Algebraic hyperbolicity for surfaces in smooth projective toric threefolds with Picard rank 2 and 3

TL;DR

This paper extends the study of algebraic hyperbolicity from projective spaces to surfaces in smooth projective toric threefolds with Picard rank 2 and 3, identifying many hyperbolic surfaces using combinatorial methods.

Contribution

It generalizes algebraic hyperbolicity results to specific toric threefolds and applies combinatorial techniques to find hyperbolic surfaces.

Findings

Many algebraically hyperbolic surfaces identified in toric threefolds with Picard rank 2 and 3

Use of combinatorial descriptions to classify toric threefolds

Application of Haase and Ilten's method to these varieties

Abstract

Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a very general surface of degree at least five in projective space is algebraically hyperbolic. We are interested in generalizing the study of surfaces in projective space to surfaces in smooth projective toric threefolds with Picard rank 2 or 3. Following Kleinschmidt and Batyrev, we explore the combinatorial description of smooth projective toric threefolds with Picard rank 2 and 3. We then use Haase and Ilten's method of finding algebraically hyperbolic surfaces in toric threefolds. As a result, we determine many algebraically hyperbolic surfaces in each of these varieties.