New constructions of nef classes on self-products of curves

TL;DR

This paper investigates the nef cone of self-products of algebraic curves, constructing new boundary classes and identifying improved nef classes for general curves, enhancing understanding of the cone’s structure.

Contribution

It introduces novel nef classes on self-products of curves, including a unique boundary class for very general curves of genus greater than 2, and extends results to multiple copies.

Findings

Constructed a nontrivial class of self-intersection 0 on the nef cone boundary for very general curves.

Identified nef classes that improve known examples for arbitrary curves.

Extended analysis to self-products of more than two copies of the curve.

Abstract

We study the nef cone of self-products of a curve. When the curve is very general of genus $g>2$, we construct a nontrivial class of self-intersection 0 on the boundary of the nef cone. Up to symmetry, this is the only known nontrivial boundary example that exists for all $g > 2$. When the curve is general, we identify nef classes that improve on known examples for arbitrary curves. We also consider self-products of more than two copies of the curve.