Families of explicit quasi-hyperbolic and hyperbolic surfaces
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Mathematische Zeitschrift
Families of explicit quasi-hyperbolic and hyperbolic surfaces Natalia Garcia-Fritz1 · Giancarlo Urzúa1 Received: 14 September 2018 / Accepted: 29 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We construct explicit families of quasi-hyperbolic and hyperbolic surfaces parametrized by quasi-projective bases. The method we develop in this paper extends earlier works of Vojta and the first author for smooth surfaces to the case of singular surfaces, through the use of ramification indices on exceptional divisors. The novelty of the method allows us to obtain new results for the surface of cuboids, the generalized surfaces of cuboids, and other explicit families of Diophantine surfaces of general type. In particular, we produce new families of smooth complete intersection surfaces of multidegrees (m 1 , . . . , m n ) in Pn+2 which are hyperbolic, for any n ≥ 8 and any degrees m i ≥ 2. As far as we know, hyperbolic complete intersection surfaces were not known for low degrees in this generality. We also show similar results for complete intersection surfaces in Pn+2 for n = 4, 5, 6, 7. These families give evidence for [6, Conjecture 0.18] in the case of surfaces.
1 Introduction The purpose of this paper is to give an explicit method to find low genus curves in a wide range of algebraic surfaces. The method extends an earlier work of Vojta [19] for smooth surfaces, which has roots in the seminal work of Bogomolov [3] (see [7]), and the recent generalization of Vojta’s method by the first author [10]. The novelty of the method in the present article is that we include the case of singular surfaces by means of considering ramification indices on exceptional divisors. This is key for the new results we present below. In addition, we show that the method allows us to test Brody hyperbolicity in this singular setting. In particular, we show new examples of families of quasi-hyperbolic and hyperbolic surfaces. This part is based on Nevanlinna theory (cf. [21]). We recall some definitions to be precise. An entire curve in a variety X is the image of a nonconstant holomorphic map C → X . A surface X is said to be Brody hyperbolic (or simply hyperbolic, for short) if it has no entire curves. A surface X is said to be quasi-hyperbolic if all entire curves are contained
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Giancarlo Urzúa [email protected] Natalia Garcia-Fritz [email protected]
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Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna 4860, Santiago, Chile
123
N. Garcia-Fritz, G. Urzúa
in a proper Zariski closed subset of X . Hence, when X is a smooth projective surface, we have that X is hyperbolic if it is in the sense of Kobayashi or in the sense of Brody; cf. [13]. A main motivation for us comes from describing the set of rational points of particular Diophantine varieties under the Bombieri-Lang conjecture. For instance, by finding all nonhyperbolic curves on suitable Diophantine surfaces, Vojta [19] shows that the “n squares probl
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