Brill-Noether theory of curves on Enriques surfaces I: the positive cone and gonality

TL;DR

This paper investigates the gonality of curves on Enriques surfaces, providing a comprehensive computation and linking it to the positive cone of line bundles, advancing understanding in algebraic geometry.

Contribution

It introduces a new method to compute gonality of curves on Enriques surfaces and relates it to the positive cone of line bundles, extending previous results.

Findings

Computed gonality for all curves on Enriques surfaces in their linear systems.

Established a new connection between the positive cone and gonality.

Extended methods below the Bogomolov-Reider range.

Abstract

We study the existence of linear series on curves lying on an Enriques surface and general in their complete linear system. Using a method that works also below the Bogomolov-Reider range, we compute, in all cases, the gonality of such curves. We also give a new result about the positive cone of line bundles on an Enriques surface and we show how this relates to the gonality.