On Algebraic Hyperbolicity of Log Surfaces

TL;DR

This paper introduces the concept of algebraic hyperbolicity for log varieties, proving that the projective plane with a very general curve of degree at least 5 is algebraically hyperbolic, with specific bounds on curves.

Contribution

It establishes algebraic hyperbolicity for (P^2, D) with D of degree ≥ 5 and provides explicit inequalities relating genus, intersection, and degree of curves.

Findings

(P^2, D) is algebraically hyperbolic for very general D of degree ≥ 5

Derived inequality: 2g(C) - 2 + i(C, D) ≥ (deg(D) - 4) deg C

For a very general quintic D, any map from P^1 to P^2 has at least deg(f)+2 points mapping to D

Abstract

We call a log variety (X, D) algebraically hyperbolic if there exists a positive number e such that 2g(C) - 2 + i(C, D) >= e deg(C) for all curves C on X, where i(C, D) is the number of the intersections between D and the normalization of C. Among other things, we proved that (P^2, D) is algebraically hyperbolic for a very general curve D of degree at least 5. More specifically, we showed that 2g(C) - 2 + i(C, D) >= (deg(D) - 4) deg C for all curves C on P^2. For example, fix a very general quintic curve D and then for any map f: C = P^1 --> P^2, there are at least deg(f) + 2 distinct points on C that map to points on D by f.