The Unirationality of Hurwitz Spaces

of 6-Gonal Curves of Small Genus

Florian Geiß

Received: September 20, 2011 Revised: April 17, 2012

Communicated by Gavril Farkas

Abstract. In this short note we prove the unirationality of Hurwitz spaces of 6-gonal curves of genus g with 5 ≤ g ≤ 28 or g = 30, 31, 35, 36, 40, 45. Key ingredient is a liaison construction in ℙ^{1} × ℙ^{2}. By semicontinuity, the proof of the dominance of this construction is reduced to a computation of a single curve over a finite field.

2010 Mathematics Subject Classification: 14H10, 14M20, 14M06, 14H51, 14Q05 Keywords and Phrases: Unirationality, Hurwitz space.

1 Introduction

The study of the birational geometry of moduli spaces of curves with additional structures such as marked points or line bundles is a central topic in algebraic geometry, see for example the books [HM98] and [ACG11]. The Hurwitz space ℋ (d, w) parametrizes d-sheeted branched simple covers of the projective line by smooth curves of genus g with branch divisor of degree w = 2g + 2d − 2 up to isomorphism,

ℋ (d, 2g + 2d − 2) = {C ⟶^{d:1} ℙ^{1} simply branched | C smooth of genus g}/ ∼ .

It is a classical result by Arbarello and Cornalba [AC81] based on a work of Segre [Seg28] that theses spaces are unirational for all d ≤ 5 and all g ≥ d − 1 and in few cases for higher gonality, namely for d = 6 and 5 ≤ g ≤ 10 or g = 12 and for d = 7 and g = 7. In this paper we present the following extension of this result to significantly higher genus for 6-gonal curves.

Theorem 1.1. Over an algebraically closed field of characteristic zero, the Hurwitz spaces ℋ (6, 2g + 10) of 6-gonal curves of genus g are unirational for

5 ≤ g ≤ 28 or g = 30, 31, 33, 35, 36, 40, 45.  (1)

Our proof is based on the observation that a general 6-gonal curve in ℙ^{1} × ℙ^{2} can be linked in two steps to the union of a rational curve and a collection of lines. It turns out that for small genera this process can be reversed by starting with a general rational curve and general lines. To show that the described construction yields a parametrization of the Hurwitz space, we only need to run the construction for a single curve over a finite field. Semicontinuity then ensures that all assumptions we made actually hold for an open dense subset of ℋ (6, 2g+10) in characteristic zero. Since the construction works a priori only for finitely many genera we settle for a computer-aided verification using the computer algebra system Macaulay2 [GS]. An immediate consequence of our approach is that in the considered cases the general 6-gonal curve has a plane model as expected from Brill-Noether theory.

Corollary 1.2. For g among (1) and d = ⌈\frac{2}{3}g + 2⌉ the Brill-Noether locus W_{d}^{2}(C) of a general curve C ∈ ℋ (6, 10 + 2g) has a smooth generically reduced component of expected dimension ρ(g, 2, d).

Acknowledgements

This work is part of my PhD thesis. I would like to thank my advisor Frank-Olaf Schreyer for valuable discussions and continuous support. I am grateful to the DFG for support through the priority program SPP 1489.

2 Preliminaries

Throughout this paper, we fix the following notation: Let ℙ = ℙ^{1} × ℙ^{2} be the product of the projective line and the projective plane over a field K with projections π_{1} : ℙ → ℙ^{1} and π_{2} : ℙ → ℙ^{2}. For a, b ∈ ℤ we write

𝒪_{ℙ}(a, b) = 𝒪_{ℙ^{1}}(a) ⊠ 𝒪_{ℙ^{2}}(b) = π_{1}^{∗}𝒪_{ℙ^{1}}(a) ⊗ π_{2}^{∗}𝒪_{ℙ^{2}}(b)

and denote with R = ⨁_{i,j} H^{0}(ℙ, 𝒪_{ℙ}(i, j)) ≅ K[x_{0}, x_{1}, y_{0}, y_{1}, y_{2}] the bihomogeneous coordinate ring of ℙ. By a curve C in ℙ, we mean an equidimensional subscheme of codimension 2 which is locally a complete intersection. We say that C is (geometrically) linked to a curve C′ ⊂ ℙ by a complete intersection X ⊂ ℙ if C and C′ have no common components and C ∪ C′ = X. The Chow ring of ℙ is generated by classes α and β which are the pullback of a point in ℙ^{1} and the pullback of a hyperplane in ℙ^{2}, respectively. The bidegree (d_{1}, d_{2}) of a curve C is given by d_{1} = [C].α and d_{2} = [C].β. As in the classical setting of liaison of subschemes in ℙ^{n}, we have the following Proposition 2.1 (Exact sequence of liaison). Let C be a curve of bidegree (d_{1}, d_{2}) that is linked to C′ via a complete intersection X defined by forms of bidegree (a_{1}, b_{1}) and (a_{2}, b_{2}). We set a = a_{1} + a_{2} and b = b_{1} + b_{2}.

(a) There is an exact sequence

0 → ω_{C} → ω_{X} → 𝒪_{C′}(a − 2, b − 3) → 0.

(b) The curve C′ has bidegree (d′_{1}, d′_{2}) = (b_{1}b_{2} − d_{1}, a_{1}b_{2} + a_{2}b_{1} − d_{2}) and arithmetic genus p_{a}(C′) = p_{a}(X)−(d_{1}(a − 2) + d_{2}(b − 3) + (1 − p_{a}(C))).

Proof. To prove the first part, consider the standard exact sequence

0 → ℐ_{C/X} → 𝒪_{X} → 𝒪_{C} → 0

and apply Hom_{𝒪_{ℙ}} (−, ω_{ℙ}). From the long exact sequence, we get

0 → ω_{C} → ω_{X} → Ext^{2}(ℐ_{C/X}, ω_{ℙ}) → 0

but Ext^{2}(ℐ_{C/X}, ω_{ℙ}) = 𝒪_{C′}(a − 2, b − 3) since C and C′ are linked by X. The formula for the genus follows immediately. To compute the bidegree, note that [C] + [C′] = [X] = (b_{1}b_{2})β^{2} + (a_{1}b_{2} + a_{2}b_{1})αβ in the Chow ring of ℙ.

Recall the following well-known fact about minimal resolutions of points in the plane.

Proposition 2.2. Let Δ be a collection of δ general points in ℙ^{2} and let k be maximal under the condition ε = δ − \binom{k+1}{2} ≥ 0. Then the minimal free resolution of 𝒪_{Δ} is of the form

0 → 𝒢 → ℱ → 𝒪_{ℙ^{2}} → 𝒪_{Δ} → 0

with ℱ = 𝒪(−k)^{k+1−ε} and 𝒢 = 𝒪(−k − 1)^{k−2ε} ⊕ 𝒪(−k − 2)^{ε} if 2ε ≤ k and ℱ = 𝒪(−k)^{k+1−ε} ⊕ 𝒪(−k − 1)^{2ε−k} and 𝒢 = 𝒪(−k − 2)^{ε} else.

Proof. [Gae51]

We also note the following simple but useful criterion for the irreducibility of plane curves. Recall that a variety over a field K is called absolutely irreducible if it is irreducible as a variety over the algebraic closure \overline{K}.

Proposition 2.3. Let C be a plane curve of degree d with δ ≤ \frac{d(d−3)}{2} ordinary double points and no other singularities. If the singular locus Δ of C has a resolution as in 2.2 then C is absolutely irreducible.

Proof. Assume that C decomposes into two curves C_{1} and C_{2} of degree d_{1} and d_{2} defined by homogeneous polynomials f_{1} and f_{2}. By assumption, C_{1} and C_{2} intersect transversely in d_{1} · d_{2} distinct points. First, we reduce to the case d_{1}, d_{2} ≤ k where k = ⌈(\sqrt{9 + 8δ} − 3)/2⌉ is the minimal degree of generators of I_{Δ}. Clearly, the case that one of the generators has degree strictly larger than k + 1 is not possible since I_{Δ} ⊂ (f_{1}, f_{2}) is generated in degree k and (possibly) k + 1. The cases d_{1} = k + 1, say, and d_{2} ≤ k + 1 can be excluded by considering the number of minimal generators of I_{Δ} in degrees k and k + 1. We are left with the case d_{1}, d_{2} ≤ k. Trivially, we can assume that δ−d_{1}d_{2} ≥ 0. A polynomial of the form sf_{1} + tf_{2} of degree k lies in I_{Δ} if it vanishes at the remaining δ − d_{1}d_{2} points. Hence,

h^{0}(ℐ_{Δ}(k)) ≥ \binom{k − d_{1} + 2}{2} + \binom{k − d_{2} + 2}{2} − δ + d_{1}d_{2}
= 2\binom{k + 2}{2} + \binom{d − 1}{2} − (dk + 1) − δ

But this is strictly larger than \binom{k+2}{2} − δ since d ≥ k + 3.

Recall from [ACGH85] the following facts from Brill-Noether theory: For a fixed smooth curve C of genus g, the Brill-Noether locus

W_{d}^{r}(C) = {L ∈ Pic^{d}(C) | h^{0}(L) ≥ r + 1}  (2)

is of dimension at least equal to the Brill-Noether number

ρ(g, r, d) = g − (r + 1)(g − d + r).  (3)

The tangent space at a linear series L ∈ W_{d}^{r}(C) ∖ W_{d}^{r+1}(C) is the dual of the cokernel of the Petri-map

μ_{L} : H^{0}(C, L) ⊗ H^{0}(C, ω_{C} ⊗ L^{−1}) → H^{0}(C, ω_{C})  (4)

Hence, W_{d}^{r}(C) is smooth of dimension ρ at L if and only if μ_{L} is injective.

Proposition 2.4. Let C be a smooth curve of genus g ≥ 3 with |D| a base point free 𝔤_{d}^{2}, d = ⌈\frac{2g}{3} + 2⌉, such that the image of C under the associated map is a plane curve with δ = \binom{d−1}{2} − g ordinary double points and no other singularities. If the singular locus Δ has a resolution as in 2.2 then |D| is a smooth point in W_{d}^{2}(C).

Proof. By adjunction, the Petri map for 𝒪(D) can be identified with

H^{0}(ℙ^{2}, 𝒪(1)) ⊗ H^{0}(ℙ^{2}, ℐ_{Δ}(d − 4)) → H^{0}(ℙ^{2}, ℐ_{Δ}(d − 3)).

Under the given assumptions the minimal degree of generators of I_{Δ} is precisely k = d − 4. As 2ε ≤ k we see from the minimal free resolution of I_{Δ} that the Petri map is injective since there are no linear relations among the generators of degree k and k + 1.

3 Liaison construcion

For g ≥ 5, let f : C → ℙ^{1} be an element of ℋ (6, 10 + 2g) and let 𝒪(D_{1}) = f^{∗}𝒪_{ℙ^{1}}(1) be the 6-gonal bundle. We assume that C has a line bundle 𝒪(D_{2}) such that |D_{2}| is a complete base point free 𝔤_{d}^{2} with d = d(g) = ⌈\frac{2g}{3} + 2⌉ minimal under the condition that the Brill-Noether number ρ(g, 2, d) ≥ 0. Suppose further that the map

φ : C ⟶^{|D_{1}|,|D_{2}|} ℙH^{0}(𝒪(D_{1})) × ℙH^{0}(𝒪(D_{2})) = ℙ  (5)

is an embedding. In particular, this is the case when we assume that the plane model has only ordinary double points and no other singularities and for any node p the points in the preimage of p under C → ℙ^{2} are not identified under the map to ℙ^{1}. Hence, we can identify C with its image under φ. Furthermore, we assume that the map H^{0}(𝒪_{ℙ}(a, 3)) → H^{0}(𝒪_{C}(a, 3)) is of maximal rank for all a ≥ 1. To simplify matters, assume g ≡ 0 (12) for the moment. By the maximal rank assumption, we have

a_{Cubic} := min\{a | H^{0}(ℐ_{C}(a, 3)) ≠ 0\} = \frac{g}{4}  (6)

and h^{0}(ℐ_{C}(a_{Cubic}, 3)) = 3. Let X = V (f_{1}, f_{2}) be the complete intersection defined by two general sections f_{i} ∈ H^{0}(ℐ_{C}(a_{i}, b_{i})) of bidegrees (a_{1}, b_{1}) = (a_{2}, b_{2}) = (a_{Cubic}, 3). The curve C′, obtained by liaison of C by X, is smooth of bidegree (3, \frac{5}{6}g − 2) and genus g′ = \frac{g}{2} − 3 with h^{0}(ℐ_{C′}(a_{Cubic}, 3)) ≥ 2. The geometric situation is understood best when thinking of C as a family of collections of plane points over ℙ^{1}. We expect the general fiber of C to be a collection of 6 points in ℙ^{2} which are cut out by 4 cubics. We expect a finite number ℓ of distinguished fibers where the points lie on a conic as this is a codimension 1 condition on the points. Since the residual three points under liaison are collinear exactly in the distinguished fibers we can compute ℓ by examining the geometry of C′. The projection of C′ to ℙ^{2} yields a divisor D′_{2} of degree d′ > g′ + 2. Our claim is that ℓ = d′ − (g′ + 2). Indeed, the image of C′ under the associated map

ψ : C′ → ℙ^{1} × ℙH^{0}(C′, 𝒪(D′_{2})) = ℙ^{1} × ℙ^{d′−g′}  (7)

lies on the graph of the projection S → ℙ^{1} where S is a 3-dimensional scroll of degree d′ − g′ − 2 swept out by the 3-gonal series |D′_{1}|, i.e.

ψ(C′) ⊂ ℙ^{1} × S = ⋃_{D∈|D′_{1}|}\{D\} × \overline{D}.  (8)

See [Sch86] for a proof of this fact. C′ is obtained from ψ(C′) by projection from a linear subspace ℙ^{1} × V ⊂ ℙ^{1} × ℙ^{d′−g′} of codimension 3. A general space V intersects S in precisely d′ − g′ − 2 points lying in distinct fibers over ℙ^{1}.

Clearly, under the projection the points of D ∈ |D′_{1}| are mapped to 3 collinear points if and only if V meets the corresponding fiber of S. To keep things neat, we consider again the case g ≡ 0 (12) which implies ℓ = \frac{1}{3}g − 1. Suppose further that ℓ ≡ 1 (3). If we assume that the map H^{0}(𝒪_{ℙ}(a, 2)) → H^{0}(𝒪_{C′}(a, 2)) is of maximal rank for all a ≥ 1 then

a_{Conic} = min\{a | H^{0}(ℐ_{C′}(a, 2)) ≠ 0\} = \frac{g′ + 2ℓ + 1}{3}  (9)

and h^{0}(ℐ_{C′}(a_{Conic}, 2)) = 2. Let X′ = V (f′_{1}, f′_{2}) be defined by two general forms f′_{i} ∈ H^{0}(ℐ (a′_{i}, b′_{i})) of bidegrees (a′_{1}, b′_{1}) = (a′_{2}, b′_{2}) = (a_{Conic}, 2) and let C′′ denote the curve that is linked to C′ via X′. The general fiber of C′′ consists of a single point. In a distinguished fiber the conics of the complete intersection are reducible and have the line spanned by the points of the fiber of C′ as a common factor. Hence, C′′ is a rational curve together with ℓ lines. The rational curve has degree

d′′ = \frac{g′ + 2ℓ − 2}{3} = \frac{7}{18}g − \frac{7}{3}.  (10)

Turning things around we see that the difficulty lies in reversing the first linkage step. Indeed, a simple counting argument shows that for any g, the union C′′ of ℓ general lines in ℙ and the graph of a general rational normal curve of degree d′′ we have

min\{a ∈ ℤ| H^{0}(ℐ_{C′′}(a, 2)) ≠ 0\} = ⌈\frac{2d′′ + 3ℓ}{5}⌉ − 1 ≤ a_{Conics}.

Hence, we always obtain a trigonal curve C′ as desired. However, for general choices of C′′ and X′ we expect that the map H^{0}(𝒪_{ℙ}(a, 3)) → H^{0}(𝒪_{C′}(a_{Cubic}, 3)) is of maximal rank. In the case g ≡ 0(12), this restriction yields h^{0}(ℐ_{C′}(a_{Cubic}, 3)) = −\frac{g}{4} + 12, hence g < 48. Checking all congruency classes of g, we expect that C′ can be linked to a general curve C exactly in the cases

5 ≤ g ≤ 28 or g = 30, 31, 33, 35, 36, 40, 45.  (11)

Table 1 lists the appearing numbers for all values of g in (11). Summarizing, we obtain for g among (11) the following unirational construction for curves in ℋ (6, 10 + 2g):

1. We start with a general rational curve of degree d′′ in ℙ together with a collection of ℓ general lines. Call the union C′′.

2. We choose two general forms f′_{i} ∈ H^{0}(ℐ_{C′′}(a′_{i}, b′_{i})), i = 1, 2, that define a complete intersection X′ and obtain a trigonal curve C′ = \overline{X′ ∖ C′′} of degree d′ and genus g′.

3. We choose two general forms f_{i} ∈ H^{0}(ℐ_{C′}(a_{i}, b_{i})), i = 1, 2, that define a complete intersection X and obtain a 6-gonal curve C = \overline{X ∖ C′}.

It remains to show that the construction actually yields a parametrization of the Hurwitz spaces.

Hurwitz Schemes of 6-Gonal Curves 633

\begin{matrix}g & d & (a_{1}, b_{1}), (a_{2}, b_{2}) & g′ & d′ & (a′_{1}, b′_{1}), (a′_{2}, b′_{2}) & ℓ & d′′ \\ 5 & 6 & (2, 3), (2, 3) & 2 & 6 & (3, 2), (2, 2) & 2 & 2 \\ 6 & 6 & (2, 3), (1, 3) & 0 & 3 & (1, 2), (1, 2) & 1 & 0 \\ 7 & 7 & (2, 3), (2, 3) & 1 & 5 & (2, 2), (2, 2) & 2 & 1 \\ 8 & 8 & (3, 3), (2, 3) & 2 & 7 & (3, 2), (3, 2) & 3 & 2 \\ 9 & 8 & (2, 3), (2, 3) & 0 & 4 & (2, 2), (2, 2) & 2 & 2 \\ 10 & 9 & (3, 3), (3, 3) & 4 & 9 & (4, 2), (4, 2) & 3 & 4 \\ 11 & 10 & (3, 3), (3, 3) & 2 & 8 & (4, 2), (4, 2) & 4 & 4 \\ 12 & 10 & (3, 3), (3, 3) & 3 & 8 & (4, 2), (3, 2) & 3 & 3 \\ 13 & 11 & (4, 3), (3, 3) & 4 & 10 & (5, 2), (4, 2) & 4 & 4 \\ 14 & 12 & (4, 3), (4, 3) & 5 & 12 & (6, 2), (5, 2) & 5 & 5 \\ 15 & 12 & (4, 3), (4, 3) & 6 & 12 & (5, 2), (5, 2) & 4 & 4 \\ 16 & 13 & (4, 3), (4, 3) & 4 & 11 & (5, 2), (5, 2) & 5 & 4 \\ 17 & 14 & (5, 3), (5, 3) & 8 & 16 & (7, 2), (7, 2) & 6 & 6 \\ 18 & 14 & (5, 3), (4, 3) & 6 & 13 & (6, 2), (6, 2) & 5 & 6 \\ 19 & 15 & (5, 3), (5, 3) & 7 & 15 & (7, 2), (7, 2) & 6 & 7 \\ 20 & 16 & (6, 3), (5, 3) & 8 & 17 & (8, 2), (8, 2) & 7 & 8 \\ 21 & 16 & (5, 3), (5, 3) & 6 & 14 & (7, 2), (6, 2) & 6 & 6 \\ 22 & 17 & (6, 3), (6, 3) & 10 & 19 & (9, 2), (8, 2) & 7 & 8 \\ 23 & 18 & (6, 3), (6, 3) & 8 & 18 & (9, 2), (8, 2) & 8 & 8 \\ 24 & 18 & (6, 3), (6, 3) & 9 & 18 & (8, 2), (8, 2) & 7 & 7 \\ 25 & 19 & (7, 3), (6, 3) & 10 & 20 & (9, 2), (9, 2) & 8 & 8 \\ 26 & 20 & (7, 3), (7, 3) & 11 & 22 & (10, 2), (10, 2) & 9 & 9 \\ 27 & 20 & (7, 3), (7, 3) & 12 & 22 & (10, 2), (10, 2) & 8 & 10 \\ 28 & 21 & (7, 3), (7, 3) & 10 & 21 & (10, 2), (10, 2) & 9 & 10 \\ 30 & 22 & (8, 3), (7, 3) & 12 & 23 & (11, 2), (10, 2) & 9 & 10 \\ 31 & 23 & (8, 3), (8, 3) & 13 & 25 & (12, 2), (11, 2) & 10 & 11 \\ 33 & 24 & (8, 3), (8, 3) & 12 & 24 & (11, 2), (11, 2) & 10 & 10 \\ 35 & 26 & (9, 3), (9, 3) & 14 & 28 & (13, 2), (13, 2) & 12 & 12 \\ 36 & 26 & (9, 3), (9, 3) & 15 & 28 & (13, 2), (13, 2) & 11 & 13 \\ 40 & 29 & (10, 3), (10, 3) & 16 & 31 & (15, 2), (14, 2) & 13 & 14 \\ 45 & 32 & (11, 3), (11, 3) & 18 & 34 & (16, 2), (16, 2) & 14 & 16 \end{matrix}

Table 1: Numerical data for all cases of the construction

Documenta Mathematica 17 (2012) 627–640

4 Proof of the dominance

Theorem 4.1. For all (g, d) as in Table 1, there is a unirational component H_{g} of the Hilbert scheme Hilb_{(6,d),g}(ℙ) of curves in ℙ of bidegree (6, d) and genus g. The generic point of H_{g} corresponds to a smooth absolutely irreducible curve C such that the map H^{0}(𝒪_{ℙ}(a, 3)) → H^{0}(𝒪_{C}(a, 3)) is of maximal for all a > 1.

Proof. The crucial part is to prove the existence of a curve with the desired properties. Code 5.1 implements the construction above for any given value of g in (11) and establishes the existence of a smooth and absolutely irreducible curve C_{p} of given genus and bidegree defined over a prime field 𝔽_{p}. This computation can be regarded as the reduction of a computation over ℚ which yields some curve C_{0}. This curve is already defined over the rationals, since all construction steps invoke only Groebner basis computations. By semicontinuity, C_{0} is also smooth, absolutely irreducible and of maximal rank. Again, by semicontinuity, there is a Zariski open neighborhood U ⊂ Hilb_{(6,d),g}(ℙ) of points corresponding to smooth absolutely irreducible curves that fulfill the maximal rank condition. Let 𝔸^{N} be the parameter-space for all the choices made in the construction, i.e. the space of coefficients of the polynomials defining C′′ and the complete intersections X and X′. The construction then translates to a rational map 𝔸^{N} −−→ U defined over ℚ and we set H_{g} to be the closure of the image of this map.

Remark 4.2. We want to point out two issues concerning the computational verification:

1. The restriction to finite fields in the Macaulay2 computation in the appendix is only due to limitations in computational power. For very small values of g, i.e. g ≤ 15, it is still possible to compute examples over the rationals if all coefficients are chosen among integers of small absolute value.

2. The reduction of C_{0} modulo p gives a curve C_{p} with desired properties for p in an open part of Spec(ℤ). Hence, the main theorem is also true in almost all characteristics p. One way to extend it to all prime numbers would be to keep trace of all denominators in a computation over the rationals and check case by case the primes where a bad reduction happens. This is computationally also out of reach at the moment.

It remains to show that there exists a dominant rational map from H_{g} to the Hurwitz-scheme.

Theorem 4.3. For g among (11) and H_{g} as in Theorem 4.1 there is a dominant rational map

H_{d} −−→ ℋ (6, 10 + 2g).

Proof. Using Code 5.1 again, we check for any given value of g in (11) the existence of a point in H_{g} corresponding to a smooth absolutely irreducible curve C ⊂ ℙ such that the projection onto ℙ^{1} is simply branched and the bundle L_{2} = φ^{∗}𝒪_{ℙ}(0, 1) is a smooth point in the corresponding W_{d}^{2}(C). By semicontinuity, the loci of curves with this property is open and dense in H_{g}. Hence, we have a rational map H_{g} −−→ ℋ (6, 10 + 2g). The locus of curves in ℋ (6, 10+2g) having a smooth component of the Brill-Noether loci of expected dimension is also open and contains the image of [C] under this map. Since ℋ (6, 10 + 2g) is irreducible this locus is dense. This proves the theorem.

We want to emphasize the last statement in the proof:

Corollary 4.4. For g among (11) and d = ⌈\frac{2}{3}g + 2⌉ the Brill-Noether locus W_{d}^{2}(C) of a general curve C ∈ ℋ (6, 10 + 2g) has a smooth generically reduced component of expected dimension ρ.

5 Computational Verification

The following Code for Macaulay2 [GS] realizes the unirational construction of a 6-gonal curve of genus g as in (11) over a finite field K = 𝔽_{p} with random choices for all parameters. In order to explain the single steps in the computation, we also print the most relevant parts of the output for the example case g = 24.

Code 5.1. We start with the following initialization:

   i1 : Fp=ZZ/32009; -- a finite field
        S=Fp[x_0,x1,y_0..y_2,Degrees=>{2:{1,0},3:{0,1}}];
           -- Cox-ring of P^1 x P^2
        m=ideal basis({1,1},S);
           -- irrelevant ideal
        setRandomSeed("HurwitzSpaces");
           -- initialization of the random number generator

The following functions handle the numerics of the construction:

   i2 : expHilbFuncIdealSheaf=(g,d,a)->
           max(0,(a_0+1)*(a_1+2)*(a_1+1)/2-(a_0*d_0+a_1*d_1+1-g))
           -- expected number of sections of the ideal sheaf
        linkedGenus=(g,d,F,G)->(
           pX:=binomial(F_0+G_0-1,1)*binomial(F_1+G_1-1,2)-
               (F_0-1)*binomial(F_1-1,2)-(G_0-1)*binomial(G_1-1,2);
               -- genus of the complete intersection
           pX-d_0*(F_0+G_0-2)-d_1*(F_1+F_1-3)-1+g)
           -- genus of the linked curve
        linkedDegree=(g,d,F,G)->{F_1*G_1-d_0,F_0*G_1+G_0*F_1-d_1}
           -- bidegree of the linked curve

The first step is to determine the degree d′′ of the rational curve and the number of lines ℓ. We start by computing the bidegrees of the forms that define the complete intersection for the linkage to the trigonal curve:

636 Florian Geiß

i3 : g=24;
d={6,ceiling(-g/3+g+2)};
   -- choose the second degree Brill-Noether general
a=for i from 0 do
    if expHilbFuncIdealSheaf(g,d,{i,3})!=0 then break i;
   -- find the minimal value a s.t. H^0(IC(a,3)) nonzero
if expHilbFuncIdealSheaf(g,d,{a,3})==1 then
   fX={{a+1,3},{a,3}} else fX={{a,3},{a,3}};
   -- choose bidegrees of forms for the complete intersection
(d,fX)
o3 = ({6, 18}, {{6, 3}, {6, 3}})

The genus and degree of the trigonal curve and the number of lines:

i4 : g'=linkedGenus(g,d,fX_0,fX_1);
d'=linkedDegree(g,d,fX_0,fX_1);
l=d'_1-g'-2;
(g',d',l)
o4 = (9,{3,18},7)

We compute the bidegrees for the complete intersection for the linkage to the rational curve

i5 : b=for i from 0 do
    if expHilbFuncIdealSheaf(g',d',{i,2})!=0 then break i;
if expHilbFuncIdealSheaf(g',d',{b,2})==1 then
   fX'={{b+1,2},{b,2}} else fX'={{b,2},{b,2}};
d''=linkedDegree(g'+2*l,d'+{0,l},fX'_0,fX'_1);
dRat={{ceiling(d''_1/2),1},{floor(d''_1/2),1}};
(fX',d'')
o5 = ({{8, 2}, {8, 2}}, {1, 7})

The second step is the actual construction: First, we choose a rational curve and random lines and compute the saturated vanishing ideal I_{C′′} of their union:

i6 : ICrat=saturate(ideal random(S^1,S^(-dRat)),m);
ILines=apply(l,i->ideal random(S^1,S^{{-1,0},{0,-1}}));
time IC''=saturate(intersect(ILines|{ICrat}),ideal(x_0*y_0));
   -- used 1.29537 seconds

Next, we choose random forms in I_{C′′} of degree b (resp. of b + 1) that define the complete intersection X′ and compute the saturated vanishing ideal I_{C′} of the trigonal curve C′.

i7 : IX'=ideal(gens IC'' * random(source gens IC'',S^(-fX')));
IC'=IX':ICrat;
time scan(l,i->IC'=IC':ILines_i);
time IC'sat=saturate(IC',ideal(x_0*y_0));
   -- used 2.06236 seconds
   -- used 23.7319 seconds

Hurwitz Schemes of 6-Gonal Curves 637

In the final step, we compute the vanishing ideal of the 6-gonal curve C by linking C′ with a complete intersection X given by random forms in I_{C′} of degree a (resp. a + 1).

i8 : IX=ideal(gens IC'sat * random(source gens IC'sat,S^(-fX)));
time IC=IX:IC';
time ICsat=saturate(IC,ideal(x_0*y_0));
-- used 15.7815 seconds
-- used 3.84807 seconds

We check that C is of maximal rank in the degrees (a, 3) by looking at the minimal generators of the saturated vanishing ideal:

i9 : tally degrees ideal mingens gb ICsat
o9 = Tally{{0, 18} => 1}
{1, 14} => 5
{1, 15} => 4
{2, 8} => 2
{2, 9} => 8
{3, 6} => 9
{4, 4} => 2
{4, 5} => 8
{5, 4} => 7
{6, 3} => 3
{7, 3} => 1

In order to check irreducibility, we compute the plane model Γ of C:

i10 : Sel=Fp[x_0,x_1,y_0..y_2,MonomialOrder=>Eliminate 2];
-- eliminination order
R=Fp[y_0..y_2]; -- coordinate ring of P^2
IGammaC=sub(ideal selectInSubring(1,gens gb sub(ICsat,Sel)),R);
-- ideal of the plane model

We check that Γ is a curve of desired degree and genus and its singular locus Δ consists only of ordinary double points:

i11 : distinctPoints=(J)->(
singJ:=minors(2,jacobian J)+J;
codim singJ==3)
i12 : IDelta=ideal jacobian IGammaC + IGammaC; -- singular locus
distinctPoints(IDelta)
o12 = true
i13 : delta=degree IDelta;
dGamma=degree IGammaC;

638 Florian Geiß

gGamma=binomial(dGamma-1,2)-delta;
(dGamma,gGamma)==(d_1,g)
o13 = true

We compute the free resolution of I_{Δ}:

i14 : time IDelta=saturate IDelta;
betti res IDelta
-- used 55.063 seconds
   0 1 2
o14 = total: 1 8 7
0: 1 . .
1: . . .
2: . . .
3: . . .
4: . . .
5: . . .
6: . . .
7: . . .
8: . . .
9: . . .
10: . . .
11: . . .
12: . . .
13: . 8 .
14: . . 7

This is the resolution as expected. Hence, C is absolutely irreducible by Proposition 2.2 and 𝒪(D_{2}) is a smooth point of the Brill-Noether loci by Proposition 2.4. It remains to verify that C is actually smooth and simply branched. We compute the vanishing ideal I_{B} ⊂ K[x_{0}, x_{1}] of the locus B in ℙ^{1} of points with non-reduced fiber.

i15 : gensICsat=flatten entries mingens ICsat;
Icubics=ideal select(gensICsat,f->(degree f)_1==3);
-- select the cubic forms
Jacobian=diff(matrix{{y_0}..{y_2}},gens Icubics);
-- compute the jacobian w.r.t. to vars of P^2
IGraphB=minors(2,Jacobian)+Icubics;
time IGraphBsat=saturate(IGraphB,ideal(x_0*y_0));
-- used 60.2963 seconds

We check that the fibers over B are disjoint from the preimages of the double points of the plane model. This shows that C is smooth:

i16 : time ISing=saturate(sub(IDelta,S)+IGraphBsat,ideal(S_0*S_2));
      degree ISing==0
o16 = true

Finally, we verify that B is reduced of expected degree 2g + 10 and hence that C is simply branched.

i17 : time IGraphBsat=saturate(IGraphB,ideal(x_0*y_0));
     gensIGraphBsat=flatten entries mingens IGraphBsat;
     IB=ideal select(gensIGraphBsat,f->(degree f)_1==0);
     degree radical IB==2*g+10
o17 = true

This code is available in form of a Macaulay2 -file from [G11] for download. It takes approximately 5 hours CPU-time on a 2.4 GHz processor to check all cases.

References

[AC81] Enrico Arbarello and Maurizio Cornalba. Footnotes to a paper of Beniamino Segre: “On the modules of polygonal curves and on a complement to the Riemann existence theorem” (Italian) [Math. Ann. 100 (1928), 537– 551; Jbuch 54, 685]. Math. Ann., 256(3):341–362, 1981. The number of g_{d}^{1}’s on a general d-gonal curve, and the unirationality of the Hurwitz spaces of 4-gonal and 5-gonal curves.

[ACGH85] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris. Geometry of algebraic curves. Vol. I, volume 267 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1985.

[ACG11] E. Arbarello, M. Cornalba, and P. A. Griffith. Geometry of algebraic curves. Vol. II, volume 268 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 2011.

[Gae51] Federigo Gaeta. Sur la distribution des degrés des formes appartenant à la matrice de l’idéal homogène attaché à un groupe de N points génériques du plan. C. R. Acad. Sci. Paris, 233:912–913, 1951.

[G11] Florian Geiß. Macaulay2 code for the proof of unirationality of 6-gonal curves of small genus, 2011. Available at the web page

http://www.math.uni-sb.de/ag/schreyer/home/6gonal.m2.

[GS] Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at the web page

http://www.math.uiuc.edu/Macaulay2/.

[HM98] Joe Harris and Ian Morrison. Moduli of curves, volume 187 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1998.

[MS86] G. Martens and F.-O. Schreyer. Line bundles and syzygies of trigonal curves. Abh. Math. Sem. Univ. Hamburg, 56:169–189, 1986.

[Sch86] Frank-Olaf Schreyer. Syzygies of canonical curves and special linear series. Math. Ann., 275(1):105–137, 1986.

[Seg28] Beniamino Segre. Sui moduli delle curve poligonali, e sopra un complemento al teorema di esistenza di Reimann. Math. Ann., 100(1):537–551, 1928.

Florian Geiß Mathematik und Informatik Universität des Saarlandes, Campus E2 4, 66123 Saarbrücken, Germany fg@math.uni-sb.de