Dyson's lemma and a theorem of Esnault and Viehweg
Proofs in diophantine approximation and transcendence theory often make use of an upper bound on the vanishing of an auxiliary polynomial or, more generally, a global section of a sheaf. One such upper bound, called Dyson's lemma, was introduced by \textit{F. J. Dyson} [Acta Math., Uppsala 79, 225-240 (1947; Zbl 0030.02101)] (where it was proved, essentially, for \(\mathbb{P}^1 \times \mathbb{P}^1\)) and generalized to \((\mathbb{P}^1 )^m\) by \textit{H. Esnault} and \textit{E. Viehweg} [Invent. Math. 78, 445-490 (1984; Zbl 0532.10020)]. The present paper proves a more general version of Dyson's lemma for a nonzero global section of an invertible sheaf \({\mathcal L}\) on a product \(C_1 \times \dots \times C_m\) of integral nonsingular projective curves over a field of characteristic zero. The main theorem of the paper is difficult to state concisely. It does, however, require one additional condition that is worth mentioning. Let \(X= C_1 \times \dots \times C_m\); for \(i=1, \dots,m\) let \(\pi_i: X\to C_i\) denote the \(i\)-th projection. For an invertible sheaf \({\mathcal L}\) on \(X\), let \(d_i ({\mathcal L})\) denote the intersection number of \({\mathcal L}\) with a curve of the form \(P_1 \times \dots \times C_i \times \dots \times P_n\), where \(P_j\in C_j\) is a closed point for all \(j\neq i\). The additional condition is that there exist a numerically effective \(\mathbb{Q}\)-divisor \(D\) on \(X\) with \(d_i ({\mathcal O} (D))\leq d_i ({\mathcal L})\) for each \(i\); a divisor \(F\) on \(X\) of the form \(F= \sum_{i=1}^m \pi_i^* F_i\), where each \(F_i\) is a divisor on \(C_i\) of degree \(d_i ({\mathcal L})\); and an integer \(N> 0\) such that \(H^0 (X, ({\mathcal L} (D- F))^{\otimes N}) \neq 0\). This condition is not a serious obstacle for most applications. The proof follows Esnault-Viehweg rather closely in places, but replaces the use of weak positivity with the derivative of a global section. This latter notion was motivated by the product theorem of \textit{G. Faltings} [Ann. Math., II. Ser. 133, 549-576 (1991; Zbl 0734.14007)]. The paper makes some interesting comparisons between the methods of Esnault-Viehweg and Faltings.
upper bound on the vanishing of a global section of an invertible sheaf
scientific article; zbMATH DE number 807308
scientific article