Nef cone volumes and discriminants of abelian surfaces

TL;DR

This paper investigates the nef cone volume of algebraic surfaces, focusing on abelian surfaces with principal polarization and products of elliptic curves, to measure the size of the nef cone and its geometric implications.

Contribution

It extends the concept of nef cone volume to abelian surfaces and products of elliptic curves, providing explicit calculations and insights into their geometric structure.

Findings

Complete nef cone volume calculations for simple abelian surfaces with principal polarization.

Explicit nef cone volume results for products of elliptic curves.

Insights into how ample line bundles can be moved within the nef cone.

Abstract

The nef cone volume appeared first in work of Peyre in a number-theoretic context on Del Pezzo surfaces, and it was studied by Derenthal and co-authors in a series of papers. The idea was subsequently extended to also measure the Zariski chambers of Del Pezzo surfaces. We start in this paper to explore the possibility to use this attractive concept to effectively measure the size of the nef cone on algebraic surfaces in general. This provides an interesting way of measuring in how big a space an ample line bundle can be moved without destroying its positivity. We give here complete results for simple abelian surfaces that admit a principal polarization and for products of elliptic curves.