A sharp vanishing theorem for line bundles on K3 or Enriques surfaces

TL;DR

This paper establishes a precise vanishing theorem for the first cohomology of line bundles on K3 and Enriques surfaces, providing necessary and sufficient geometric conditions, crucial for Brill-Noether theory and Enriques-Fano threefolds.

Contribution

It introduces a new vanishing theorem that offers exact geometric criteria for cohomology vanishing on these surfaces, advancing the understanding of their line bundle properties.

Findings

Provides necessary and sufficient conditions for $H^1(L)=0$

Enables detailed study of Brill-Noether theory on Enriques surfaces

Facilitates analysis of Enriques-Fano threefolds

Abstract

Let $L$ be a line bundle on a K3 or Enriques surface. We give a vanishing theorem for $H^1(L)$ that, unlike most vanishing theorems, gives necessary and sufficient geometrical conditions for the vanishing. This result is essential in our study of Brill-Noether theory of curves on Enriques surfaces (reference [KL1]) and of Enriques-Fano threefolds (reference [KLM]).