Dyson's Lemma and a Theorem of Esnault and Viehweg

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Dyson’s Lemma can be viewed as a statement about certain derivatives of sections of linebundles. In the case of P it is clear how to take derivatives of the polynomial P(ξ 1 , . . .,ξ m ).To fix notation, let D i = ∂/∂ξ i andD α =m∏i=1D α ii , for an m–tuple α = (α 1 , . . .,α m ).It will be helpful in some instances not only to take derivatives locally on A but also globally.If s ∈ H 0 [P, O P (d)] and Z ⊂ Z(s) is some integral subvariety contained in the zero schemeof s, then one can define D i (s) as a meromorphic section of O P (d) ⊗ πi ∗K P 1 |Z: locally, oniA say, D i (s) is given by taking the derivative in the naive sense of polynomials. In practice,however, it will be necessary to have an everywhere regular section of O P (d) ⊗πi ∗K P 1 |Z andiso a slight modification will be necessary; we will describe this modification at the end ofthe section.The notion of partial derivatives can now be extended to products of curves of arbitrarygenus (cf. [F1] p. 560 and p. 566 for a more general treatment). Fix projective embeddingsC i ֒→ P n iand generic projections p i : C i → P 1 . Then one can pull back the derivation onP 1 via p i , with poles along the ramification divisor R of p i and so obtain:Definition 1.1 Let s ∈ H 0 (X, L) and suppose Z ⊂ Z(s). Then D i (s)|Z is the meromorphicsection of L[π ∗ i (p ∗ iK P 1 +R)]|Z ≃ L(π ∗ i K Ci )|Z obtained locally by pulling back the derivationon O P 1 via the composition p i · π i . Higher order derivatives are defined analogously so forα = (α 1 , . . .,α m )( m)∣D α ∑ ∣∣∣∣(s) is a meromorphic section of L α i πi ∗ K C iZi=iprovided all partial derivatives D β (s) with β < α vanish identically along Z.Having defined derivatives we can reformulate Definition 0.1. For s ∈ H 0 (X, L) and aclosed point ζ ∈ X we have{ m∣ }∑ ∣∣∣∣ m∏ind a (ζ, s) = min a i α i D α (s)|ζ ≠ 0 where D α = D α ii .i=1i=1When X = P and L = O P (d), ind a (ζ, s) ≥ t if and only if s has a zero of type (a, t) at ζ inthe language of [EV1] Definition 0.2.Fix a finite set of points S = {ζ i } M i=1 ⊂ X and an m–tuple a = (a 1 , . . .,a m ) as inDefinition 0.1. We want to define an ideal sheaf I(s) on X so that s ∈ H 0 [X, L ⊗ I(s)] andso that I(s) characterizes the index of s at the points ζ i ∈ S. For a fixed point ζ ∈ X andt ∈ R + , define an ideal sheaf I ζ,d,t as follows: write ζ j = π j (ζ) and consider the local ringm⊗O ζ,X ≃ O ζj ,C j.j=16
Let z j be a local parameter in O ζj ,C jso that {z 1 , . . .,z m } form a regular system of parametersfor O ζ,X . First we define an ideal I ζ,d,t ⊂ O ζ,X :⎛⎞m∏I ζ,d,t := ⎝M α = z α jjα j ≤ d j for all j and ind a (ζ, M α ) ≥ t⎠.∣j=1The definition of I ζ,d,t does not depend on the choice of local parameters {z 1 , . . .,z m }. AlthoughI ζ,d,t does depend on the choice of a used to define the index, we omit this from thenotation as a will be fixed throughout. Choose a small product affine open subsetm∏U = U j = Specj=1m⊗A jj=1with z i ∈ A i for all i so that the only zero of z i in U i is ζ i . We will continue to denote by I ζ,d,tthe ideal sheaf on U associated to I ζ,d,t . Letting f : U ֒→ X denote the natural inclusion,defineI ζ,d,t := f ∗ I ζ,d,t ∩ O X (cf. [EV1] 2.2).It is not difficult to check that the definition does not depend on the choice of U.Definition 1.2 Fix a set S ⊂ X as in Theorem 0.4 and let t i = ind a (ζ i , s). ThenM⋂I(s) := I ζi ,d,t i.i=1It is clear from the definition that s ∈ H 0 [X, L ⊗ I(s)]. In the special case where X = Pand L = O P (d), the sheaf I(s) corresponds with L ′ as defined in [EV1] 2.6. Note also (cf.[EV1] 2.3) that the Riemann–Roch theorem for curves implies thatI ζ,nd,nt ⊗ F ⊗nd is generated by global sections for all n ≫ 0.There is some subtlety here because when t = m, this statement is false for an arbitrarychoice of divisors D i on C i in the definition of F d . The problem is not serious, however,because one can always add the Q–Cartier divisor ǫ ∑ mi=1 π ∗ i D i to O X (D) and F d ; one canthen verify that taking the limit as ǫ → 0 yields Theorem 0.4. We will frequently encounterthis procedure. In any case, we will always have t < m so this is not important.We will need an auxilliary sheaf M(L, S), depending on L and the finite subset S ⊂ X.This will be used to make adjustments after taking derivatives of the section s ∈ H 0 (X, L).WriteS i = π i (S) = {ζ i1 , . . ., ζ iNi }.Also if g(C i ) = 0 and N i = 1 let S i ′ = {ζ i1, ζ i2 } for a general point ζ i2 ∈ C i . Otherwise letS i ′ = S i. Finally set{maxN i ′ {Ni , 2}, if g(C=i ) = 0,N i , otherwise.7
Page 1 and 2: Dyson’s Lemma and a Theorem of Es
Page 3 and 4: Then define∫Vol(d, a, t) :=I(d,a,
Page 5: Lang for his encouragement and guid
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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