Dyson's Lemma and a Theorem of Esnault and Viehweg

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Corollary 2.3 ([EV1] Lemma 2.8) Let W i be the support of the subscheme defined by I ζi ,s.Then W i⋂ Wj is empty for all i ≠ j.Proof: Corollary 2.3 is a straightforward consequence of Lemma 2.2 and is left as anexercise; the proof in [EV1] Lemma 2.8 goes through unchanged as [EV1] Lemma 2.5 easilyextends to our situation.Since we have assumed N = 1, the assumptions in Theorem 0.4 imply that there is aneffective divisor Γ such thatO X (Γ) ≃ L(D) ⊗ Fd −1 .Lemma 2.4 The invertible sheaf O X (Γ) ⊗ M(δ) is nef.Lemma 2.4 will be proved at the end of §3; it is similar to the proof of Theorem 2.1except that the inductive step is significantly simpler.Lemma 2.5 Let ζ ∈ X be a closed point and let Z ζ,n be the subscheme defined by I ⊗nζ,d,t fort = ind a (ζ, s). Thenh ( 0 X, Z ζ,n ⊗ [L(D) ⊗ M(δ)] ⊗n)limn→∞ n m ∏ m≥ Vol(d, a, t).i=1 d iProof: By [EV1] Lemma 1.9 and the assumption made at the beginning of this sectionBy Definition 0.3 one obtains (cf. [EV1] 2.4)I ⊗nζ,d,t = I ζ,nd,nt⎛ ( )lim ⎝ h0 X, F ( )d⊗n − h0X, Fd ⊗n ⎞⊗ I ζ,nd,ntn→∞n m ∏ ⎠m= Vol(d, a, t).i=1 d iFrom the long exact cohomology sequence associated toone obtainsSince0 → I ζ,nd,nt ⊗ F ⊗nd→ F ⊗nd→ Z ζ,n ⊗ F ⊗nd → 0⎛ ( )lim ⎝ h0 X, Z ζ,n ⊗ Fd⊗n ⎞n→∞ n m ∏ ⎠m≥ Vol(d, a, t). (2.5.1)i=1 d i[L(D) ⊗ M(δ)] ⊗n ≃ [O X (Γ) ⊗ M(δ)] ⊗n ⊗ F ⊗nd .Lemma 2.5 now follows from (2.5.1) and Lemma 2.4 by adding ǫA to L where A is an ampledivisor on X and taking the limit as ǫ → 0.Proof of Theorem 0.4 from Theorem 2.1: first we need to show, in the language of [EV1]Definition 3.3, that the ideal sheaf I(s) is full. This can be done as in [EV1] §3, using the10
covering construction of [V1] Lemma 3.1 instead of the simple ramified cover of P used in[EV1] 3.9. We will not reproduce the argument here. The rest of the argument is purelycohomological and the fact that I(s) is full will be used in order to apply the Leray spectralsequence. The argument is given in [EV1] 5.11–5.12.Let f : Y → ˜X be a desingularization of ˜X and let g = τ · f : Y → X. LetO Y (−F) = Im : g ∗ I(s) → O Y .Then O Y (−F) ≃ f ∗ O ˜X(−E) and so by Theorem 2.1, g ∗ [L(D) ⊗ M(δ)](−F) is nef. Henceby [EV1] Corollary 4.7, it follows that there exists a polynomial P(n) of degree ≤ m − 1such thath 1 ( X, [g ∗ [L(D) ⊗ M(δ)](−F)] ⊗n) ≤ P(n) for all n ≥ 0.Using the fact that I(s) is full and the Leray spectral sequence it follows thath 1 ( X, [L(D) ⊗ M(δ) ⊗ I(s)] ⊗n) ≤ P(n) for all n ≥ 0. (2.6)Suppose Z n ⊂ X is the subscheme defined by the ideal sheaf I(s) ⊗n and consider the exactsequence0 → [L(D) ⊗ M(δ) ⊗ I(s)] ⊗n → [L(D) ⊗ M(δ)] ⊗n → Z n ⊗ [L(D) ⊗ M(δ)] ⊗n → 0. (2.7)By Theorem 2.1, τ ∗ [L(D) ⊗ M(δ)](−E) is nef on ˜X and hence L(D) ⊗ M(δ) is nef on X.Perturbing L by ǫA for a rational ǫ ≪ 1 and taking the limit as ǫ → 0, we can assume thatL(D) ⊗ M(δ) is ample. Thus for n ≫ 0h 0 ( X, [L(D) ⊗ M(δ)] ⊗n) = χ ( [L(D) ⊗ M(δ)] ⊗n) = nmm! [L ⊗ M(δ)]top + O(n m−1 ). (2.8)On the other hand Corollary 2.3 imples that Z n = ⋃ Z ζi ,n is a disjoint union of the subschemesZ ζi ,n defined in Lemma 2.5. From Lemma 2.5 it follows thatlimn→∞⎛⎝ h0 ( Z n ⊗ [L(D) ⊗ M(δ)] ⊗n)n m ∏ mi=1 d i⎞M∑⎠ ≥ Vol(d, a, t i ). (2.9)i=1Now it follows from (2.6), (2.8), (2.9), and the long exact cohomology sequence associatedto (2.7) thatM∑ [L(D) ⊗ M(δ)]topVol(d, a, t i ) ≤ ,m!i=1and Definition 1.3 finishes the proof of Theorem 0.4 from Theorem 2.1.11
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 and 6: Lang for his encouragement and guid
Page 7 and 8: Let z j be a local parameter in O
Page 9: To show how Theorem 2.1 implies The
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

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