A note on stability of syzygy bundles on Enriques and bielliptic surfaces

TL;DR

This paper proves the cohomological stability of syzygy bundles on Enriques and bielliptic surfaces, extending known results and removing previous restrictions, with implications for stability on minimal surfaces of Kodaira dimension zero.

Contribution

It establishes the cohomological stability of syzygy bundles on Enriques and bielliptic surfaces over certain fields, improving prior results by removing Clifford index restrictions.

Findings

Syzygy bundle $M_L$ is cohomologically stable on Enriques surfaces.

The result extends to bielliptic surfaces over specified fields.

Implication that $M_L$ is stable on all minimal complex surfaces of Kodaira dimension zero.

Abstract

In this note, we prove that the syzygy bundle $M_L$ is cohomologically stable with respect to $L$ for any ample and globally generated line bundle $L$ on an Enriques (resp. bielliptic) surface over an algebraically closed field of characteristic $\neq 2$ (resp. $\neq 2,3$). In particular our result on complex Enriques surfaces improves a result of Torres-L\'opez and Zamora by removing a condition on Clifford index. Together with the results of Camere and Caucci--Lahoz, it implies that $M_L$ is stable with respect to $L$ for an ample and globally generated line bundle $L$ on any smooth minimal complex projective surface $X$ of Kodaira dimension zero.