Families of explicit quasi-hyperbolic and hyperbolic surfaces

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 $(m_1,\ldots ,m_n)$ in ${\mathbb {P}}^{n+2}$ which are hyperbolic, for any $n \ge 8$ and any degrees $m_i \ge 2$. 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 ${\mathbb {P}}^{n+2}$ for $n=4,5,6,7$. These families give evidence for [6, Conjecture 0.18] in the case of surfaces.