Dyson's lemma with moving parts
In its original form, Dyson's Lemma bounds the order of vanishing of a polynomial in two variables at a finite set of points. It was first used as a tool in diophantine approximation: Dyson himself used his lemma in order to sharpen Siegel's lower bound on the rational approximations to algebraic numbers. \textit{E. Bombieri} [Acta Math. 148, 255-296 (1982; Zbl 0505.10015)] conjectured a several variables generalization of Dyson's Lemma, soon proved by \textit{H. Esnault} and \textit{E. Viehweg} [Invent. Math. 78, 445-490 (1984; Zbl 0545.10021)]. A natural generalization to products of curves of arbitrary genus was proved by \textit{P. Vojta} in dimension two [Invent. Math. 98, 107-113 (1989; Zbl 0666.10024)]; it is an essential step in his proof of Mordell's Conjecture, both in the function field case [Invent. Math. 98, 115-138 (1998; Zbl 0662.14019)] and in the number field case [Ann. Math. (2) 133, 509-548 (1991; Zbl 0774.14019)]. The general case was established by \textit{M. Nakamaye} [Invent. Math. 121, 355-377 (1995; Zbl 0855.11036)]. Let \({\mathbb P}\) be the product of \(m\) copies of the projective line. For an \(m\)-tuple of integers \((d_1,\ldots,d_m)\) let \({\mathcal O}_{\mathbb P}(d)\) be the line bundle: \(\bigotimes_{i=1}^m \pi_i^*{\mathcal O}_{\mathbb P_1}(d_i)\). Let \( \delta_i =\sum_{j=i+1}^m d_j(L). \) for \(i=1,\ldots,m\). For a section \(s\in H^0(X,L)\) and a point \(\zeta\in X\), call index of \(s\) at \(\zeta\) the weighted multiplicity of the zero of \(s\) at \(\zeta\) with respect to the weights \((d_1^{-1},\ldots d_m^{-1})\). Let for \(t>0\), \(V(t)\) be the measure of the set \(\{(x_1,\ldots,x_m)\mid 0\leq x_i\leq 1\), \(\sum_{j=1}^m x_j\leq t\}\). In the paper under review it is proved (Theorem 1.2) that, given a finite set \(\{\zeta_1,\ldots,\zeta_M\}\) of \(X\) satisfying standard technical conditions, and a non-zero section \(s\) of \({\mathcal O}_{\mathbb P}(d)\) having index at least \(t_i\) at the point \(\zeta_i\) for \(i=1,\ldots,M\), the following inequality holds: \[ \prod_{i=1}^m d_i\sum_{j=1}^M V(t_j)\leq {c_1({\mathcal O}_{\mathbb P}(d+\delta))^{m-1} \cap c_1({\mathcal O}_{\mathbb P}(d))\over m!}. \] The theorem above slightly improves on the quoted Theorem 0.4 of Esnault and Viehweg. The author also provides a counterexample to the natural generalization of the above inequality to the product of curves of arbitrary genus.
product of copies of the projective line
scientific article; zbMATH DE number 1111632