Some Remarks on Morphisms

between Fano Threefolds

Ekaterina Amerik

Received: June 7, 2004 Revised: October 11, 2004

Communicated by Thomas Peternell

Abstract. Let X, Y be Fano threefolds of Picard number one and such that the ample generators of Picard groups are very ample. Let X be of index one and Y be of index two. It is shown that the only morphisms from X to Y are double coverings. In fact nearly the whole paper is the analysis of the case where Y is the linear section of the Grassmannian G(1,4), since the other cases were more or less solved in another article. This remaining case is treated with the help of Debarre’s connectedness theorem for inverse images of Schubert cycles.

2000 Mathematics Subject Classification: 14J45

Keywords and Phrases: Fano threefolds, connectedness

Some twenty-five years ago, Iskovskih classified the smooth complex Fano threefolds with Picard number one. Apart from ℙ^{3} and the quadric, his list includes 5 families of Fano varieties of index two and 11 families of varieties of index one (for index one threefolds, the cube of the anticanonical divisor takes all even values from 2 to 22, except 20). Recently, the author ([A]) and C. Schuhmann ([S]) made some efforts to classify the morphisms between such Fano threefolds, the starting point being a question of Peternell: let f : X → Y be a non-trivial morphism between Fano varieties with Picard number one, is it then true that the index of X does not exceed the index of Y? In particular, Schuhmann ([S]) proved that there are no morphisms from indextwo to index-one threefolds, and that any morphism between index-two threefolds is an isomorphism (under certain additional hypotheses, some of which were handled later in [A], [IS]). As for morphisms from index-one to index-two Fano threefolds, such morphisms do exist: an index-two threefold has a double covering (branched along an anticanonical divisor) which is of index one. It is therefore natural to ask if every morphism from index-one Fano threefold X with Picard number one to index-two Fano threefold Y with Picard number one is a double covering. In [A], I proved a theorem (Theorem 3.1) indicating that the answer should be yes, however not settling the question completely. The essential problem was that the methods of [A] would never work for Y = V_{5}, the linear section of the Grassmannian G(1, 4) in the Plücker embedding (all smooth three-dimensional linear sections of G(1, 4) are isomorphic). Though there are several ways to obtain bounds for the degree of a morphism between Fano threefolds with second Betti number one ([HM], [A]), these bounds are still too rough for our purpose. This paper is an attempt to handle this problem. The main result is the following

Theorem Let X be a smooth complex Fano threefold of index one and such that P ic(X) = ℤ. Suppose moreover that X is anticanonically embedded. Let f : X → V_{5} be a non-trivial morphism. Then X is of degree 10 (“X is of type V_{10}”) and f is a double covering. In other words, X is a hyperquadric section of a cone over V_{5} in ℙ^{7}.

I believe that the extra assumption made on X is purely technical and can be ruled out if one refines the arguments below. This assumption excludes two families of Fano threefolds: sextic double solids and double coverings of the quadric branched along a hyperquartic section. Together with Theorem 3.1 of [A] and a few remarks, this theorem implies that any morphism from an index-one to an index-two threefold with cyclic Picard group is a double covering, at least under an additional assumption of the very ampleness of the generator of the Picard group of the two threefolds (see Theorem 4.1 of Section 4). The proof, somewhat unexpectedly, relies on a connectedness theorem due to Debarre, which enables one to show that the inverse images of certain special lines on V_{5} must be connected; at the same time it is well-known that, if f : X → V_{5} is as in our theorem, then the inverse image of a general line on V_{5} is a disjoint union of conics. Starting from this, we use some Hilbert scheme manipulations to show that the connected inverse images must have very special properties, and deduce the theorem. A smooth anticanonically embedded Fano threefold of index one and Picard number one is sometimes called a prime Fano threefold. We shall also call it thus throughout this paper.

Acknowledgements: It is a pleasure to thank A. Van de Ven, with whom I had some helpful discussions at the early stage of this work. I am very grateful to L. Gruson for his preliminary reading of this text and for reassuring me on certain points; and also to F. Campana for moral support.

Some Remarks on Morphisms between Fano Threefolds 473

1. Preliminaries: the geometry of V_{5}

Let us recall some more or less classical facts on the threefold V_{5} ⊂ ℙ^{6}, most of which can be found in [I] or [FN]. First of all, as any Fano threefold of index two and Picard number 1, it has a two-dimensional family of lines. A general line has trivial normal bundle (call it a (0, 0)-line), whereas there is a one-dimensional subfamily of lines with normal bundle O_{ℙ^{1}}(−1) ⊕ O_{ℙ^{1}}(1) (call them (−1, 1)-lines). The Hilbert scheme of lines on V_{5} is isomorphic to ℙ^{2}, the curve of (−1, 1)-lines is a conic in this ℙ^{2}, and there are 3 lines through a general point of V_{5}. More precisely, the (−1, 1)-lines form the tangent surface D to a rational normal sextic B on V_{5} (in particular, they never intersect), and there are three lines through any point away from D, two lines through a point on D but not on B, and one line through a point of B. The surface D is of degree 10, thus a hyperquadric section of V_{5}. We shall denote by U resp. Q the restriction to V_{5} of the universal bundle U_{G} resp. the universal quotient bundle Q_{G} on the Grassmannian G(1, 4). The cohomology groups related to those bundles are computed starting from the cohomologies of vector bundles on the Grassmannian. In particular the bundles U and Q remain stable. We shall also use the following result from [S]: let X be a prime Fano threelold, and let f : X → V_{5} be a finite morphism. Let m be such that f^{∗}O_{V_{5}}(1) = O_{X}(m). Then the inverse image of a general line consists of \frac{m^{2}deg(X)}{10} disjoint conics; in general, if one replaces V_{5} by another Fano threefold Y of index two with Picard number one, the inverse image of a general line shall consist of \frac{m^{2}deg(X)}{2deg(Y)} disjoint conics. Here by deg(Y) we mean the self-intersection number of the ample generator of P ic(Y). Our starting point is the following fact, which shall be proved in the end of this section:

Proposition 1.1 For l a (−1, 1)-line on V_{5}, any irreducible projective variety X and a surjective morphism f : X → V_{5}, f^{−1}(l) is connected.

To prove this proposition, we shall need some further details on the geometry of V_{5}. Remark that the Schubert cycles of type σ_{1,1} , which are sets of points of G(1, 4) corresponding to lines lying in a fixed hyperplane, and are also caracterized as zero-loci of sections of the bundle dual to the universal, are 4-dimensional quadrics in the Plücker G(1, 4), so each of them intersects V_{5} along a conic. Conversely, every smooth conic on V_{5} is an intersection with such a Schubert cycle. Indeed, every conics on a Grassmannian is obviously contained in some G(1, 3); and if this conic is strictly contained in G(1, 3) ∩ V_{5}, then G(1, 3) ∩ V_{5} is a surface, so the bundle U^{∗} has a section vanishing along a surface; but this contradicts the stability of U^{∗}. The same is (by the same argument) true for pairs of intersecting lines on V_{5}. Moreover, the correspondence between the Schubert cycles and the conics is one-to-one (it is induced by the restriction map on the global sections H^{0}(G(1, 4), U_{G}^{∗}) → H^{0}(V_{5}, U^{∗}) which is an isomorphism). Let us show that among these conics, there is a one-dimensional family of double lines.

Proposition 1.2 Fix an embedding V_{5} ⊂ G(1, 4) ⊂ ℙ^{9}. There is a onedimensional family of Schubert cycles Σ_{t} such that for each t, the intersection of V_{5} and Σ_{t} is a (double) line. Moreover, lines on V_{5} which are obtained as a set-theoretic intersection with a Schubert cycle of type σ_{1,1}, are exactly (−1, 1)-lines.

Proof: The three-dimensional linear sections of G(1, 4) in the Plücker embedding are parametrized by the Grassmann variety G(6, 9); let, for P ∈ G(6, 9), V_{P} denote the intersection of G(1, 4) with the corresponding linear subspace (which we will denote also by P). The Schubert cycles are parametrized by G(3, 4) = ℙ^{4}; likewise, denote by Σ_{t} the Schubert cycle corresponding to t ∈ ℙ^{4}. Consider the following incidence subvariety I ⊂ G(6, 9) × ℙ^{4}:

I = {(P, t) ∈ G(6, 9) × ℙ^{4}|V_{P} ∩ Σ_{t} is a line}.

The fiber I_{t} of I over any t ∈ ℙ^{4} parametrizes the six-dimensional subspaces P of ℙ^{9} intersecting Σ_{t} along a line. Σ_{t} is a quadric in ℙ^{5} ⊂ ℙ^{9}, and P intersects it along a line l if and only if the plane H = P ∩ ℙ^{5} is tangent to Σ_{t} along l, i.e. lies in every T_{x}Σ_{t}, x ∈ l. The intersection of all tangent spaces to Σ_{t} ⊂ ℙ^{5} along l is a three-dimensional projective space (the tangent spaces form a pencil of hyperplanes in ℙ^{5}, because Σ_{t} is a quadric). This means that for every l, the planes tangent to Σ_{t} along l form a one-dimensional family. The family of lines on a 4-dimensional quadric (= G(1, 3)) is a 5-dimensional flag variety, so the planes in ℙ^{5} tangent to Σ_{t} along a line are parametrized by a six-dimensional irreducible variety (a ℙ^{1}-bundle over a flag variety). This implies that I_{t} is irreducible of codimension 3 in G(6, 9), so I is irreducible of codimension 3 in G(6, 9) × ℙ^{4}. We must show that the first projection p_{1} : I → G(6, 9) is surjective and its general fiber is of dimension one. First of all, remark that there are points P in the image of p_{1} such that the corresponding V_{P} is smooth (so, is a V_{5}). Indeed, fix, as above, Σ_{t}, l ⊂ Σ_{t}, H a plane in ℙ^{5} =< Σ_{t} > such that H ∩ Σ_{t} = l; the remark will follow if we show that for a general ℙ^{6} = P ⊂ ℙ^{9} containing H, G(1, 4) ∩ P is smooth. We have H ∩ G(1, 4) = H ∩ Σ_{t} = l (because G(1, 4)∩ < Σ_{t} >= Σ_{t}), so the smoothness away from l is obvious, and one checks, again by standart dimension count, that for x ∈ l, the set A_{x} = {P |H ⊂ P, G(1, 4) ∩ P is singular at x} is of codimension two in the space of all P ’s containing H. Therefore for P general in the image of p_{1}, V_{P} is smooth. It is clear that if a smooth V_{P} = G(1, 4) ∩ P is such that V_{P} ∩ Σ_{t} = l, then the corresponding plane H is tangent along l not only to Σ_{t}, but also to V_{P}.

Thus the normal bundle N_{l,V_{P}} has a subbundle N_{l,H} of degree 1, and so l is of type (−1, 1) on V_{P}. Since we have only one-dimensional family of (−1, 1)- lines on a smooth V_{P}, we deduce that a fiber of p_{1} over a point P such that V_{P} is smooth, is at most one-dimensional. The irreducibility of I now implies that p_{1} is surjective and its general fiber is of dimension one. This proves the Proposition.

Proof of Proposition 1.1:

Let us recall the following result of Debarre ([D], partial case of Théorème 8.1, Exemple 8.2 (3)): Let X be an irreducible projective variety, and let f : X → G(d, n) be a morphism. Let Σ be a Schubert cycle of type σ_{m}. If in the cohomologies of G(d, n), [f(X)] · σ_{m+1} ≠ 0, then f^{−1}(Σ) is connected. Let X be an irreducible projective variety and f : X → V_{5} be a surjective morphism. Composing with the embedding i : V_{5} → G(1, 4), we can view f as a morphism to G(1, 4). By Proposition 1.2, each (−1, 1)-line is the intersection of f(X) with a Schubert cycle of type σ_{1,1} on our grassmannian G(1, 4) = G(1, ℙ(U)), where U is a five-dimensional vector space. By duality, we can view this Grassmannian as G(2, ℙ(U^{∗})), and this point of view transforms the Schubert cycles of type σ_{1,1} into Schubert cycles of type σ_{2}. The condition [f(X)] · σ_{3} ≠ 0 is obviously satisfied because f(X) = V_{5} is cut out by three hyperplanes. Thus Proposition 1.1 follows from Proposition 1.2 and Debarre’s theorem.

Remark 1.3 If we knew that the inverse image of a general line is always connected, this would immediately solve our problem; indeed, for a Fano threefold X of index and Picard number one, the equality \frac{m^{2}deg(X)}{10} = 1 implies that m = 1, deg(X) = 10 and f is a double covering. However, as shows an example of Peternell and Sommese, this is false in general, even if one supposes that X is a Fano threefold. In the example of [PS], X is the universal family of lines on V_{5}, which turns out to be a Fano threefold (of Picard number two, of course), and f is the natural triple covering. The inverse image of a general line has two connected components.

Remark 1.4 One can ask if there is a similar connectedness statement for other Fano threefolds of Picard number one and index two. Recall that these are the following: intersection of two quadrics in ℙ^{5}; cubic in ℙ^{4}; double covering of ℙ^{3} branched in a quartic; double covering of the cone over Veronese surface branched in a hypercubic section. Smooth quadrics in ℙ^{5} are Grassmannians G(1, 3), and a smooth intersection of two quadrics in ℙ^{5} is a quadric line complex. It is classically known (see [GH], Chapter 6) that on a quadric line complex, there is a finite (and non-zero) number of lines obtained as set-theoretic intersection with a plane in G(1, 3). These lines are obviously (−1, 1)-lines, since the corresponding plane is tangent to the quadric line complex along this line. Our intersection of two quadrics is contained in a pencil of such Grassmannians, so there is a one-dimensional family of lines on it such that each line is the intersection with a plane lying on some Grassmannian of the pencil. The curve of (−1, 1)-lines is irreducible (it follows from the results in [GH], Chapter 6, that it is smooth and that it is an ample divisor on the Fano surface of lines, in particular, it is connected). Thus it is just the closure of that family. So that it follows again from Debarre’s paper that the inverse image of a general (−1, 1)-line is connected. As for the cubic, even if such a connectedness statement could hold, it would not, as far as I see, follow from any well-known general result. One can, though, remark that in the examples of Peternell-Sommese type “(universal family of lines on Y)→ Y ”, the inverse image of a (−1, 1)-line has a tendency to be connected, whereas the inverse image of a (0, 0)-line is certainly not connected. Indeed, it is observed in the literature that, on the threefolds as above (the cubic, the quadric line complex, V_{5}), a line l is in the closure of the curve C_{l} ={lines intersecting l but different from l} on the Hilbert scheme if and only if l is a (−1, 1)-line.

2. A Hilbert scheme argument

The previous considerations show that on our Fano threefold X, a disjoint union of conics degenerates flatly to a connected l.c.i. scheme. Recall the following classical example: if one degenerates a disjoint union of two lines in the projective space into a pair of intersecting lines, the pair of intersecting lines shall have an embedded point at the intersection. So if one wants the limit to be a connected l.c.i., this limit must be a double line. This suggests to ask if a similar phenomenon can occur in our situation, that is: can it be true that a connected l.c.i. limit of disjoint conics is necessarily a multiple conic? In any case it is easily checked that, say, a connected limit of pairs of disjoint conics does not have to have embedded points when the two conics become reducible and acquire a common component. So this is very probably false, and in any case there is no simple argument. In this paragraph we shall prove, though, that the inverse image of a sufficiently general (−1, 1)-line is either a multiple conic, or supported on a union of lines, and in fact even slightly more (Proposition 2.5). Let T be the Hilbert scheme of lines on V_{5} and let M ⊂ T ×V_{5} be the universal family. We have the “universal family of the inverse images of lines under f”

S = M ×_{V_{5}} X ⊂ T × X.

Since f is flat and M is flat over T, S is flat over T. Let H′ be the Hilbert scheme of conics on X. Consider the irreducible components of H′ which are relevant for our problem, that is, the components such that their sufficiently general points correspond to conics which are in the inverse image of a sufficiently general line on V_{5}. Denote by H the union of all such components (each of them is, of course, two-dimensional).

For every point x ∈ H, the image of the corresponding conic C_{x} is a line. Indeed, “f(C) is a line” is a closed condition on conics C because f is a finite morphism (for f arbitrary, “f(C) is contained in a line” would be a closed condition on C). This allows to construct a morphism p : H → T taking every conic to its image under f. Indeed,

L = {(C, f(x))|x ∈ C, C ∈ H} ⊂ H × V_{5}

is a family of lines over H; though apriori it is not clear that it is flat, this is a “well-defined family of algebraic cycles” in the sense of Kollar ([K], Chapter I) and so corresponds to a morphism from H to the Chow variety of lines on V_{5}, and this is the same as T. We claim that p is finite. Indeed, it is clear that the only obstruction to the finiteness of p could be the existence of infinitely many double structures of arithmetic genus zero on some lines on X (“non-finiteness of the Hilbert-Chow morphism for the family of conics on X”). This obviously happens if one considers conics in ℙ^{3} rather than conics on X. In our situation, however, this is impossible, and the Hilbert-Chow morphism is even one-to-one. Indeed, by [I], the normal bundle of a line in a prime Fano threefold is either O_{ℙ^{1}}⊕O_{ℙ^{1}}(−1), or O_{ℙ^{1}}(1) ⊕ O_{ℙ^{1}}(−2), and there is the following

Lemma 2.1 Let l ⊂ X be a line on a prime Fano threefold. If N_{l,X} = O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(−1), then there is no locally Cohen-Macaulay double structure of arithmetic genus 0 on l. If N_{l,X} = O_{ℙ^{1}}(1) ⊕ O_{ℙ^{1}}(−2), then such a structure is unique.

Proof: All locally Cohen-Macaulay double structures on smooth curves in a threefold are obtained by a construction due to Ferrand (see for example [BF], or else [N] for details): if Y ⊂ V is a smooth curve on a smooth threefold, and Ỹ is a double structure on Y, write L for I_{Y} /IỸ; in fact L is a locally free rang-one O_{Y} -module and IỸ contains I_{Y}^{2} . The double structure is thus determined by the natural surjection from the conormal bundle of Y in V to L, up to a scalar. Now take Y = l, V = X and let L be as above; we have an exact sequence

0 → L → Ol̃ → O_{l} → 0,

from which it is clear that p_{a}(l̃) = 0 if and only if L = O_{ℙ^{1}}(−1). Now in the first part of our assertion, there is no non-trivial surjection from N_{l,X}^{∗} to O_{ℙ^{1}}(−1), and in the second part, such a surjection is unique up to a scalar.

Note that we do not have to consider curves which are not locally Cohen-Macaulay, since, for example, the above argument shows that there are no higher genus locally Cohen-Macaulay double structures, and an embedded point decreases the genus.

Thus, for any t ∈ T, p^{−1}(t) is a finite set {h_{1}, …, h_{k}}, and to each h_{i} there corresponds one conic C_{i} on X, mapped to l_{t} by f. The next step is to show that f and p “agree with each other”:

Lemma 2.2 Let t ∈ T be any point and l_{t} ∈ V_{5} be the corresponding line. Let h_{1}, …, h_{k} be the points of p^{−1}(t) and C_{1}, …, C_{k} the corresponding conics on X. Then the support of f^{−1}(l_{t}) is ⋃_{i} C_{i}.

Proof: Indeed, for a general t ∈ T, it is true: f^{−1}(l_{t}) = ⋃_{i} C_{i}. For a special t ∈ T, choose a curve V ⊂ T through t, such that t is the only “non-general” point of V in the above sense, and let U = p^{−1}(V). Denote by C_{U} ⊂ U × X the restriction to U of the universal family of conics over H. The support of the fiber over t of (p × id)(C_{U}) ⊂ V × X is equal to ⋃_{i} C_{i}. But the family S|_{V} coincides with (p × id)(C_{U}) except at t. S|_{V} being flat, it must be the scheme-theoretic closure of (p × id)(C_{U})|_{V −{t}} in V × X, and thus the support of S|_{V} is (p × id)(C_{U}), q.e.d.

Let now t ∈ T be a point corresponding to a sufficiently general (−1, 1)-line. We know that f^{−1}(l_{t}) is connected. Suppose that the number k from the Lemma is > 1, so that there are several conics in the Supp(f^{−1}(l_{t})). Decompose the set of those conics into two disjoint non-empty subsets Σ_{1} and Σ_{2}.

Proposition 2.3 There exists a conic in Σ_{1} which has a common component with a conic in Σ_{2}; in other words, (⋃_{C∈Σ_{1}} C) ⋂(⋃_{C∈Σ_{2}} C) cannot be zerodimensional.

Proof Choose a suitable small 1-dimensional disc (V, 0) centered at t. The inverse image p^{−1}V is a disjoint union of two analytic sets U_{1} and U_{2} (U_{i} consists of points corresponding to conics near those of Σ_{i}). Repeat the procedure of the previous lemma: consider the universal families C_{i} of conics over U_{i} and their images S_{i} = (p × id)(C_{i}) ⊂ V × X. Let S^{0}, S_{i}^{0} denote the restriction of our families S, S_{i} to the punctured disc V^{0} = V − {0}. The family S^{0} is just the disjoint union of S_{i}^{0}. Now take the closure of all those (as analytic spaces) in V × X: the closure of S^{0} is just S|_{V}, by flatness, and the closure S′_{i} of S_{i}^{0} has the same support as S_{i}, is contained in S|_{V} and is flat over V. The fiber of S′_{i} over 0, denoted S_{i}, is contained in the fiber S of S, since the tensor multiplication preserves the surjectivity. So f^{−1}(l_{t}) = S contains S_{1} ∪ S_{2}. By construction, S_{i} are flat limits of disjoint unions of a_{i} conics and S is a flat limit of disjoint unions of a_{1} + a_{2} (= \frac{m^{2}deg(X)}{10}) conics. If S_{1} and S_{2} do not have common components, then, since by flatness deg(S) = deg(S_{1})+deg(S_{2}), this implies S = S_{1}∪S_{2}, because S is purely one-dimensional (being an inverse image of a line under a finite morphism). But then we can apply the exact sequence

0 → O_{S} → O_{S_{1}} ⊕ O_{S_{2}} → O_{S_{1}∩S_{2}} → 0 and get a contradiction, since by flatness χ(O_{S}) = χ(O_{S_{1}}) + χ(O_{S_{2}}), S_{1} ∩ S_{2} is non-empty and it is zero-dimensional by assumption. Thus S_{1} and S_{2} must have common components, and, as S_{i} is supported on ⋃_{C∈Σ_{i}} C, the Proposition is proved.

Corollary 2.4 In the situation as above, f^{−1}(l_{t}) is supported either on a single conic, or on a union of lines.

Indeed, the proposition shows that if f^{−1}(l_{t}) contains more than one conic, then any conic from f^{−1}(l_{t}) must have a common component with the rest of these conics, that is, it must be singular.

Some results from commutative algebra allow to prove a stronger (“local”) version of Proposition 2.3:

Proposition 2.5 In the situation of Proposition 2.3, through each intersection point P of ⋃_{C∈Σ_{1}} C and ⋃_{C∈Σ_{2}} C passes some common component of ⋃_{C∈Σ_{1}} C and ⋃_{C∈Σ_{2}} C.

Proof: The family S is flat over T which is smooth, and the fibers are l.c.i., thus locally Cohen-Macaulay. It follows ([EGA], 6.3.1, 6.3.5) that S is locally Cohen-Macaulay, and that the same it true for the restriction of S to any smooth curve in T. Suppose that Proposition 2.5 is not true for some intersection point P. Let x = (t, P) ∈ T × X be the point corresponding to P in S. Consider the restriction of S to a general curve through t, and an analytic neighbourhood of x in this restriction. Clearly, if one removes x, this neighbourhood becomes disconnected: there are at least two branches corresponding to SuppS_{i} as in Proposition 2.3. But this is impossible by Hartshorne’s connectedness ([H]), which implies that a connected Cohen-Macaulay neighbourhood remains connected if one removes a subvariety of codimension at least two.

Remark 2.6 The argument of the Proposition is more or less the following: “if we have a disjoint union of certain smooth curves A and B, which degenerates flatly into a certain connected C in such a way that A and B do not acquire common components in the limit, then C will have embedded points at the intersection points of the limits of A and B, so this is impossible if we know that C is purely one-dimensional”. Examples show that one cannot say anything reasonable if one allows A and B to acquire common components. But in fact our “C”, that is, f^{−1}(l_{t}), is more than just purely one-dimensional: it is a locally complete intersection. I do not know if its being a flat limit of disjoint unions of conics can impose stronger restrictions on its geometry.

To illustrate how we shall apply this, let us handle the case when f^{−1}(l_{t}) is supported on a single conic.

Proposition 2.7 In this case X = V_{10} and f is a double covering.

Proof: As the degree of the subscheme f^{−1}(l_{t}) of X is \frac{m^{2}deg(X)}{5}, this conic is of multiplicity \frac{m^{2}deg(X)}{10} in f^{−1}(l_{t}). That is, the local degree of f near a general point of such a conic is also \frac{m^{2}deg(X)}{10}. Now this is the local degree of f along a certain divisor, because we have chosen the line l_{t} to be “sufficiently general among the (−1, 1) lines”: it varies in a one-dimensional family. This divisor is thus a component of the ramification divisor of f, and \frac{m^{2}deg(X)}{10} − 1 is its ramification multiplicity. Now the ramification divisor of f is an element of |O_{X}(2m − 1)|, and so the local degree of f at its general point is at most 2m, and if it is 2m, then the ramification divisor is the inverse image of the surface covered by the (−1, 1)- lines and set-theoretically a hyperplane section of X. So we have:

\frac{m^{2}deg(X)}{10} ≤ 2m, mdeg(X) ≤ 20,

and if the equality holds, then f is unramified outside the inverse image of the surface of (−1, 1)-lines. Also, \frac{m^{2}deg(X)}{10} must be an integer. The inequality thus only holds for deg(X) = 10 and m = 1 (this is a double covering) or m = 2 (in this case it is an equality), and for deg(X) = 4 and m = 5 (also an equality). Let us exclude the last two cases. If f is unramified outside the inverse image of the surface of (−1, 1)-lines, then p is \frac{m^{2}deg(X)}{10}-to-one everywhere except over the conic parametrizing the (−1, 1)-lines on T = ℙ^{2}. It is thus a topological covering of the complement to this conic in T. But the latter is simply-connected; so that H has \frac{m^{2}deg(X)}{10} irreducible components and each one maps one-to-one on T. Notice that the number \frac{m^{2}deg(X)}{10} is superiour to three in both cases. But this is impossible. Indeed, on V_{5} one has only 3 lines through a general point; whereas, if H has k components, each component would give at least one conic through a general point of X. Those conics are mapped to different lines through f(x), because they intersect; thus k ≤ 3.

3. Proof of the Theorem

We have seen that the inverse image of a general (−1, 1)-line is supported either on one conic, or on a union of lines, and settled the first case in the end of the second section. Let us now settle the remaining case, using Proposition 2.5. The following lemma is standart (and follows e.g. from the arguments of [M], Chapter 3):

Lemma 3.1 Let g : X_{1} → X_{2} be a proper morphism of complex quasiprojective varieties, which is finite of degree d. Suppose that X_{2} is smooth. Then the inverse image of any point x ∈ X_{2} consists of d points at most, and if there are exactly d points in the inverse image of all x ∈ X_{2}, then topologically g is a covering.

Let H be as in the last section, and let C be the universal family of conics over H. Each conic of H is contained in the inverse image of some line on V_{5}, and set-theoretically such an inverse image is a union of conics of H. Denote by D the surface covered by (−1, 1)-lines on V_{5}. Recall that through each point of the complement to D in V_{5} there are three lines, that D is a tangent surface to a rational normal sextic and that there are two lines (one (−1, 1)-line and one (0, 0)-line) through any point of D away from this sextic and a single line through each point of the sextic. Since the inverse image of a general (0, 0)- line is a disjoint union of conics of H, there are three conics of H through a general point of X, and at least three through any point away from f^{−1}(D). The natural morphism q : C → X is proper and finite of degree three. Lemma 3.1 has thus an obvious corollary:

Corollary 3.2 There are at most three conics of H through any point of X, and exactly three conics of H through any point of X away from f^{−1}(D).

Let l be a general (−1, 1)-line on V_{5}. Consider the case when Z = f^{−1}(l) is a set-theoretic union of degenerate conics C_{1}, ..., C_{k} of H.

Lemma 3.3 Z contains a line which belongs to a single C_{i} (say C_{1}).

Proof: Suppose the contrary, that is, that any component of Z is contained in at least two conics of H. Through a general point x of this component there is at least one more conic of H, coming from the inverse image of the (0, 0)-line through f(x). This implies that the morphism q : C → X is three-to-one outside an algebraic subset A of codimension at least two in X. That is, C−q^{−1}(A) is, topologically, a covering of X −A. But X −A is simply-connected because X is Fano and thus simply-connected. This means that C is reducible, consists of three components and each of them maps one-to-one to X. Since X is smooth, it must be isomorphic to each of those components (by Zariski’s Main Theorem). But this is impossible because the components are fibered in conics and X has cyclic Picard group.

Before continuing our argument, let us recall some well-known facts on lines on prime Fano threefolds ([I]). Lines on our Fano threefold X are parametrized by a curve, which may of course be reducible or non-reduced. Its being reduced or not influences the geometry of the surface covered by lines on X. Namely, if a component of the Hilbert scheme of lines on X is reduced, then the natural morphism from the corresponding component of the universal family to X is an immersion along a general line; and there is a classical computation ([I], [T]) which says that if its image M is an element of |O_{X}(d)|, then a general line of M intersects d + 1 other lines of M. If a component of the Hilbert scheme of lines is non-reduced, then the surface M covered by the corresponding lines is either a cone (but this can happen only on a quartic), or a tangent surface to a curve. One knows only one explicit example of a Fano threefold as above such that the surface covered by lines on it is a tangent surface to a curve, it is constructed by Mukai and Umemura (having been overlooked by Iskovskih) and has degree 22. The surface itself is a hyperplane section of this threefold and its lines never intersect. The following Proposition, due to Iliev and Schuhmann, is the main result of [IS] slightly reformulated:

Proposition 3.4 Let X be a prime Fano threefold, L a complete onedimensional family of lines on X and M the surface on X covered by lines of L. If X is different from the Mukai-Umemura threefold, then a general line of L intersects at least one other line of L.

An outline of the proof: If not, then, by what we have said above, the surface M must be a tangent surface to a curve. Studying its singularities, Iliev and Schuhmann prove that it must be a hyperplane section of X. Then they show, by case-by-case analysis (of which certain cases appear already in [A]), that the only prime Fano threefold containing a tangent surface to a curve as a hyperplane section, is the Mukai-Umemura threefold.

“Lines contained in a single C_{i}” cover a divisor on X as Z varies (this is the branch divisor of q). Since (−1, 1)-lines on V_{5} never intersect, Proposition 3.4 implies that if X is not the Mukai-Umemura threefold, then in Z there are at least two lines contained in a single conic (say, l_{1} ⊂ C_{1} and l_{2} ⊂ C_{2}), and that they intersect, say at the point P. Notice that C_{1} is necessarily different from C_{2}: otherwise we get a contradiction with Proposition 2.5 by considering Σ_{1} = {l_{1} ∪ l_{2}}, Σ_{2} the set of all the other C_{i} and the intersection point P.

Claim 3.5 Both C_{1} and C_{2} are pairs of lines intersecting at the point P, and Z is supported on C_{1} ∪ C_{2}. Thus Z is, set-theoretically, the union of three or four lines through P.

Proof:

1) If C_{1} is a double line, we get a contradiction with Proposition 2.5 by considering Σ_{1} = {C_{1}} and the point P; the same is true for C_{2}.

2) Let C_{1} = l_{1} ∪ l′_{1}. If l′_{1} does not pass through P, we get the contradiction in the same way, thus P ∈ l′_{1}. Also, P ∈ l′_{2}, where C_{2} = l_{2} ∪ l′_{2}. 3)There are two possibilities: a)If l′_{1} ≠ l′_{2}, then there must be another conic from Z through P, containing l′_{1}. Indeed, otherwise we again get a contradiction with Proposition 2.5. In the same way, there is a conic from Z through P which contains l′_{2}. In fact it is the same conic, because otherwise there are at least four conics through P, contradicting Corollary 3.2. Denote it by C_{3}. No other conic from Z passes through P. So C_{3} = l′_{1} ∪ l′_{2}, and l′_{1}, l′_{2} are not contained in conics others than C_{1}, C_{2}, C_{3}.

b) If l′_{1} = l′_{2}, then no other conic from Z contains this line (otherwise through its general point there will pass at least four conics from H, the fourth one coming from the inverse image of the correspondent (0, 0)-line).

4) Now the union C_{1} ∪ C_{2} ∪ C_{3} in the case a), resp. the union C_{1} ∪ C_{2} in the case b), cannot have any points in common with the other components of Z; otherwise, taking Σ_{1} = {C_{1}, C_{2}, C_{3}}, resp. Σ_{1} = {C_{1}, C_{2}}, we obtain a contradiction with Proposition 2.5. But Z is connected, so Z is supported on the lines l_{1}, l′_{1}, l_{2}, l′_{2}, q.e.d.

We are now ready to finish the proof of the theorem stated in the introduction.

Proof of the theorem: If X is the Mukai-Umemura threefold, then the lines on X never intersect at all, so that f^{−1}(l) must be supported on a single conic. Proposition 2.7 shows that a morphism from X to V_{5} is impossible. (It should be, however, said at this point that the paper [HM] contains a better proof of the non-existence of morphisms from the Mukai-Umemura threefold onto any other smooth variety, besides ℙ^{3}!). If X is not the Mukai-Umemura threefold and f^{−1}(l) is not supported on a single conic, then we know by Claim 3.5 how f^{−1}(l) looks. Remark that f^{−1}(D) is a reducible divisor: it has two components, one swept out by the lines l_{1} and l_{2} as Z varies, another by l′_{1} and l′_{2} (the components are really different because, by construction, l_{1} and l_{2} are lines contained in a single conic of H, whereas l′_{1} and l′_{2} are not). Neither component is a hyperplane section: indeed, if a hyperplane section of X is covered by lines, then it is either a cone (impossible in our situation), or a general line intersects two other lines on the surface by the classical computation from [T] mentioned above, since a hyperplane section cannot be a tangent surface to a curve by [IS]. Let k be the multiplicity of the component corresponding to l_{i} and k′ be the multiplicity of the component corresponding to l′_{i}. As f^{∗}(D) is a divisor from |O_{X}(2m)|, k + k′ ≤ m. At the same time, Z must be of degree \frac{m^{2}deg(X)}{5}, and thus 2k + 2k′ = \frac{m^{2}deg(X)}{5}, so m^{2}deg(X) ≤ 10, leavng the only possibility m = 1, deg(X) = 10.

4. Concluding remarks

In this section, we shall make a further (minor) precision on Theorem 3.1 from [A]. In that theorem, it was proved that if X, Y are Fano threefolds with Picard number one and very ample generator of the Picard group, X is of index one, Y is of index two different from V_{5} (that is, Y is a cubic or a quadric line complex), and f : X → Y is a surjective morphism, then f is a “projection”, that is, f^{∗}O_{Y} (1) = O_{X}(1). The argument of the theorem also worked for Y a quartic double solid, whereas there were some problems (hopefully technical ones) for Y a double Veronese cone and for X not anticanonically embedded. Even in the “good” cases, the theorem proves a little bit less than one would like; that is, we want f to be a double covering and we prove only that f^{∗}O_{Y} (1) = O_{X}(1). This still leaves the following additional possibilities:

(1) If Y is a cubic, X can be V_{12}, deg(f) = 4 (X cannot be V_{18} because of the Betti numbers: b_{3}(V_{18}) < b_{3}(Y));

(2) If Y is an intersection of two quadrics, X can be V_{16}, deg(f) = 4 (here V_{12} is impossible since in this case the inverse image of a general line would consist of 3/2 conics). The first possibility can be excluded by using an inequality of [ARV]: it says that for a finite morphism f : X → Y and a line bundle L on Y such that Ω_{Y} (L) is globally generated, deg(f)c_{top}Ω_{Y} (L) ≤ c_{top}Ω_{X}(f^{∗}L), so, for X and Y of dimension three, deg(f)(c_{3}(Ω_{Y}) + c_{2}(Ω_{Y})L + c_{1}(Ω_{Y})L^{2}) must not exceed c_{3}(Ω_{X}) + c_{2}(Ω_{X})f^{∗}L + c_{1}(Ω_{X})f^{∗}L^{2}. Consider the situation of (1): we may take L = O_{Y} (2), and we know that c_{3}(Ω_{Y}) = 6 and c_{3}(Ω_{X}) = 10. Using the equalities c_{2}(X)c_{1}(X) = c_{2}(Y)c_{1}(Y) = 24, we arrive at 4(6 + 24 − 24) ≤ 10 + 48 − 48, which is false. So the case (1) cannot occur. This inequality does not work in the case (2): indeed, now c_{3}(Ω_{Y}) = 0, c_{3}(Ω_{Y}) = 2 and the inequality reads as follows: 4(0 + 24 − 32) ≤ 2 + 48 − 64, so does not give a contradiction. However we can rule out this case by our connectedness argument. Indeed, the inverse image of a general (−1, 1)-line is connected (Remark 1.4) and the inverse image of a general (0, 0)-line consists of two disjoint conics. The results of Section 2 apply, of course, to our situation; it follows that the inverse image of a general (−1, 1)-line is either a double conic, or a union of two reducible conics which have a common component. In both cases, it is clear that the ramification locus of f projects onto the surface covered by (−1, 1)-lines. But the ramification divisor is a hyperplane section of V_{16}, and thus can project onto a surface from |O_{Y} (4)| at most. Whereas it is well-known (and follows for example from the results in [GH], Chapter 6) that the surface covered by (−1, 1)-lines on Y is an element of |O_{Y} (8)|. All this put together gives the following

Theorem 4.1 Let X, Y be smooth complex Fano threefolds of Picard number one, X of index one, Y of index two. Assume further that the ample generators of P ic(X) and P ic(Y) are very ample. Then any morphism from X to Y is a double covering.

I would like to mention that the verification of this statement without the very ampleness hypothesis amounts to a very small number of particular cases; for instance, if Y is a double Veronese cone, then already the formula of [ARV] combined with the knowledge of Betti numbers implies that for any morphism f : X → Y with X Fano of index one with cyclic Picard group, deg(f) = 2 and X is a sextic double solid. It seems that one could be able to work out the remaining cases without any essentially new ideas.

References

[A] E. Amerik, Maps onto certain Fano threefolds, Doc. Math. 2 (1997), 195–211.

[ARV] E. Amerik, M. Rovinsky, A. Van de Ven, A boundedness theorem for morphisms between threefolds, Ann. Inst. Fourier 49 (1999), 405-415.

[BF] C. Bănică, O. Forster, Multiplicity structures on space curves, The Lefschetz centennial conference, Part I (Mexico City, 1984), Contemp. Math., 58 (1986), 47-64.

[D] O. Debarre, Théorèmes de connexité pour les produits d’espaces projectifs et les grassmanniennes, Amer. J. Math. 118 (1996), no. 6, 1347– 1367.

[EGA] A. Grothendieck, Elements de la géométrie algébrique, IV, seconde partie, Publ. Math. IHES 24 (1965).

[FN] M. Furushima, N. Nakayama: The family of lines on the Fano threefold V_{5}. Nagoya Math. J. 116 (1989), 111–122.

[GH] Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley, 1978.

[H] R. Hartshorne, Complete intersections and connectedness, Amer. J. Math. 84 (1962), 497-508.

[HM] J.M. Hwang, N. Mok, Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles, J. Algebraic Geom. 12 (2003), no. 4, 627–651.

[I] V.A. Iskovskih, Fano threefolds I, II, Math. USSR Izv. 11 (1977), 485– 527, and 12 (1978), 469-506.

[IS] A. Iliev, C. Schuhmann: Tangent scrolls in prime Fano threefolds, Kodai Math. J. 23 (2000), no.3, 411-431.

[K] J. Kollar, Rational curves on algebraic varieties, Springer-Verlag, 1996.

[M] D. Mumford, Algebraic geometry I. Complex projective varieties. Springer-Verlag, 1976.

[N] S. Nollet, The Hilbert scheme of degree three curves, Ann. Sci. ENS 30 (1997), 367-384.

[PS] Th. Peternell, A.J. Sommese, Ample vector bundles and branched coverings II, preprint math.AG/0211220.

[S] C. Schuhmann, Morphisms between Fano threefolds, J. Algebraic Geom. 8 (1999), no. 2, 221–244.

[T] A.N. Tyurin, Five lectures on three-dimensional varieties, Uspekhi Mat. Nauk 27 (1972), no. 5 (167), 3–50; translation in Russian Math. Surveys 27 (1972), no. 5, 1–53.

Ekaterina Amerik Université Paris-Sud Laboratoire des Mathématiques Bâtiment 425 91405 Orsay, France Ekaterina.Amerik@math.u-psud.fr