Algebraic hyperbolicity of ramified covers of $\mathbb{G}^2_m$ (and integrap points on affine subsets of $\mathbb{P}_2$)

Abstract

Let $X$ be a smooth affine surface, $X \to \mathbb{G}^2_m$ be a finite morphism. We study the affine curves on $X$, with bounded genus and number of points at infinity, obtaining bounds for their degree in terms of Euler characteristic.

A typical example where these bounds hold is represented by the complement of a three-component curve in the projective plane, of total degree at least 4.

The corresponding results may be interpreted as bounding the height of integral points on $X$ over a function field. In the language of Diophantine Equations, our results may be rephrased in terms of bounding the height of the solutions of $f(u, v, y) = 0$, with $u, v, y$ over a function field, $u, v$ $S$-units.

It turns out that all of this contain some cases of a strong version of a conjecture of Vojta over function fields in the split case. Moreover, our method would apply also to the nonsplit case.

We remark that special cases of our results in the holomorphic context were studied by M. Green already in the seventies, and recently in greater generality by Noguchi, Winkelmann, and Yamanoi; however, the algebraic context was left open and seems not to fall in the existing techniques.