On morphisms onto quadrics

Ekaterina Amerik

March 21, 2006

The purpose of this note is to prove the following result:

Theorem Let X be a smooth projective variety of dimension n with cyclic Néron-Severi group, and let Q denote an n-dimensional quadric (n ≤ 3). The degree of a morphism f : X → Q is bounded in terms of the discrete invariants of X.

More precisely, let L denote an ample generator of NS(X), and let k, r be such that K_{X} ≡ kL and rL is very ample. If f^{∗}O_{Q}(1) ≡ mL, then m < 2k + (2n + 5)r when n ≥ 4, and m < 3k + 16r for n = 3.

The existence of such a result was briefly mentioned in [A], Remark 4.1.2; however, the proof (very much analogous to that of [A], Theorem 4.1.1) was never made explicit. Recently, J.-M. Hwang, to whom I am grateful for his interest in that work, suggested that I should write down this proof, which I am glad to do here.

Let f : X → Q, NS(X) ≅ ℤ, be a morphism; f is finite if non-trivial. Take l a sufficiently general line on Q, and let S ⊂ Q be a general 2-dimensional linear section of Q ⊂ ℙ^{n+1} containing l (so S is a two-dimensional quadric). Write S = Q ∩ H_{1} ∩ H_{2} ∩ … ∩ H_{n−2}, with H_{i} sufficiently general hyperplanes.

One has: N_{l,S} = O_{ℙ^{1}} and N_{S,Q}|_{l} = O_{ℙ^{1}}(1)^{⊕n−2} That is, the exact sequence of normal bundles

0 → N_{l,S} → N_{l,Q} → N_{S,Q}|_{l} → 0

splits. Let C = f^{−1}l and M = f^{−1}S. As l and S are chosen to be sufficiently general, C and M are smooth by Kleiman’s version of Bertini theorem ([H], chapter III, section 10). The exact sequence of normal bundles

0 → N_{C,M} → N_{C,X} → N_{M,X}|_{C} → 0

is just the pull-back of the corresponding sequence on Q, that is, it splits. So the induced map α : H^{0}(C, N_{C,X}) → H^{0}(C, N_{M,X}|_{C}) is surjective.

In particular, Im(α) contains the image of the restriction map β : H^{0}(M, N_{M,X} → H^{0}(C, N_{M,X}|_{C}). This is interpreted as follows: any infinitesimal deformation of M in X contains an infinitesimal deformation of C. That is, the class of C in H^{1,1}(M) is “infinitesimally fixed” according to the terminology of [CGGH].

Now there are various results of “infinitesimal Noether-Lefschetz” type, which state that if M is sufficiently ample on X, then all infinitesimally fixed classes in H^{1,1}(M) are restrictions of (1,1)-classes on X. Our class [C] obviously cannot be such a restriction: indeed, N_{C,M} = O_{C}, that is, [C]^{2} = 0; but, thanks to the hypothesis that NS(X) = ℤ, any (1,1)-class which has zero square on M and is a restriction of a (1,1)-class on X, must be numerically trivial, which is obviously not the case for [C]. So such results give the non-existence of f as soon as M is sufficiently ample, that is, deg(f) is sufficiently high.

In order to make this more precise, we shall prove the following proposition, which is a very direct generalization of Proposition 3.4 from [EL]:

Proposition Let X be a smooth projective n-fold and Y a smooth complete intersection of ample divisors D_{1}, D_{2}, …, D_{n−2} on X (so that dim(Y)=2). Suppose that for some very ample A, one has D_{i} ∈ |K_{X} + (n + 1)A + B_{i}| and D = D_{1} + D_{2} + … + D_{n−2} ∈ |nK_{X} + (n + 1)^{2}A + E| with B_{i} and E nef. Then the infinitesimal Noether-Lefschetz theorem holds for Y ⊂ X, that is, any (1,1)-class on Y which remains of type (1,1) under all infinitesimal deformations of Y in X, is a restriction of a class on X.

Proof: We follow the exposition of [EL]. Let N = ⊕_{i}O_{Y} (D_{i}) be the normal bundle of Y in X. bundle According to the theory of [CGGH], the space of infinitesimally fixed classes is the kernel of the map

α : H^{1}(Y, Ω_{Y}^{1} ) ⟶^{γ} H^{2}(Y, N^{∗}) ⟶^{β} Hom(H^{0}(Y, N), H^{2}(Y, O_{Y}),

γ being the usual coboundary map, and β coming from the multiplication map μ : H^{0}(Y, N) ⊗ H^{0}(Y, K_{Y}) → H^{0}(Y, N ⊗ K_{Y}) by Serre duality. We shall be done if we check that

1) β is injective, that is, μ is surjective;

2) the restriction map H^{1}(X, Ω_{X}^{1}) → H^{1}(Y, Ω_{X}^{1}|_{Y}) is surjective.

To check the surjectivity of μ, it is enough to check the surjectivity of each “component”

μ_{i} : H^{0}(Y, O_{Y} (D_{i})) ⊗ H^{0}(Y, K_{Y}) → H^{0}(Y, O_{Y} (D_{i}) ⊗ K_{Y}).

From the resolution

(∗) 0 → O_{X}(−D) → … → ⊕_{i<j}O_{X}(−D_{i} − D_{j}) → ⊕_{i}O_{X}(−D_{i}) → O_{X} → O_{Y} → 0

one sees that H^{0}(X, O_{X}(D_{i}) ⊗ K_{X} ⊗ O_{X}(D)) surjects onto H^{0}(Y, O(D_{i}) ⊗ K_{Y}) (indeed, the D_{i} are ample, so this is implied by the Kodaira vanishing). That is, the surjectivity of μ_{i} would, by restriction, follow from the surjectivity of the multiplication map on X:

H^{0}(X, O_{X}(D_{i})) ⊗ H^{0}(X, O_{X}(D) ⊗ K_{X}) → H^{0}(X, O_{X}(D) ⊗ K_{X} ⊗ O_{X}(D_{i})).

This, in turn, follows from the case p = 0 of [EL], Theorem 2, by taking (in the notations of [EL]) L_{d} = O_{X}(D_{i}), W = H^{0}(X, L_{d}), N_{f} = K_{X} + D. Indeed, N_{f} = K_{X} + (n + 1)A + n(K_{X} + (n + 1)A) + E, and n(K_{X} + (n + 1)A) is nef; and D_{i} is globally generated because it’s of the form K_{X} + (n + 1)A + B_{i} with A very ample.

As for the surjectivity of the restriction map, one sees from the resolution (∗) and the Kodaira vanishing that it is enough to check that H^{n−1}(X, Ω_{X}^{1}(−D)) = 0, or, dually, H^{1}(X, Ω_{X}^{n−1}(D)) = 0. Now the bundle Ω_{X}^{n−1}(nA) is globally generated, so one can apply Griffiths’ vanishing: H^{i}(X, E ⊗F) = 0 for i > 0, where E is a globally generated vector bundle and F is a line bundle such that F ⊗ (det(E) ⊗ K_{X})^{−1} is ample. As det(Ω_{X}^{n−1}(nA)) = n^{2}A + (n − 1)K_{X}, the required vanishing follows.

Let us return to our M ⊂ X, M = D_{1} ∩ D_{2} ∩ … ∩ D_{n−2} with D_{i} ∈ |f^{∗}O_{Q}(1)|, violating the infinitesimal Noether-Lefschetz condition. It is clear that if f^{∗}O_{Q}(1) is sufficiently ample, then the D_{i} shall satisfy the two conditions of our proposition, leading to a contradiction. More precisely, suppose that D_{i} is of the form 2K_{X}+(2n+ 5)A for some very ample A; an easy calculation (using the fact that K_{X} +(n+1)A is nef) shows that the conditions of the proposition are satisfied provided that n ≥ 4. (For n = 3, these hold when D = D_{1} = 3K_{X} + 16A; this is already in [EL]). So D_{i} cannot get that ample; in other words, if m is such that f^{∗}O_{Q}(1) ≡ mL, then m < 2k + (2n + 5)r for n ≥ 4 and m < 3k + 16r for n = 3.

Remark: This can be slightly improved by remarking that X is not ℙ^{n}, and if X is a quadric, then f is an isomorphism; so one may suppose that X is neither ℙ^{n} nor a quadric, in which case already K_{X} + (n − 1)A is nef.

References

[A] E. Amerik, On a problem of Noether-Lefschetz type, Compositio Math. 112 (1998), 255-271.

[CGGH] Carlson, M. Green, Ph. Griffiths, J. Harris: Infinitesimal variations of Hodge structures, Compositio Math., 50 (1983).

[EL] L. Ein, R. Lazarsfeld: Syzygies and Koszul cohomologies of smooth projective varieties, Inv. Math. 111 (1993), 53-67

[H] R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.