}, δ̃_{i}^{2} : k_{d,g} | 2i − 1 and i ≠ g/2} (for 0 ≤ i ≤ g/2) are linearly independent on \overline{J ac}_{d,g}. Suppose there is a linear relation

(4.9)  O ( ∑_{or i=g/2^{k_{d,g} ∤ 2i−1}} a_{i}δ̃_{i} + ∑_{and i≠g/2^{k_{d,g} |2i−1}}(a_{i}^{1}δ̃_{i}^{1} + a_{i}^{2}δ̃_{2}^{i})) = O,

in the Picard group of \overline{J ac}_{d,g}. We want to prove that all the above coefficients a_{i}, a_{i}^{1} and a_{i}^{2} are zero. Pulling back the above relation (4.9) to the curve F̃ → \overline{J ac}_{d,g} and using the formulas (4.3), we get that a_{0} = 0. Pulling back (4.9) to the curves F̃′_{1} → \overline{J ac}_{d,g} and F̃′_{2} → \overline{J ac}_{d,g} (in the range of degrees in which they are defined) and using the formulas (4.4), we get that a_{1} = 0 if k_{d,g} ∤ 1 (i.e. if d ≠ g − 1) or that a_{1}^{1} = a_{1}^{2} = 0 if k_{d,g} | 1 (i.e. if d = g − 1). Finally, by pulling back the relation (4.9) to the families F̃_{h,1} → \overline{J ac}_{d,g} and F̃_{h,2} → \overline{J ac}_{d,g} (for any 1 ≤ h ≤ (g − 2)/2) and using the formulas (4.8), we get that a_{h+1} = 0 if k_{d,g} ∤ (2h + 1) or h + 1 = g/2 and a_{h+1}^{1} = a_{h+1}^{2} = 0 if k_{d,g} | (2h + 1) and h + 1 ≠ g/2, which concludes the proof. ■

As a corollary of the above Theorem 4.1, we can prove that the boundary line bundles of \overline{J}_{d,g} are linearly independent.

Corollary 4.6. We have an exact sequence (4.10)

0 → ⨁_{or i=g/2^{k_{d,g} ∤ 2i−1}}⟨O(\overline{δ}_{i})⟩ ⨁_{and i≠g/2^{k_{d,g} |2i−1}}⟨O(\overline{δ}_{i}^{1})⟩⊕⟨O(\overline{δ}_{i}^{2})⟩ → Pic(\overline{J}_{d,g}) → Pic(J_{d,g}) → 0,

where the right map is the natural restriction morphism and the left map is the natural inclusion.

Proof. As observed before, the only thing to prove is that the above sequence is exact on the left, or in other words that the boundary line bundles \{O(\overline{δ}), O(\overline{δ}_{i}^{1}), O(\overline{δ}_{i}^{2})\} are linearly independent in Pic(\overline{J}_{d,g}). This follows from Theorem 4.1 using Corollary 3.3(ii) and the fact that the pull-back map ν_{d}^{∗} : Pic(\overline{J}_{d,g}) → Pic(\overline{J ac}_{d,g}) is injective, as observed in the introduction (see diagram (1.1)).

■

5. Tautological line bundles

The aim of this section is to introduce some natural line bundles on \overline{J ac}_{d,g}, which we call tautological line bundles, and to determine the relations among them. Let π : \overline{J ac}_{d,g,1} → \overline{J ac}_{d,g} be the universal family over \overline{J ac}_{d,g} (see [Mel10] for a modular description of \overline{J ac}_{d,g,1}). The stack \overline{J ac}_{d,g,1} comes equipped with two natural line bundles: the universal line bundle L_{d} and the relative dualizing sheaf ω_{π}. Since π is a representable, flat and proper morphism whose geometric fibers are nodal curves, we can apply the formalism of the determinant of cohomology and of the Deligne pairing (see 2.6) to produce some natural line bundles on \overline{J ac}_{d,g} which we call tautological line bundles:

(5.1)  \begin{matrix}K_{1,0} := ⟨ω_{π}, ω_{π}⟩_{π}, \\ K_{0,1} := ⟨ω_{π}, L_{d}⟩_{π}, \\ K_{−1,2} := ⟨L_{d}, L_{d}⟩_{π}, \\ Λ(n, m) = d_{π}(ω_{π}^{n} ⊗ L_{d}^{m}) & for m, n ∈ ℤ. \end{matrix}

By abuse of notation, we use the same notation for the restriction of a tautological class to the open substack J ac_{d,g}. Using Facts 2.4 and 2.6, the first Chern classes of the above tautological line bundles are given by

(5.2)  \begin{matrix}κ_{1,0} := c_{1}(K_{1,0}) = π_{∗}(c_{1}(ω_{π})^{2}), \\ κ_{0,1} := c_{1}(K_{0,1}) = π_{∗}(c_{1}(ω_{π}) · c_{1}(L_{d})), \\ κ_{−1,2} := c_{1}(K_{−1,2}) = π_{∗}(c_{1}(L_{d})^{2}), \\ λ(n, m) = c_{1}(Λ(n, m)) = c_{1}(π_{!}(ω_{π}^{n} ⊗ L_{d}^{m})) for any n, m ∈ ℤ. \end{matrix}

Note that, if k = ℂ, the image of the classes κ_{i,j} via the natural map A^{1}(J ac_{d,g}) → H^{2}(J ac_{d,g}, ℤ) → H^{2}(Hol_{g}^{d}, ℤ) are, up to sign, the κ_{i,j} classes that were considered by Erbert and Randal-Williams in [ERW12, Sec. 2.4]. The pull-back of the Hodge line bundle (2.7) of \overline{M}_{g} via the natural map Φ̃_{d} : \overline{J ac}_{d,g} → \overline{M}_{g} is a tautological line bundle on \overline{J ac}_{d,g}.

Lemma 5.1. We have that Φ̃_{d}^{∗}(Λ) = Λ(1, 0).

Proof. Consider the diagram

(5.3)  \xymatrix{\overline{J ac}_{d,g,1} \ar[r]^{Φ̃_{d,1}} \ar[d]_{π} & \overline{M}_{g,1} \ar[d]^{\overline{π}} \\ _{\overline{J ac}_{d,g}} \ar[r]^{Φ̃_{d}} & _{\overline{M}_{g}}}

Recall from Section 2.1 that the map Φ̃_{d} sends an element (C → S, L) ∈ \overline{J ac}_{d,g}(S) into the stabilization C^{st} → S ∈ \overline{M}_{g}(S). Now it is well-known that for every quasi-stable (or more generally semistable) curve X with stabilization morphism ψ : X → X^{st}, the pull-back via ψ induces an isomorphism ψ^{∗} : H^{0}(X^{st}, ω_{X^{st}}) →^{≅} H^{0}(X, ω_{X}). Therefore, the relative dualizing sheaves of the families π and \overline{π} are related by

(5.4)  Φ̃_{d,1}^{∗}(ω\overline{π}) = ω_{π}.

We conclude by using the functoriality of the determinant of cohomology. ■

There are some relations between the tautological line bundles on \overline{J ac}_{d,g}, as shown in the following.

Theorem 5.2. The tautological line bundles on \overline{J ac}_{d,g} satisfy the following relations in the rational Picard group Pic(\overline{J ac}_{d,g}) ⊗ ℚ:

(i) K_{1,0} = Λ(1, 0)^{12} ⊗ O(−δ̃), (ii) K_{0,1} = Λ(1, 1) ⊗ Λ(0, 1)^{−1}, (iii) K_{−1,2} = Λ(0, 1) ⊗ Λ(1, 1) ⊗ Λ(1, 0)^{−2}, (iv) Λ(n, m) = Λ(1, 0)^{6n^{2}−6n−m^{2}+1} ⊗ Λ(1, 1)^{mn+\binom{m}{2}} ⊗ Λ(0, 1)^{−mn+\binom{m+1}{2}} ⊗

O (−\binom{n}{2} · δ̃) .

Proof. Since the first Chern class map c_{1} : Pic(\overline{J ac}_{d,g}) → A^{1}(\overline{J ac}_{d,g}) is an isomorphism by Fact 2.9(i), it is enough to prove the above relations in the rational Chow group A^{1}(\overline{J ac}_{d,g}) ⊗ ℚ.

Following the same strategy as in the proof of Mumford’s relations among the tautological classes of M_{g} (see [ACG11, Chap. 13, Sec. 7]), we apply the Grothendieck-Riemann-Roch Theorem to the morphism π : \overline{J ac}_{d,g,1} → \overline{J ac}_{d,g}:

(5.5)  ch (π_{!} (ω_{π}^{n} ⊗ L_{d}^{m})) = π_{∗} (ch(ω_{π}^{n} ⊗ L_{d}^{m}) · Td(Ω_{π})^{−1}) ,

where ch denotes the Chern character, Td denotes the Todd class and Ω_{π} is the sheaf of relative Kähler differentials. Using (2.4), we can compute the degree one part of the left hand side of (5.5):

(5.6) ch (π_{!} (ω_{π}^{n} ⊗ L_{d}^{m}))_{1} = c_{1} (π_{!} (ω_{π}^{n} ⊗ L_{d}^{m})) = c_{1} (d_{π} (ω_{π}^{n} ⊗ L_{d}^{m})) = λ(n, m).

Let us now compute the degree one part of the right hand side of (5.5). Note that, as proved in [ACG11, p. 383], we have that c_{1}(Ω_{π}) = c_{1}(ω_{π}) and that c_{2}(Ω_{π}) is the class of the nodal locus of the morphism π. In particular, we have that

(5.7)  π_{∗}(c_{2}(Ω_{π})) = δ̃ ∈ A^{1}(\overline{J ac}_{d,g}),

where δ̃ is the total boundary divisor (3.1) of \overline{J ac}_{d,g}. The first three terms of the inverse of the Todd class of Ω_{π} are equal to

(5.8)

Td(Ω_{π})^{−1} = 1− \frac{c_{1}(Ω_{π})}{2} + \frac{c_{1}^{2}(Ω_{π}) + c_{2}(Ω_{π})}{12} +… = 1− \frac{c_{1}(ω_{π})}{2} + \frac{c_{1}(ω_{π})^{2} + c_{2}(Ω_{π})}{12} +…

Using the multiplicativity of the Chern character, we get

ch(ω_{π}^{n} ⊗ L_{d}^{m}) (5.9)

= (1 + c_{1}(ω_{π}) + \frac{c_{1}(ω_{π})^{2}}{2} + …)^{n} · (1 + c_{1}(L_{d}) + \frac{c_{1}(L_{d})^{2}}{2} + …)^{m}

= (1 + nc_{1}(ω_{π}) + \frac{n^{2}c_{1}(ω_{π})^{2}}{2} + …) · (1 + mc_{1}(L_{d}) + \frac{m^{2}c_{1}(L_{d})^{2}}{2} + …)

= 1 + [nc_{1}(ω_{π}) + mc_{1}(L_{d})]+

+ [\frac{n^{2}c_{1}(ω_{π})^{2}}{2} + nmc_{1}(ω_{π}) · c_{1}(L_{d}) + \frac{m^{2}c_{1}(L_{d})^{2}}{2}] + …

Combining (5.8) and (5.9) and using (5.2) together with (5.7), we can compute the degree one part of the right hand side of (5.5)

[π_{∗} (ch(ω_{π}^{n} ⊗ L_{d}^{m}) · Td(Ω_{π})^{−1})]_{1} = π_{∗} ([ch(ω_{π}^{n} ⊗ L_{d}^{m}) · Td(Ω_{π})^{−1}]_{2})

=π_{∗}[\frac{6n^{2} − 6n + 1}{12}c_{1}(ω_{π})^{2} + \frac{2nm − m}{2}c_{1}(ω_{π}) · c_{1}(L_{d}) + \frac{m^{2}}{2} c_{1}(L_{d})^{2} + \frac{c_{2}(Ω_{π})}{12}]

(5.10)  = \frac{6n^{2} − 6n + 1}{12}κ_{1,0} + \frac{2nm − m}{2}κ_{0,1} + \frac{m^{2}}{2} κ_{−1,2} + \frac{δ̃}{12}.

Putting together (5.6) and (5.10), we get the relation

(5.11)  λ(n, m) = \frac{6n^{2} − 6n + 1}{12}κ_{1,0} + \frac{2nm − m}{2}κ_{0,1} + \frac{m^{2}}{2} κ_{−1,2} + \frac{δ̃}{12}.

Formula (5.11) for n = 1 and m = 0 gives that

(*)  λ(1, 0) = \frac{κ_{1,0}}{12} + \frac{δ̃}{12},

which proves part (i). By substituting (*) into (5.11), we get

(5.12) λ(n, m) = (6n^{2} − 6n + 1)λ(1, 0) + \frac{2nm − m}{2}κ_{0,1} + \frac{m^{2}}{2} κ_{−1,2} − \binom{n}{2}δ̃.

Formula (5.12) for (n, m) = (0, 1) and (n, m) = (1, 1) gives that

(**)  \{ \begin{matrix}λ(0, 1) = λ(1, 0) − \frac{κ_{0,1}}{2} + \frac{κ_{−1,2}}{2} , \\ λ(1, 1) = λ(1, 0) + \frac{κ_{0,1}}{2} + \frac{κ_{−1,2}}{2} , \end{matrix}

The system of equations (**) is equivalent to the system

(***)  \{ \begin{matrix}κ_{0,1} = λ(1, 1) − λ(0, 1), \\ κ_{−1,2} = −2λ(1, 0) + λ(0, 1) + λ(1, 1), \end{matrix}

which also proves parts (ii) and (iii). Substituting (***) into (5.12), we get the following relation

(5.13) λ(n, m) = (6n^{2} − 6n + 1 − m^{2})λ(1, 0)+

+ [−mn + \binom{m + 1}{2}] λ(0, 1) + [mn + \binom{m}{2}] λ(1, 1) − \binom{n}{2}δ̃,

which proves part (iv). ■

By a slight generalization of Lemma 5.1, it is easy to see that the relations in Theorem 5.2(i) and in Theorem 5.2(iv) with m = 0 are the pull-back to \overline{J ac}_{d,g} of Mumford’s relations among the tautological classes of \overline{M}_{g} (see [ACG11, Chap. 13, Thm. (7.6)]).

Remark 5.3. The proof of Theorem 5.2 works a priori only in the rational Picard group of \overline{J ac}_{d,g}, since it uses the Grothedieck-Riemann-Roch theorem which is valid only in the rational Chow group. However, since the Picard group of \overline{J ac}_{d,g} is torsion-free (as it follows from Theorem A(ii), to be proved in §7), the relations in the above Theorem hold true a posteriori also in the integral Picard group of \overline{J ac}_{d,g}.

Motivated by Theorem 5.2, we can now define the tautological subgroup of the Picard group of the stacks \overline{J ac}_{d,g} and J ac_{d,g}.

Definition 5.4. The tautological subgroup Pic^{taut}(\overline{J ac}_{d,g}) ⊆ Pic(\overline{J ac}_{d,g}) is the subgroup generated by the line bundles associated to the boundary divisors of \overline{J ac}_{d,g} (see Section 3) and by the tautological line bundles Λ(1, 0), Λ(0, 1) and Λ(1, 1). The image of Pic^{taut}(\overline{J ac}_{d,g}) ⊆ Pic(\overline{J ac}_{d,g}) via the natural restriction map Pic(\overline{J ac}_{d,g}) → Pic(J ac_{d,g}) is defined to be Pic^{taut}(J ac_{d,g}); hence, Pic^{taut}(J ac_{d,g}) ⊆ Pic(J ac_{d,g}) is the subgroup generated by the tautological line bundles Λ(1, 0), Λ(0, 1) and Λ(1, 1).

6. Comparing the Picard groups of J ac_{d,g} and J_{d,g}

The aim of this Section is to study the pull-back map

ν_{d}^{∗} : Pic(J_{d,g}) → Pic(J ac_{d,g})

induced by the map ν_{d} : J ac_{d,g} → J_{d,g} (see Section 2.1). To this aim, consider the Leray spectral sequence for the étale sheaf 𝔾_{m} with respect to the map ν_{d}:

E_{2}^{p,q} = H_{ét}^{p}(J_{d,g}, (R^{q}ν_{d})_{∗}𝔾_{m}) ⟹ H_{ét}^{p+q}(J ac_{d,g}, 𝔾_{m}).

The first terms of the above spectral sequence give rise to the exact sequence

0 → H_{ét}^{1}(J_{d,g}, (R^{0}ν_{d})_{∗}𝔾_{m}) ⟶ H_{ét}^{1}(J ac_{d,g}, 𝔾_{m}) ⟶ ⟶H_{ét}^{0}(J_{d,g}, (R^{1}ν_{d})_{∗}𝔾_{m}) ⟶ H_{ét}^{2}(J_{d,g}, (R^{0}ν_{d})_{∗}𝔾_{m}).

Since ν_{d} is a 𝔾_{m}-gerbe, we have that (R^{0}ν_{d})_{∗}𝔾_{m} = 𝔾_{m} and (R^{1}ν_{d})_{∗}𝔾_{m} = Pic B𝔾_{m}, where Pic B𝔾_{m} is canonically identified with the group (𝔾_{m})^{∗} ≅ ℤ of characters of 𝔾_{m}. By plugging these isomorphisms into the above long exact sequence, we get the exact sequence

(6.1)  0 → Pic(J_{d,g}) ⟶^{ν_{d}^{∗}} Pic(J ac_{d,g}) ⟶^{res} ℤ ⟶^{obs} Br(J_{d,g}),

where the above maps admits the following interpretation (which one can easily check via standard cocycle computations): ν_{d}^{∗} is the pull-back map induced by ν_{d}; res is the restriction to the fibers of ν_{d} (it coincides with the weight map defined in [Hof07, Def. 4.1] and with the character appearing in the decomposition in [Lie08, Prop. 3.1.1.4]) and obs (the obstruction map) sends 1 ∈ ℤ = (𝔾_{m})^{∗} into the class [ν_{d}] of the 𝔾_{m}-gerbe ν_{d} in the (cohomological) Brauer group Br(J_{d,g}) := H_{ét}^{2}(J_{d,g}, 𝔾_{m}) (see [Gir71, Chap. IV.3]). Since ν_{d}^{∗} is injective, we can define a tautological subgroup of Pic(J_{d,g}) by intersecting Pic(J_{d,g}) (which we identify with its image via ν_{d}^{∗}) with the tautological subgroup Pic^{taut}(J ac_{d,g}), as follows.

Definition 6.1. The tautological subgroup of Pic(J_{d,g}) is defined as

Pic^{taut}(J_{d,g}) := Pic^{taut}(J ac_{d,g}) ∩ Pic(J_{d,g}) ⊆ Pic(J ac_{d,g}).

In order to compute generators for Pic^{taut}(J_{d,g}), we need first to compute the map res from (6.1) on the generators of Pic^{taut}(J ac_{d,g}).

Lemma 6.2. We have that

res(Λ(1, 0)) = 0,
res(Λ(0, 1)) = d − g + 1,
res(Λ(1, 1)) = d + g − 1.

Proof. Using the functoriality of the determinant of cohomology, we get that the fiber of Λ(1, 0) = d_{π}(ω_{π}) over a point (C, L) ∈ J ac_{d,g} is canonically isomorphic to det H^{0}(C, ω_{C})⊗det^{−1} H^{1}(C, ω_{C}). Since 𝔾_{m} acts trivially on H^{0}(C, ω_{C}) and on H^{1}(C, ω_{C}), we get that res(Λ(1, 0)) = 0. Similarly, the fiber of Λ(0, 1) over a point (C, L) ∈ J ac_{d,g} is canonically isomorphic to det H^{0}(C, L) ⊗ det^{−1} H^{1}(C, L). Since 𝔾_{m} acts with weight one on the vector spaces H^{0}(C, L) and H^{1}(C, L), Riemann-Roch gives that

res(Λ(0, 1)) = dim H^{0}(C, L) − dim H^{1}(C, L) = χ(C, L) = d + 1 − g.

Finally, the fiber of Λ(1, 1) over a point (C, L) ∈ J ac_{d,g} is canonically isomorphic to det H^{0}(C, L ⊗ ω_{C}) ⊗ det^{−1} H^{1}(C, L ⊗ ω_{C}). Since 𝔾_{m} acts with weight one on the vector spaces H^{0}(C, ω_{C} ⊗ L) and H^{1}(C, ω_{C} ⊗ L), Riemann-Roch gives that

res(Λ(1, 1)) = dim H^{0}(C, ω_{C} ⊗ L) − dim H^{1}(C, ω_{C} ⊗ L) = = χ(C, ω_{C} ⊗ L) = d + 2g − 2 + 1 − g = d − 1 + g.

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Combining the above Lemma 6.2 with Definition 5.4, we get the following

Corollary 6.3.

(i) The image of Pic^{taut}(J ac_{d,g}) via the map res of (6.1) is the subgroup generated by (d + g − 1, d − g + 1) = (d + g − 1, 2g − 2).

(ii) Pic^{taut}(J_{d,g}) is generated by Λ(1, 0) and

(6.2)  Ξ := Λ(0, 1)^{\frac{d+g−1}{(d+g−1,d−g+1)}} ⊗ Λ(1, 1)^{− \frac{d−g+1}{(d+g−1,d−g+1)}} .

Corollary 6.3(i) combined with the exact sequence (6.1) gives that the order of [ν_{d}] in the Brauer group Br(J_{d,g}) divides (d + g − 1, 2g − 2). Indeed the following is true:

Theorem 6.4. The order of [ν_{d}] in Br(J_{d,g}) is equal to (d + 1 − g, 2g − 2).

In order to prove the theorem, we will reinterpret the order of [ν_{d}] in terms of the existence of a (generalized) Poincaré bundle. Consider the universal family π : J ac_{d,g,1} → J ac_{d,g}. The 𝔾_{m}-rigidification of J ac_{d,g,1}, denoted by J_{d,g,1} := J ac_{d,g,1} 𝔾_{m}, has a natural map π̃ : J_{d,g,1} → J_{d,g} which is indeed the universal family over J_{d,g}. However, the universal (or Poincaré) line bundle L_{d} on J ac_{d,g,1} does not necessarily descend to a line bundle on J_{d,g,1}. Instead, it turns out that there always exists on J_{d,g,1} an m-Poincaré line bundle as in the definition below.

Definition 6.5. Let m ∈ ℤ. An m-Poincaré line bundle for J_{d,g} is a line bundle L on J_{d,g,1} such that the restriction of L to the fiber π^{−1}(C, L) ≅ C over a geometric point (C, L) of J_{d,g} is isomorphic to L^{m}.

The above definition generalizes the classical definition of Poincaré line bundle, which corresponds to the case m = 1.

Proposition 6.6. The order of [ν_{d}] in the group Br(J_{d,g}) is equal to the smallest number m ∈ ℕ such that there exists an m-Poincaré line bundle for J_{d,g}.

Proof. In order to prove the statement, we need to introduce some auxiliary stacks. Given m ∈ ℤ, consider the stack J ac_{d,g}^{m} whose fiber J ac_{d,g}^{m}(S) over a scheme S consists of families C → S of smooth curves of genus g endowed with a line bundle L of relative degree d and whose morphisms between two objects (C′ → S′, L′) and (C → S, L) are given by a triple (g, ϕ, η) where

[figure: C′; ϕ; □; C; S′; g; S]

is a Cartesian diagram and η : L^{′m} → ϕ^{∗}(L^{m}) is an isomorphism of line bundles on C′. Note that J ac_{d,g}^{1} ≅ J ac_{d,g}. The multiplicative group 𝔾_{m} injects into the automorphism group of every object (C → S, L) ∈ J ac_{d,g}^{m}(S) as multiplication by scalars on L. The rigidification J ac_{d,g}^{m} 𝔾_{m} is isomorphic to J_{d,g} and the natural map ν_{d}^{m} : J ac_{d,g}^{m} → J_{d,g} is a 𝔾_{m}-gerbe. By construction, the class of [ν_{d}^{m}] in Br(J_{d,g}) is equal to [ν_{d}^{m}] = m · [ν_{d}]. Consider the universal family π^{m} : J ac_{d,g,1}^{m} → J ac_{d,g}^{m}. The fiber of J ac_{d,g,1}^{m} over a scheme S consists of the triples (C → S, σ, L), where (C → S, L) ∈ J ac_{d,g}(S) and σ is a section of the morphism C → S. The morphisms between two objects (C′ → S′, σ′, L′) ∈ J ac_{d,g,1}^{m}(S′) and (C → S, σ, L) ∈ J ac_{d,g,1}^{m}(S) are given by the isomorphisms (g, ϕ, η) as above satisfying the relation σ∘g = ϕ∘σ′. The 𝔾_{m}-rigidification of J ac_{d,g,1}^{m} is isomorphic to J_{d,g,1} and therefore we get a Cartesian diagram:

(6.3)  [figure: J ac_{d,g,1}^{m}; π^{m}; ν_{d}^{′m}; □; J ac_{d,g}^{m}; ν_{d}^{m}; J_{d,g,1}; π̃; J_{d,g}]

On the stack J ac_{d,g,1}^{m} there is a universal line bundle N_{m}, defined as follows: to every morphism from a scheme f : S → J ac_{d,g,1}^{m}, which corresponds to an object (C → S, σ, L) ∈ J ac_{d,g,1}^{m}(S) as above, we associate the line bundle

N_{m}(f) := σ^{∗}(L^{m}) ∈ Pic(S); to every morphism S′ →^{g} S →^{f} J ac_{d,g,1}^{m}, corresponding to the morphism (g, ϕ, η) between two objects (C → S, σ, L) and (C′ → S′, σ′, L′) as above, we associate the isomorphism

N_{m}(f ∘ g) = σ^{′∗}(L^{′m}) ⟶^{σ^{′∗}(η)} σ^{′∗}ϕ^{∗}(L^{m}) = g^{∗}σ^{∗}(L^{m}) = g^{∗}N_{m}(f).

We have now the tools that we need to prove the result. Since [ν_{d}^{m}] = m[ν_{d}] ∈ Br(J_{d,g}), the period of [ν_{d}] is equal to the smallest m ∈ ℕ such that the 𝔾_{m}-gerbe ν_{d}^{m} is trivial and this happens precisely when there exists a section σ_{d}^{m} : J_{d,g} → J ac_{d,g}^{m} of ν_{d}^{m}. Since the diagram (6.3) is Cartesian, the existence of a section σ_{d}^{m} of ν_{d}^{m} is equivalent to the existence of a section σ_{d}^{′m} of ν_{d}^{′m}. If such a section exists, then the pull-back (σ_{d}^{′m})^{∗}N_{m} is an m-Poincaré line bundle on J_{d,g}, by the above description of N_{m}. Conversely, the existence of a Poincaré line bundle on J_{d,g} allows us to define a section σ_{d}^{′m} of ν_{d}^{′m} by the above description of J ac_{d,g,1}^{m}.

■

Proof of Theorem 6.4. Consider the group

A_{d,g} := {m ∈ ℤ : there exists an m-Poincaré line bundle L on J ac_{d,g,1}}

Proposition 6.6 gives that the positive generator of A_{d,g} is equal to the order of [ν_{d}] in Br(J_{d,g}). On the other hand, the positive generator of A_{d,g} is equal to (d + g − 1, 2g − 2) by [Kou93, Application at p. 514]. This concludes the proof.

■

Remark 6.7. From Proposition 6.6 and Theorem 6.4, we recover the following well-known result due to Mestrano-Ramanan ([MR85, Cor. 2.9]): there exists a Poincaré line bundle on J_{d,g,1} if and only if (d + 1 − g, 2g − 2) = 1.

Remark 6.8. It is possible to prove that the index of [ν_{d}] is equal to (d + g − 1, 2g − 2) (recall that the index of [ν_{d}] is the smallest m ∈ ℕ such that [ν_{d}] is represented by a projective bundle over J_{d,g} of relative dimension m − 1). Since we will not need this result, we do not include a proof here.

We make the following

Conjecture 6.9. The cohomological Brauer group Br(J_{d,g}) of J_{d,g} is generated by the class [ν_{d}] of the 𝔾_{m}-gerbe ν_{d} : J ac_{d,g} → J_{d,g}.

Using the notation of Section 1.1, the above conjecture must be compared with the result of Ebert and Randal-Williams who proved in [ERW12, Thm. B] that, for g ≥ 6, H^{3}(Pic_{g}^{d}, ℤ) is cyclic of order (2g − 2, d + g − 1) and generated by the Dixmier-Douday class of the ℂ^{∗}-gerbe ϕ_{g}^{d} : Hol_{g}^{d} → Pic_{g}^{d}. From the diagram (1.7) and the coboundary map coming from the exponential sequence of locally constant sheaves 0 → ℤ → ℂ ⟶^{exp} ℂ^{∗} → 0, we get a map cl : Br(J_{d,g}) → H^{2}(Pic_{g}^{d}, ℂ^{∗}) → H^{3}(Pic_{g}^{d}, ℤ) which clearly sends the class of ν_{d} into the class of ϕ_{g}^{d}. A positive answer to Conjecture 6.9 together with Theorem 6.4 would imply that the above map cl is an isomorphism for g ≥ 6. From the above Theorem 6.4, we deduce the following Corollary 6.10.

(i) The image of Pic(J ac_{d,g}) via the map res of (6.1) is the subgroup generated by (d + g − 1, 2g − 2).

(ii) The pull-back map ν_{d}^{∗} induces an isomorphism

ν_{d}^{∗} : Pic(J_{d,g})/ Pic^{taut}(J_{d,g}) ⟶^{≅} Pic(J ac_{d,g})/ Pic^{taut}(J ac_{d,g}).

Proof. Part (i) follows from the exact sequence (6.1) together with Theorem 6.4. Part (ii): using Corollary 6.3(i) and part (i), we get the following commutative diagram with exact rows:

[figure: 0; Pic(J_{d,g}); ν_{d}^{∗}; Pic(J ac_{d,g}); res; ℤ · ⟨(d + g − 1, 2g − 2)⟩; 0; 0; Pic^{taut}(J_{d,g}); ν_{d}^{∗}; Pic^{taut}(J ac_{d,g}); res; ℤ · ⟨(d + g − 1, 2g − 2)⟩; 0]

The conclusion follows from the snake lemma. ■

7. The Picard group of J_{d,g}

In this subsection we will determine the Picard group of the stack J_{d,g}, using a strategy similar to the one used by Kouvidakis [Kou91] to determine the Picard group of J_{d,g}^{0}, the open subset of J_{d,g} consisting of pairs (C, L) where C is a smooth curve without non-trivial automorphisms. Consider the representable morphism Φ_{d} : J_{d,g} → M_{g}. Clearly the fiber of Φ_{d} over C ∈ M_{g} is the degree-d Jacobian J^{d}(C) of C. Since Φ_{d} has connected fibers, the pull-back map Φ_{d}^{∗} : Pic(M_{g}) → Pic(J_{d,g}) is injective. The cokernel of Φ_{d}^{∗} is denoted by RPic(J_{d,g}) and is called classically the group of rationally determined line bundles of the family J_{d,g} → M_{g} (see e. g. [Cil87]). Therefore, we have the following exact sequence

(7.1)  0 → Pic(M_{g}) →^{Φ_{d}^{∗}} Pic(J_{d,g}) → RPic(J_{d,g}) → 0.

Since the fiber of Φ_{d} over C ∈ M_{g} is the degree-d Jacobian J^{d}(C) of C, we have a natural map

(7.2)  ρ_{C} : Pic(J_{d,g}) → Pic(J^{d}(C)) → NS(J^{d}(C)),

where the first map is the restriction to the fiber Φ_{d}^{−1}(C) = J^{d}(C) and the second map is the projection of the Picard group of J^{d}(C) onto the Néron-Severi group of J^{d}(C), which parametrizes divisors on J^{d}(C) up to algebraic equivalence. We will use additive notation for the group law on NS(J^{d}(C)). Consider now the theta divisor Θ(C) ⊂ J^{g−1}(C) and denote by θ_{C} ∈ NS(J^{g−1}(C)) its algebraic equivalence class. By choosing an isomorphism t_{M} : J^{d}(C) →^{≅} J^{g−1}(C) given by sending L ∈ J^{d}(C) into L ⊗ M ∈ J^{g−1}(C) for some M ∈ J^{g−1−d}(C), we can pull-back θ_{C} to get a well-defined (i.e. independent of the chosen isomorphism t_{M}) class in NS(J^{d}(C)) which, by a slight abuse of notation, we will still denote by θ_{C}. Since, for a very general curve C ∈ M_{g}, NS(J^{d}(C)) is generated by θ_{C} (see e. g. [Kou91, Lemma 2]), it follows that there is a morphism of groups

(7.3)  χ_{d} : Pic(J_{d,g}) ⟶ ℤ

sending L ∈ Pic(J_{d,g}) to the integer m such that ρ_{C}(L) = mθ_{C} for every C ∈ M_{g} (see also [Kou91, p. 840]). We will need the following two results of Kouvidakis, describing the image and the kernel of the above map χ_{d}. Actually, Kouvidakis proves these results in [Kou91] for the variety J_{d,g}^{0}, but a close inspection reveals that the same proof works for J_{d,g}.

Theorem 7.1 (Kouvidakis).

(i) ker χ_{d} = Im Φ_{d}^{∗}.

(ii) Im χ_{d} ⊆ \frac{2g − 2}{(2g − 2, d + g − 1)} · ℤ ⊆ ℤ.

Part (i) follows from [Kou91, Thm. 3]; part (ii) follows from [Kou91, Formula (*), p. 844]. Note that part (i) implies (and it is indeed equivalent to) that the map χ_{d} factors as

(7.4)  χ_{d} : Pic(J_{d,g}) ↠ RPic(J_{d,g}) ↪ ℤ.

We now compute the image of the map χ_{d} on the tautological subgroup Pic^{taut}(J_{d,g}) of Pic(J_{d,g}) (see Definition 6.1).

Theorem 7.2. We have that

χ_{d}(Pic^{taut}(J_{d,g})) = \frac{2g − 2}{(2g − 2, d + g − 1)} · ℤ ⊆ ℤ.

Proof. According to Corollary 6.3(ii), Pic^{taut}(J_{d,g}) is generated by the tautological classes Λ(1, 0) and Ξ. Lemma 5.1 gives that Λ(1, 0) = Φ_{d}^{∗}(Λ); hence clearly χ_{d}(Λ(1, 0)) = 0 (this is the easy inclusion in Theorem 7.1(i)). Therefore, the proof will follow if we show that

(7.5)  χ_{d}(Ξ) = \frac{2g − 2}{(2g − 2, d + g − 1)},

or equivalently that

(7.6)  ρ_{C}(Ξ) = \frac{2g − 2}{(2g − 2, d + g − 1)}θ_{C}

Documenta Mathematica 19 (2014) 457–507

for any C ∈ M_{g}. In order to prove this, consider the following diagram (7.7)

[figure: L_{C}; L_{d}; C × J ac^{d}(C); p; id ×ν_{C}; _{J ac_{d,g,1}}; π; J ac^{d}(C); ν_{C}; □; _{J ac_{d,g}}; ν_{d}; □; C × J^{d}(C); p_{2}; J_{d,g,1}; π̃; J^{d}(C); J_{d,g}; Φ_{d}; C; M_{g}]

where the Cartesian square on the left is the fiber of the Cartesian square on the right over the point C ∈ M_{g} and L_{C} is the fiber of the universal line bundle L_{d} over C ∈ M_{g}. In particular, the stack J ac^{d}(C) is the degree-d Jacobian stack of C (i.e. the stack whose fiber over a scheme S is the groupoid of line bundles on C × S of relative degree d over S) and L_{C} is the universal (or Poincaré) line bundle for J ac^{d}(C). The map ν_{C} : J ac^{d}(C) → J^{d}(C) is a 𝔾_{m}-gerbe which is well-known to be trivial, or in other words J ac^{d}(C) ≅ J^{d}(C) × B𝔾_{m}. Therefore, there exists a section s of ν_{C} and we can define L̃_{C} := (id ×s)^{∗}(L_{C}). By construction, we have that L̃_{|C×{M}} = M for any M ∈ J^{d}(C). Any line bundle on C × J^{d}(C) with this property is called a Poincaré line bundle for J^{d}(C). Indeed, any Poincaré line bundle for J^{d}(C) is isomorphic to (id ×s)^{∗}(L_{C}) for a uniquely determined section s of ν_{C}. Moreover, two Poincaré line bundles for J^{d}(C) differ by the tensor product with the pull-back of a line bundle on J^{d}(C). Note that for any Poincaré line bundle L̃_{C} = (id ×s)^{∗}(L_{C}) for J^{d}(C), we have that (id ×ν_{C})^{∗}(L̃_{C}) = (id ×ν_{C})^{∗}((id ×s)^{∗}(L_{C})) = L_{C}. Recalling the definition of Ξ from Corollary 6.3(ii) and applying the functoriality of the determinant of cohomology to the above diagram (7.7), we get that

(7.8) ρ_{C}(Ξ) = \frac{d + g − 1}{(d + g − 1, d − g + 1)}[d_{p_{2}}(L̃_{C})]−

− \frac{d − g + 1}{(d + g − 1, d − g + 1)}[d_{p_{2}}(L̃_{C} ⊗ p_{1}^{∗}(ω_{C}))],

where p_{1} : C × J^{d}(C) denotes the projection onto the first factor and L̃_{C} is any Poincaré line bundle for J^{d}(C). Note that the fact that Ξ ∈ Pic(J_{d,g}) guarantees that the right hand side of (7.8) is independent of the choice L̃_{C}.

In order to compute the right hand side of (7.8), we can choose a Poincaré line bundle L̃_{C} for J^{d}(C) that satisfies the following \underline{Condition (*)}: [(L̃_{C})_{|p_{1}^{−1}(r)}] = 0 ∈ NS(J^{d}(C)) for any r ∈ C. Indeed, since L̃_{C} can be seen as a family of line bundles on J^{d}(C) parametrized by C, if condition (*) holds for a certain point r_{0} ∈ C then it holds for all points r ∈ C. However, up to tensoring L̃_{C} with the pull-back of a line bundle on J^{d}(C), we can always assume that (L̃_{C})_{|p_{1}^{−1}(r_{0})} is the trivial line bundle on J^{d}(C), q.e.d. With the above condition on L̃_{C}, we can prove the following two claims. \underline{Claim 1}: If L̃_{C} satisfies condition (*) then

[d_{p_{2}}(L̃_{C} ⊗ p_{1}^{∗}(M))] = [d_{p_{2}}(L̃_{C})] ∈ NS(J^{d}(C)) for any M ∈ J(C).

Indeed, write M = O_{C}(−γ + δ) with γ = ∑_{i} a_{i}r_{i} and δ = ∑_{j} b_{j}r_{j} effective divisors on C. From the exact sequences defining the structure sheaves of p_{1}^{−1}(δ) ⊂ C × J^{d}(C) and p_{1}^{−1}(γ) ⊂ C × J^{d}(C), we get

\{ \begin{matrix}0 → L̃_{C} ⊗ p_{1}^{∗}O_{C}(−γ) → L̃_{C} → (L̃_{C})_{|p_{1}^{−1}(γ)} → 0, \\ 0 → L̃_{C} ⊗ p_{1}^{∗}O_{C}(−γ) → L̃_{C} ⊗ p_{1}^{∗}(M) → (L̃_{C})_{|p_{1}^{−1}(δ)} → 0. \end{matrix}

From the multiplicativity of the determinant of cohomology applied to the above exact sequences, we get

d_{p_{2}}(L̃_{C} ⊗ p_{1}^{∗}M) ⊗ d_{p_{2}}(L̃_{C})^{−1} = d_{p_{2}}((L̃_{C})_{|p_{1}^{−1}(δ)}) ⊗ d_{p_{2}}((L̃_{C})_{|p_{1}^{−1}(γ)})^{−1} =

= ⨂_{j}(L̃_{C})_{p_{1}^{−1}(r_{j})}^{b_{j}} ⨂_{i}(L̃_{C})_{p_{1}^{−1}(r_{i})}^{−a_{i}}.

Claim 1 follows now by condition (*). \underline{Claim 2}: If L̃_{C} satisfies condition (*) then

[d_{p_{2}}(L̃_{C})] = θ_{C} ∈ NS(J^{d}(C)).

Indeed, choose a line bundle M ∈ J^{d−g+1}(C) and consider the Cartesian diagram

[figure: (id ×t_{M})^{∗}(L̃_{C}); L̃_{C}; C × J^{g−1}(C); id ×t_{M}; p′_{2}; C × J^{d}(C); p_{2}; J^{g−1}(C); t_{M}; J^{d}(C),]

where t_{M} is the map sending L ∈ J^{g−1}(C) into L⊗N ∈ J^{d}(C). The line bundle L̃′_{C} := (id ×t_{M})^{∗}(L̃_{C}) ⊗ p_{1}^{∗}(M)^{−1} is clearly a Poincaré line bundle for J^{g−1}(C) and it satisfies condition (*) since L̃_{C} satisfies condition (*) by assumption.

Therefore, using the functoriality of the determinant of cohomology and Claim 1, we get the following equality in NS(J^{g−1}(C)):

(7.9) [t_{M}^{∗}d_{p_{2}}(L̃_{C})] = [d_{p′_{2}}((id ×t_{M})^{∗}(L̃_{C}))] = [d_{p′_{2}}(L̃′_{C} ⊗ p_{1}^{∗}(M))] = [d_{p′_{2}}(L̃′_{C})].

Claim 2 now follows from the well-known fact that d_{p′_{2}}(L̃′_{C}) ∈ Pic(J^{g−1}(C)) is the line bundle associated to the theta divisor Θ(C) ⊂ J^{g−1}(C) for any Poincaré line bundle L̃′_{C} for J^{g−1}(C). Now choosing a Poincaré line bundle L̃_{C} that satisfies condition (*), formula (7.8) together with Claim 1 and Claim 2 gives that

ρ_{C}(Ξ) = \frac{d + g − 1}{(d + g − 1, d − g + 1)}θ_{C} − \frac{d − g + 1}{(d + g − 1, d − g + 1)}θ_{C} =

= \frac{2g − 2}{(2g − 2, d + g − 1)}θ_{C},

which proves (7.6). ■

By combining the above results, we can now prove the main Theorems A and B from the introduction.

Proof of Theorem B. Let us first prove Theorem B(i). By combining Theorem 7.1(ii) with Theorem 7.2, we get that χ_{d}(Pic(J_{d,g})) = χ_{d}(Pic^{taut}(J_{d,g})). By Theorem 7.1(i), the kernel of χ_{d} is equal to Φ_{d}^{∗}(Pic(M_{g})), which is generated by Λ(1, 0) = Φ_{d}^{∗}(Λ) by Theorem 2.12 and Lemma 5.1; hence Im Φ_{d}^{∗} ⊂ Pic^{taut}(J_{d,g}). We deduce that

(7.10)  Pic^{taut}(J_{d,g}) = Pic(J_{d,g}).

Therefore, Pic(J_{d,g}) is generated by Λ(1, 0) and by Ξ by Corollary 6.3(ii). Consider now the exact sequence (7.1). Combining the factorization of χ_{d} provided by (7.4) with formula (7.5), we get that RPic(J_{d,g}) is free of rank one. On the other hand, using Theorem 2.12 (since g ≥ 3 by assumption), we know that Pic(M_{g}) is free of rank one. Therefore the exact sequence (7.1) gives that Pic(J_{d,g}) is free of rank two, which concludes the proof of part (i). Theorem B(ii) follows now from part (i) and Corollary 4.6. ■

Proof of Theorem A. Let us first prove Theorem A(i). From (7.10) and Corollary 6.10(ii), we deduce that

(7.11)  Pic^{taut}(J ac_{d,g}) = Pic(J ac_{d,g}).

Therefore, Pic(J ac_{d,g}) is generated by Λ(1, 0), Λ(0, 1) and Λ(1, 1) by Definition 5.4. Moreover, the exact sequence (6.1) together with Theorem B(i) implies that Pic(J ac_{d,g}) is free of rank three. Part (i) is now proved. Theorem A(ii) follows now from part (i) and Theorem 4.1.

■

We can now compare our computation of Pic(J_{d,g}) (see Theorem B(i)) with the computation of Pic(J_{d,g}^{0}) carried out by Kouvidakis in [Kou91].

Remark 7.3. Assume that g ≥ 3. Then the natural map Ψ_{d} : J_{d,g} → J_{d,g} is an isomorphism over the open subset J_{d,g}^{0} ⊂ J_{d,g} parametrizing pairs (C, L) ∈ J_{d,g} such that C does not have non-trivial automorphisms. In other words, the map Ψ_{d} induces an isomorphism

Ψ_{d} : J_{d,g}^{0} := Ψ_{d}^{−1}(J_{d,g}^{0}) ⟶^{≅} J_{d,g}^{0}.

Therefore, we get a natural homomorphism

(7.12)  ψ : Pic(J_{d,g}) → Pic(J_{d,g}^{0}) ⟶_{Ψ_{d}^{∗}}^{≅} Pic(J_{d,g}^{0}),

where the first homomorphism is the natural restriction map. If g ≥ 4, then the codimension of J_{d,g} \J_{d,g}^{0} inside J_{d,g} is at least two and hence the map ψ is an isomorphism by Fact 2.9(iii). Hence Theorem B(i) recovers [Kou91, Thm. 4]. However, this does not hold anymore if g = 3 since in this case J_{d,g} \J_{d,g}^{0} is a divisor inside J_{d,g}, namely the pull-back of the hyperelliptic (irreducible) divisor in M_{3}, whose class in A^{1}(M_{g}) is equal to 9λ (see [HM98, Chap. 3, Sec. E]). Therefore, by Fact 2.9(ii), we get that Pic(J_{d,g}^{0}) ≅ Pic(J_{d,g}^{0}) is the quotient of Pic(J_{d,g}) by the relation Λ(1, 0)^{9} = 0.

7.1. Relation between Ξ and the universal theta divisor. There is a close relationship between the line bundle Ξ ∈ Pic(J_{d,g}) ⊂ Pic(J ac_{d,g}) and the universal theta divisor Θ ⊂ J ac_{g−1,g}, which is the closed substack parametrizing pairs (C, L) ∈ J ac_{g−1,g} such that h^{0}(C, L) > 0. Observe that Θ naturally descends to a divisor on the rigidification J_{g−1,g}, which we denote by \overline{Θ} and we call the universal theta divisor on J_{g−1,g}. By construction, the restriction of \overline{Θ} to any fiber Φ_{d}^{−1}(C) = J^{g−1}(C) is isomorphic to the theta divisor Θ(C) ⊂ J^{g−1}(C). Consider first the special case d = g−1. From the definition (6.2) of Ξ and using the definition (5.1) of the tautological line bundles, we get that Ξ = Λ(0, 1) = d_{π}(L_{g−1}), where L_{g−1} is the universal line bundle on the universal family over J ac_{g−1,g}. It is well know that d_{π}(L_{g−1}) is the line bundle associated to the universal theta divisor, or in other words we have that

(7.13)  Ξ = O(Θ) if d = g − 1.

For an arbitrary d, we consider the stack S_{g}^{1/k_{d,g}} of k_{d,g}-spin curves, where as usual

k_{d,g} = \frac{2g − 2}{(2g − 2, d + 1 − g)}.

Recall that S_{g}^{1/k_{d,g}} is the stack whose fiber over a scheme S consists of the groupoid of families of smooth curves C → S of genus g, plus a line bundle η on C of relative degree (d − g + 1, 2g − 2) over S endowed with an isomorphism η^{⊗k_{d,g}} ≅ ω_{C/S}. The stack S_{g}^{1/k_{d,g}} is a smooth Deligne-Mumford stack endowed with a (forgetful) finite and étale map S_{g}^{1/k_{d,g}} → M_{g} of degree (2g)^{k_{d,g}}. We have a diagram

(7.14)  [figure: F; π; s̃; p̃_{2}; □; □; _{J ac_{d,g,1}}; π_{2}; _{J ac_{g−1,g,1}}; π_{1}; S_{g}^{1/k_{d,g}} ×_{M_{g}} J ac_{d,g}; p_{2}; s; _{J ac_{d,g}}; _{J ac_{g−1,g}}]

where p_{2} is the projection onto the second factor and s sends the element (C → S, η, L) ∈ S_{g}^{1/k_{d,g}} ×_{M_{g}} J ac_{d,g}(S) into (C → S, L⊗η^{−e_{d,g}}) ∈ J ac_{g−1,g}(S), where

e_{d,g} := \frac{d − g + 1}{(d − g + 1, 2g − 2)}.

The universal family F is endowed with a universal line bundle L_{d} of relative degree d which is the pulled-back from J ac_{d,g,1} and a universal spin line bundles η_{k_{d,g}} which is pulled-back from the universal family above S_{g}^{1/k_{d,g}}. By the definition of the morphism s, we get that

(7.15)  s^{∗}(L_{g−1}) = η_{k_{d,g}}^{−e_{d,g}} ⊗ L_{d}.

The relation between the line bundle Ξ ∈ Pic(J ac_{d,g}) and the universal theta divisor Θ ⊂ J ac_{g−1,g} is provided by the following.

Lemma 7.4. We have that

p_{2}^{∗}(Ξ) = s^{∗}O(k_{d,g} · Θ) ⊗ ⟨η_{k_{d,g}}, η_{k_{d,g}}⟩_{π}^{− \frac{k_{d,g} (k_{d,g} +e_{d,g})e_{d,g}}{2}} .

Proof. By the definition (6.2) of Ξ and the standard properties of the determinant of cohomology, we compute

(7.16) p_{2}^{∗}(Ξ) = d_{π}(L_{d})^{\frac{d+g−1}{(2g−2,d+1−g)}} ⊗ d_{π}(ω_{π} ⊗ L_{d})^{− \frac{d−g+1}{(2g−2,d+1−g)}} =

= d_{π}(L_{d})^{k_{d,g}+e_{d,g}} ⊗ d_{π}(η_{k_{d,g}}^{k_{d,g}} ⊗ L_{d})^{−e_{d,g}}.

Using (7.13) and (7.15) together with standard properties of the determinant of cohomology, we get that

(7.17)  s^{∗}(O(k_{d,g} · Θ)) = s^{∗}(d_{π_{1}}(L_{g−1})^{k_{d,g}}) = d_{π}(η_{k_{d,g}}^{−e_{d,g}} ⊗ L_{d})^{k_{d,g}}.

In order to compare (7.16) and (7.17), we apply the Grothedieck-Riemann-Roch theorem to the sheaf η_{k_{d,g}}^{n} ⊗ L_{d}^{m} on the universal family π : F → S_{g}^{1/k_{d,g}} ×_{M_{g}} J ac_{d,g}. After some easy computations similar to the ones done in the proof of Theorem 5.2 which we leave to the reader, we get that

(7.18) c_{1}(d_{π}(η_{k_{d,g}}^{n} ⊗ L_{d}^{m})) = \frac{6n^{2} − 6k_{d,g}n + k_{d,g}^{2}}{12} c_{1}(⟨η_{k_{d,g}}, η_{k_{d,g}}⟩_{π})+

+ \frac{2mn − k_{d,g}m}{2}c_{1}(⟨η_{k_{d,g}}, L_{d}⟩_{π}) + \frac{m^{2}}{2} c_{1}(⟨L_{d}, L_{d}⟩_{π}).

Using the above formula (7.18), we can compute the difference between the first Chern classes of the line bundles in (7.16) and in (7.17):

c_{1}(p_{2}^{∗}(Ξ)) − c_{1}(s^{∗}(O(k_{d,g} · Θ))) =

= (k_{d,g} + e_{d,g})c_{1}(d_{π}(L_{d})) − e_{d,g}c_{1}(d_{π}(η_{k_{d,g}}^{k_{d,g}} ⊗ L_{d})) − k_{d,g}c_{1}(d_{π}(η_{k_{d,g}}^{−e_{d,g}} ⊗ L_{d})) =

= −\frac{k_{d,g}(k_{d,g} + e_{d,g})e_{d,g}}{2} c_{1}(⟨η_{k_{d,g}}, η_{k_{d,g}}⟩_{π}).

The result now follows since c_{1} : Pic(S_{g}^{1/k_{d,g}} ×_{M_{g}} J ac_{d,g}) → A^{1}(S_{g}^{1/k_{d,g}} ×_{M_{g}} J ac_{d,g}) is an isomorphism (see Fact 2.9(i)).

■

Remark 7.5. Using the computation of the Picard group of the moduli stacks of spin curves by Jarvis [Jar01], it can be proved that the pull-back morphism p_{2}^{∗} : Pic(J ac_{d,g}) → Pic(S_{g}^{1/k_{d,g}} ×_{M_{g}} J ac_{d,g}) is injective. Therefore, Lemma 7.4 uniquely determines the line bundle Ξ. However, while the definition (6.2) extends naturally to \overline{J ac}_{d,g}, we do not know how to extend the formula of Lemma 7.4 to \overline{J ac}_{d,g}. The problem is that we do not know how to extend the correspondence between J ac_{d,g} and J ac_{g−1,g} given in diagram (7.14) to a correspondence between \overline{J ac}_{d,g} and \overline{J ac}_{g−1,g}.

7.2. Relation between J ac_{d,g} and the universal d-th symmetric product. The referee pointed out to us an interesting connection between the Picard groups of J ac_{d,g} and of the d-th symmetric product Sym^{d} M_{g,1} of the universal curve M_{g,1} → M_{g}, when d > 2g − 2. The fiber of the stack Sym^{d} M_{g,1} (for d ≥ 1) over a scheme S is the groupoid whose objects are families of smooth curves C → S of genus g together with an effective divisor D ⊂ C of relative degree d over S, and whose arrows are the obvious isomorphisms. Consider the universal Abel-Jacobi morphism

(7.19)  \begin{matrix}Ã_{d} : Sym^{d} M_{g,1} & ⟶ J ac_{d,g} \\ (C → S, D) & ↦ (C → S, O_{C}(D)), \end{matrix}

and the induced commutative diagram

(7.20)  \xymatrix{Sym^{d} M_{g,1} ×_{M_{g}} M_{g,1} \ar[d]_{π̃} \ar[rr]_{□}^{Â_{d}} &  & J_{d,g,1} \ar[d]^{π} \\ Sym^{d} M_{g,1} \ar[rr]^{Ã_{d}} \ar[dr]_{A_{d}} &  & _{J ac_{d,g}} \ar[dl]^{ν_{d}} \\  & J_{d,g} \ar[d]^{Φ_{d}} \\  & M_{g}}

If d > 2g − 2 then A_{d} is a projective bundle of relative dimension d − g whose class [A_{d}] in the Brauer group Br(J_{d,g}) is equal to the class [ν_{d}] of the 𝔾_{m}-gerbe ν_{d}, as it follows easily from [MR85, Lemma 2.1]. Therefore, the exact sequence (6.1) induced by the 𝔾_{m}-gerbe ν_{d} maps into the analogous exact sequence for the projective bundle A_{d}:

(7.21) [figure: 0; Pic(J_{d,g}); ν_{d}^{∗}; Pic(J ac_{d,g}); res; Ã_{d}^{∗}; ℤ; ≅; obs; Br(J_{d,g}); 0; Pic(J_{d,g}); A_{d}^{∗}; Pic(Sym^{d} M_{g,1}); r̃es; ℤ; õbs; Br(J_{d,g})]

where the maps in the second exact sequence of the above diagram admit the following interpretation (which one can easily check via standard cocycle computations): A_{d}^{∗} is the pull-back map induced by A_{d}; r̃es is the restriction to the generic fiber of A_{d} and õbs (the obstruction map) sends 1 ∈ ℤ into the class [A_{d}] of the projective bundle A_{d} in the (cohomological) Brauer group Br(J_{d,g}) := H_{ét}^{2}(J_{d,g}, 𝔾_{m}). The above diagram (7.21) implies that the pullback map Ã_{d}^{∗} is an isomorphism. Moreover the pullback of the tautological line bundles on J ac_{d,g} can be expressed as tautological line bundles on Sym^{d} M_{g,1}. Indeed, from the Cartesian square at the top of diagram (7.21), we get that

(7.22)  Â_{d}^{∗}(ω_{π}) = ωπ̃  and  Â_{d}^{∗}(L_{d}) = O(D_{d}),

where ωπ̃ is the relative dualizing line bundle for π̃ and D_{d} is the universal degree-d divisor on Sym^{d} M_{g,1} ×_{M_{g}} M_{g,1}. Using the functoriality of the determinant of cohomology, we get

(7.23)  \begin{matrix}Ã_{d}^{∗}(Λ(1, 0)) = dπ̃(ωπ̃) := Λ̃(1, 0), \\ Ã_{d}^{∗}(Λ(0, 1)) = dπ̃(O(D_{d})) := Λ̃(0, 1), \\ Ã_{d}^{∗}(Λ(1, 1)) = dπ̃(ωπ̃(D_{d})) := Λ̃(1, 1). \end{matrix}

Therefore, combining Theorem A(i), (7.21) and (7.23), we deduce the following

Corollary 7.6. Assume that g ≥ 3 and that d > 2g − 2. The Picard group of Sym^{d} M_{g,1} is freely generated by Λ̃(1, 0), Λ̃(0, 1) and Λ̃(1, 1).

Remark 7.7. The referee pointed out to us that Corollary 7.6 could be proved independently from Theorem A(i), using the computations contained in [Kou94]. In turn, this can be used to give an alternative proof of Theorems A(i) and B(i) (at least for d > 2g −2). However, this alternative approach does not give a modular description of the generators of the Picard groups of \overline{J ac}_{d,g} and of \overline{J}_{d,g}, since it is not known how to extend the Abel-Jacobi morphism over the boundary of \overline{M}_{g}.

8. Relation with the moduli space \overline{J_{d,g}}

The aim of this section is to relate the Picard group of the stack \overline{J}_{d,g} with the divisor class group and the rational Picard group of its moduli space \overline{J_{d,g}}, computed by Fontanari in [Fon05, Thm. 5, Cor. 1], based upon the results of Kouvidakis [Kou91]. Recall that, given a variety Y, the divisor class group Cl(Y) is the group of Weil divisors modulo rational equivalence. If Y is normal, denoting by Y_{reg} the open subset of regular points of Y, then we have that

(8.1)  Pic(Y) ↪ Cl(Y) ≅ Cl(Y_{reg}) ≅ Pic(Y_{reg}).

Recall that \overline{J}_{d,g} is a normal variety (see Theorem 2.5) and it is endowed with a morphism ϕ_{d} : \overline{J}_{d,g} → \overline{M}_{g} into the coarse moduli space of stable curves of genus g (see diagram (2.2)).

Theorem 8.1 (Fontanari). Set Δ̃_{i} := ϕ_{d}^{−1}(Δ_{i}) ⊂ \overline{J}_{d,g} for i = 0, ⋯, [g/2].

(i) The divisors Δ̃_{i} are irreducible and we have an exact sequence

0 → ⨁_{i=0}^{[g/2]} ℤ · Δ̃_{i} → Cl(\overline{J}_{d,g}) → Cl(J_{d,g}) → 0.

(ii) The natural inclusion Pic(\overline{J}_{d,g}) ↪ Cl(\overline{J}_{d,g}) is of finite index, i.e. every Weil divisor on \overline{J}_{d,g} is ℚ-Cartier.

We have therefore a commutative diagram with exact rows:

\xymatrix{0 \ar[r] & ⨁_{i=0}^{[g/2]} ℤ · Δ̃_{i} \ar[r] \ar[d]^{α_{d}} & Cl(\overline{J}_{d,g}) \ar[r] \ar[d]^{Ψ_{d}^{∗}} & Cl(J_{d,g}) \ar[r] \ar[d]^{β_{d}} & 0 \\ 0 \ar[r] & ⨁_{or i=g/2^{k_{d,g} ∤ 2i−1}} ⟨O(\overline{δ}_{i})⟩ ⨁_{and i≠g/2^{k_{d,g} |2i−1}} ⟨O(\overline{δ}_{i}^{1}), O(\overline{δ}_{i}^{2})⟩ \ar[r] & Pic(\overline{J}_{d,g}) \ar[r] & Pic(J_{d,g}) \ar[r] & 0,}

where the map Ψ_{d}^{∗} is the pull-back map induced by Ψ_{d} : \overline{J}_{d,g} → \overline{J}_{d,g}. We can now prove Theorem C from the introduction.

Proof of Theorem C. In order to prove part (i) of Theorem C, consider the commutative diagram, obtained by pulling back divisors along the two fibrations J_{d,g} → M_{g} and J_{d,g} → M_{g}:

[figure: Cl(M_{g}); Cl(J_{d,g}); 0; Pic((M_{g})_{reg}); γ_{d}; Pic((J_{d,g})_{reg}); β_{d}; RPic((J_{d,g})_{reg}); \overline{β_{d}}; 0; 0; Pic(M_{g}); Pic(J_{d,g}); RPic(J_{d,g}); 0,]

The map γ_{d} is well-known to be an isomorphism (see e. g. [AC87, Prop. 2]). The map \overline{β_{d}} is an isomorphism since the group of rational determined line bundles RPic of a fibration is birational on the base (see [Cil87, Lemma 1.3]) and the map J_{d,g} → M_{g} is representable. Since the rows of the above diagram are exact, we conclude that β_{d} is an isomorphism, q.e.d. In order to prove part (ii) of Theorem C, we need a local description of the morphism Ψ_{d} : \overline{J}_{d,g} → \overline{J}_{d,g} at the general point of Δ̃_{i}. This was carried on in [BFV12, Proof of Thm. 1.5] for the morphism ν_{d} ∘ Ψ_{d} : \overline{J ac}_{d,g} → \overline{J}_{d,g}, but it is very easy to adapt the description in loc. cit. to the morphism Ψ_{d} (simply by passing to the 𝔾_{m}-rigidification). If k_{d,g} ∤ (2i − 1) (which corresponds to the cases (1) and (2) of loc. cit.) then the morphism Ψ_{d} is an isomorphism locally at the general point of Δ̃_{i} (see [BFV12, p. 25]). Therefore Ψ_{d}^{∗}(Δ̃_{i}) = O(\overline{δ}_{i}). If k_{d,g} | (2i − 1) (which corresponds to the case (3) of loc. cit.) then the morphism Ψ_{d} looks like (after neglecting trivial coordinates)

X := [Spf k[[x, y]]ˆ⊗A/𝔾_{m}] ⟶^{p} X := Spf k[[x, y]]/𝔾_{m} ˆ⊗A = Spf k[[xy]]ˆ⊗A,

where A = Spf k[[y_{1}, ⋯, y_{4g−4}]], 𝔾_{m} acts via λ · (x, y) = (λx, λ^{−1}y) and trivially on A (see [BFV12, p. 26]). In this local description, the divisor Δ̃_{i} corresponds to the divisor (xy = 0) on X and the divisors δ̃_{i}^{1} and δ̃_{i}^{2} correspond to the divisors (x = 0) and (y = 0) on X (note that in the particular case i = g/2 and k_{d,g} | (g − 1), the divisor δ̃_{g/2}, even though irreducible, locally analytically splits into two components, which we can call δ̃_{g/2}^{1} and δ̃_{g/2}^{2}, so that the above description remains valid also in this case). From the explicit form of the map p, it is clear that p^{∗}(xy = 0) = (x = 0) + (y = 0), from which we deduce that

Ψ_{d}^{∗}(Δ̃_{i}) = \begin{cases}O(\overline{δ}_{i}^{1} + \overline{δ}_{i}^{2}) & if i < g/2, \\ O(2\overline{δ}_{g/2}) & if i = g/2. \end{cases}

Part (ii) is now proved.

■

References

[ACV03] D. Abramovich, A. Corti, A. Vistoli: Twisted bundles and admissible covers. Special issue in honor of Steven L. Kleiman. Comm. Algebra 31 (2003), no. 8, 3547–3618.

[AGV09] D. Abramovich, T. Graber, A. Vistoli: Gromov-Witten theory of Deligne-Mumford stacks. Amer. J. Math. 130 (2008), no. 5, 1337– 1398.

[Alp1] J. Alper: Good moduli spaces for Artin stacks. To appear in Annales de l’Institute de Fourier (preprint arXiv:0804.2242v2).

[Alp2] J. Alper: Adequate moduli spaces and geometrically reductive group schemes. Preprint arXiv:1005.2398.

[AC87] E. Arbarello and M. Cornalba: The Picard groups of the moduli spaces of curves. Topology 26 (1987), no. 2, 153–171.

[ACG11] E. Arbarello, M. Cornalba, P.A. Griffiths: Geometry of algebraic curves. Volume II. With a contribution by Joseph Daniel Harris. Grundlehren der Mathematischen Wissenschaften 268. Springer, Heidelberg, 2011.

[BL94] A. Beauville, Y. Laszlo: Conformal blocks and generalized theta functions. Comm. Math. Phys. 164 (1994), 385-419.

[BLS98] A. Beauville, Y. Laszlo and C. Sorger: The Picard group of the moduli of G-bundles on a curve. Compositio Math. 112 (1998), 183–216.

[BFV12] G. Bini, C. Fontanari, F. Viviani: On the birational geometry of the universal Picard variety. Int. Math. Res. Not. 2012, No. 4, 740-780 (2012).

[BFMV] G. Bini, F. Felici, M. Melo, F. Viviani: GIT for polarized curves. Preprint arXiv:1109.6908v3.

[BMV12] G. Bini, M. Melo, F. Viviani: On GIT quotients of Hilbert and Chow schemes of curves. Electron. Res. Announc. Math. Sci. 19 (2012), 33– 40.

[BH10] I. Biswas, N. Hoffmann: The line bundles on moduli stacks of principal bundles on a curve. Doc. Math. 15 (2010), 35–72.

[BK05] A. Boysal, S. Kumar: Explicit determination of the Picard group of moduli spaces of semistable G-bundles on curves. Math. Ann. 332 (2005), 823-842.

[Cap94] L. Caporaso: A compactification of the universal Picard variety over the moduli space of stable curves. J. Amer. Math. Soc. 7 (1994), no. 3, 589–660.

[Cap05] L. Caporaso: Néron models and compactified Picard schemes over the moduli stack of stable curves. Amer. J. Math. 130 (2008), no. 1, 1–47.

[CMKVa] S. Casalaina-Martin, J. L. Kass, F. Viviani: The Local Structure of Compactified Jacobians. Preprint arXiv:1107.4166v2.

[CMKVb] S. Casalaina-Martin, J. L. Kass, F. Viviani: The singularities and birational geometry of the universal compactified Jacobian. In preparation.

[Cil87] C. Ciliberto: On rationally determined line bundles on a family of projective curves with general moduli. Duke Math. J. 55 (1987), no. 4, 909–917.

[Cor91] M. Cornalba: A remark on the Picard group of spin moduli space. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 2 (1991), no. 3, 211–217.

[Cor07] M. Cornalba: The Picard group of the moduli stack of stable hyperelliptic curves. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007), no. 1, 109–115.

[Del87] P. Deligne: Le déterminant de la cohomologie. Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), 93177, Contemp. Math., 67, Amer. Math. Soc., Providence, RI, 1987.

[DN89] J. M. Drezet, M. S. Narasimhan: Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques. Invent. Math. 97 (1989), 53-94.

[Edi12] D. Edidin: Equivariant geometry and the cohomology of the moduli space of curves. To appear in Handbook of Moduli, edited by G. Farkas and I. Morrison, Advanced Lectures in Mathematics (2012). Preprint arXiv:1006.2364.

[EG98] D. Edidin, W. Graham: Equivariant intersection theory. Invent. Math. 131 (1998), 595-634.

[ERW12] J. Ebert, O. Randal-Williams: Stable cohomology of the universal Picard varieties and the extended mapping class group. Doc. Math. 17 (2012), 417–450.

[Fal03] G. Faltings: Algebraic loop groups and moduli spaces of bundles. J. Eur. Math. Soc. 5 (2003), 41–68.

[FV13] G. Farkas, S. Verra: The classification of universal Jacobians over the moduli space of curves. Comment. Math. Helv. 88 (2013), no. 3, 587–611.

[Fon05] C. Fontanari: On the geometry of moduli of curves and line bundles. Ren. Mat. Acc. Lincei 16 (2005), no. 1, 45–59.

[Ful98] W. Fulton: Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 2. Springer-Verlag, Berlin, 1998.

[Gie82] D. Gieseker: Lectures on moduli of curves. Tata Inst. Fund. Res. Lectures on Math. and Phys. 69, Springer-Verlag, Berlin-New York, 1982.

[Gir71] J. Giraud: Cohomologie non abélienne. Die Grundlehren der mathematischen Wissenschaften, Band 179. Springer-Verlag, Berlin-New York, 1971.

[GV08] S. Gorchinskiy, F. Viviani: Picard group of moduli of hyperelliptic curves. Math. Z. 258 (2008), no. 2, 319–331.

[Har83] J. Harer: The second homology group of the mapping class group of an orientable surface. Inven. Math. 72 (1983), 221–239.

[HM98] J. Harris, I. Morrison: Moduli of curves. Graduate Texts in Mathematics, 187. Springer-Verlag, New York, 1998.

[HH09] B. Hassett, D. Hyeon: Log canonical models for the moduli space of curves: first divisorial contraction. Trans. Amer. Math. Soc. 361 (2009), 4471–4489.

[Hof07] N. Hoffmann: Rationality and Poincaré families for vector bundles with extra structure on a curve. Int. Math. Res. Not. 2007, no. 3, Art. ID rnm010, 30 pp.

[KM76] F.F. Knudsen, D. Mumford: The projectivity of the moduli space of stable curves. I. Preliminaries on det and Div. Mathematica Scandinavica 39 (1976), 19–55.

[GIT65] D. Mumford: Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34 Springer-Verlag, Berlin-New York 1965.

[Jar01] T. J. Jarvis: The Picard group of the moduli of higher spin curves. New York J. Math. 7 (2001), 23–47.

[Kou91] A. Kouvidakis: The Picard group of the universal Picard varieties over the moduli space of curves, J. Differential Geom. 34 (1991), no. 3, 839–850.

[Kou93] A. Kouvidakis: On the moduli space of vector bundles on the fibers of the universal curve. J. Differential Geom. 37 (1993), no. 3, 505–522.

[Kou94] A. Kouvidakis: Picard groups of Hilbert schemes of curves. J. Algebraic Geom. 3 (1994), no. 4, 671–684.

[KN97] S. Kumar and M. S. Narasimhan: Picard group of the moduli spaces of G-bundles. Math. Ann. 308 (1997), 155–173.

[LS97] Y. Laszlo, C. Sorger: The line bundles on the moduli of parabolic Gbundles over curves and their sections. Ann. Sci. École Norm. Sup.

(4) 30 (1997), 499–525.

[Lie08] M. Lieblich: Twisted sheaves and the period-index problem. Compositio Math. 144 (2008), 1–31.

[Mel09] M. Melo: Compactified Picard stacks over \overline{M}_{g}. Math. Z. 263 (2009), no. 4, 939–957.

[Mel10] M. Melo: Compactified Picard stacks over the moduli stack of stable of stable curves with marked points. Adv. Math. 226 (2011), no. 1, 727-763.

[MR85] N. Mestrano, S. Ramanan: Poincaré bundles for families of curves. J. Reine Angew. Math. 362 (1985), 169–178.

[Mor01] A. Moriwaki: The ℚ-Picard group of the moduli space of curves in positive characteristic. Internat. J. Math. 12 (2001), no. 5, 519–534.

[Mum65] D. Mumford: Picard groups of moduli functors, in Arithmetical algebraic geometry Proceedings of a conference held at Purdue University, O. Schilling, editor (1965), 33–81.

[Mum77] D. Mumford: Stability of projective varieties. L’Ens. Math. 23 (1977), 1–39.

[Mum83] D. Mumford: Towards an enumerative geometry of the moduli space of curves. Arithmetic and geometry, Vol. II, 271-328, Progr. Math., 36, Birkhäuser Boston, Boston, MA, 1983.

[Pan96] R. Pandharipande: A compactification over \overline{M}_{g} of the universal moduli space of slope-semi-stable vector bundles. J. Amer. Math. Soc. 9 (1996), 425–471.

[Put12] A. Putman: The Picard group of the moduli space of curves with level structures. Duke Math. J. 161 (2012), 623–674.

[Rom05] M. Romagny: Group actions on stacks and applications. Michigan Math. J. 53 (2005), no. 1, 209–236.

[Sor99] C. Sorger: On moduli of G-bundles of a curve for exceptional G. Ann. Sci. École Norm. Sup. 32 (1999), 127–133.

[Vis98] A. Vistoli: The Chow ring of M_{2}. Invent. Math. 131 (1998), 635–644.

Margarida Melo Departamento de Matemática da Universidade de Coimbra Largo D. Dinis Apartado 3008 3001 Coimbra Portugal mmelo@mat.ucpt Filippo Viviani Dipartimento di Matematica Università Roma Tre Largo S. Leonardo Murialdo 1 00146 Roma Italy filippo.viviani@gmail.com 508

Documenta Mathematica 19 (2014)