On surfaces of general type with maximal Albanese dimension

TL;DR

This paper establishes an optimal bound on the canonical degree of rational and elliptic curves on certain algebraic surfaces, leading to finiteness results for such curves on surfaces of general type with specific properties.

Contribution

It provides the first optimal bound on the canonical degree for rational and elliptic curves on surfaces with maximal Albanese dimension, and proves finiteness of these curves under certain conditions.

Findings

Optimal bound on canonical degree for rational and elliptic curves.

Finiteness of rational and elliptic curves on surfaces with two independent regular 1-forms.

Applicable to surfaces of general type with maximal Albanese dimension.

Abstract

Given a minimal surface equipped with a generically finite map to an Abelian variety, we give an optimal bound on the canonical degree of a rational or an elliptic curve. As a corollary, we obtain the finiteness of rational and elliptic curves on any surface of general type with two linearly independent regular one forms.