Dyson's Lemma and a Theorem of Esnault and Viehweg

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Corollary 0.6 will follow readily from the general method used to tackle Theorem 0.4.Vojta’s result [V1] is better by a factor of two on the second term on the right; for an analysisof this situation, see [EV1] §10.The drawback with Theorem 0.4 is the dependence on the Q-divisor D. It is not clear,given an arbitrary line bundle L on X, how to compute the “best” or minimal D. In someinstances, however, such as the line bundles used by Faltings [F1] and Vojta [V4], one hascontrol over D and it would perhaps be interesting to carry out the proof of the Mordellconjecture in [V4] using Theorem 0.4.Our proof of Theorem 0.4 combines ideas of Esnault and Viehweg [EV1], Vojta [V1], andFaltings [F1]. In broad outline, the proof follows [EV1] §1 – §5 very closely. In particular,we derive Theorem 0.4 from the positivity of a certain sheaf constructed from L and thepositivity statement is shown by induction on the dimension of product subvarieties. But ourconstruction is more direct in that instead of working with weak positivity of direct imagesheaves we take “derivatives” of the section s and use a result resembling Faltings’ ProductTheorem. Most of the results of [EV1] §1 – §5 carry over to the more general setting withlittle or no modification. An exception, however, is [EV1] §2 where the proofs, relying onexplicit constructions of multihomogeneous polynomials, need to be replaced with somewhatmore abstract methods. In particular, the Q–divisor D in the statement of Theorem 0.4 isnecessary in order to derive the analogue of [EV1] Lemma 2.9 (ii).In simplifying the logic of the proof, we hope to make the more technical sections of [EV1]available to a wider audience. A fundamental insight of [EV1], exploited most fully by Vojta[V1, V2, V3], is that lower bounds on vanishing come from arithmetic conditions while theupper bounds are purely geometric. The full significance of the methods of weakly positivesheaves and their relation to diophantine problems have not been completely understood. Inparticular, one might hope to use weak positivity in order to give a more “geometric” proofof the results in [F2].The stucture of the paper is as follows. In order to prove Theorem 0.4 it suffices to showthat a certain sheaf is positive; in practice, this means producing more sections s ′ ∈ H 0 (X, L)with index at least t i at ζ i . If there were enough such sections to generate L off of the pointsζ i , Theorem 0.4 would follow immediately. In full generality, however, s might be the onlyglobal section of L with index at least t i at ζ i . In order to produce more sections, we takederivatives of s. This is straightforward in the case when X is a product of projective linesbut in general more care is necessary and we introduce the necessary definitions in section 1.In section 2 we show, following [EV1] §5, how to deduce Theorem 0.4 from the positivity ofan invertible sheaf. The positivity result is shown in two steps, occupying sections 3 and 4respectively. First one shows that after taking a “bounded” number of derivatives of s, theextra sections generate except on a finite union of proper product subvarieties. This allowsone to induct on the dimension of X and the inductive step is carried out in section 4. Atthe end of section 4 we show how to derive Vojta’s result [V1] with the techniques developedto prove Theorem 0.4.Acknowledgments.This paper forms part of my Ph.D. thesis and it is a pleasure to thank my advisor Serge4
Lang for his encouragement and guidance. I would also like to thank H. Esnault, E. Viehweg,P. Vojta, and especially R. Lazarsfeld for many stimulating conversations without which thiswork could not have been completed.Notation and Conventions• On a variety X, a Q-Cartier divisor is an element of Div(X) ⊗Q where Div(X) is thegroup of Cartier divisors on X.• A Q-Cartier divisor on X is said to be numerically effective or nef for short if D·C ≥ 0for all integral curves C ⊂ X (of course the product D·C is just the intersection producton Cartier divisors and is extended by linearity to Div(X) ⊗ Q).• If F is a coherent sheaf on X then h i (X, F) = dim H i (X, F).• When X is smooth K X denotes the canonical bundle on X.• If f : X → Y is a morphism of schemes and I is an ideal sheaf on Y , then we writef −1 I for the inverse image ideal sheaf on X (cf. [H] p. 163).1 PreliminariesAs in the introduction, fix once and for all a product of smooth projective curvesX = C 1 × . . . × C m .An important special case (the case handled by [EV1]), which we will consider in some depthas it has the advantage of being very concrete, is that of a product of m projective lines:LetP = P 1 1 × . . . × P1 m .R = k[X 1 , Y 1 , . . .,X m , Y m ]denote the projective coordinate ring of P. If I ⊂ R is a homogeneous ideal, then denoteby V (I) ⊂ P the associated subscheme and by Z(I) = V (I) red the underlying point set. Wewill often work on a fixed product affine open subset A = ∏ mi=1 A 1 i ⊂ P where each A 1 i isgiven by, say, Y i ≠ 0. Let ξ i denote the affine coordinate on A 1 i . For an m–tuple of integersd = (d 1 , . . .,d m ) writem⊗O P (d) = πi ∗ O P 1(d i).ii=1In what follows we will not distinguish between a global section P ∈ H 0 [O P (d)] and theassociated polynomial in several variables P A .5
Page 1 and 2: Dyson’s Lemma and a Theorem of Es
Page 3: Then define∫Vol(d, a, t) :=I(d,a,
Page 7 and 8: Let z j be a local parameter in O
Page 9 and 10: To show how Theorem 2.1 implies The
Page 11: covering construction of [V1] Lemma
Page 14 and 15: In order to prove Lemma 3.3 it will
Page 16 and 17: Since h −1 I(s) is an invertible
Page 18 and 19: Two ingredients are needed to prove
Page 20 and 21: Lemma 4.7 Let X = C 1 × C 2 with s
Page 22: References[B1] E. Bombieri, On the

Dyson's Lemma and a Theorem of Esnault and Viehweg

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