A functorial approach to the stability of vector bundles

TL;DR

This paper investigates the stability properties of vector bundles on algebraic varieties, showing that the locus of stable bundles that remain stable under certain covers is large and open in the moduli space, with specific results for curves of genus at least 2.

Contribution

It introduces a functorial framework to analyze the stability of vector bundles and demonstrates the openness and size of the stable locus under Galois covers.

Findings

The locus of μ-stable bundles stable under all prime-to-characteristic Galois covers is open in the moduli space.

On smooth projective curves of genus ≥ 2, this locus is large with a complement of codimension at least 2.

The stability behavior is characterized in a functorial manner across covers.

Abstract

On a normal projective variety the locus of $\mu$-stable bundles that remain $\mu$-stable on all Galois covers prime to the characteristic is open in the moduli space of Gieseker semi-stable sheaves. On a smooth projective curve of genus at least 2 this locus is big in the moduli space of stable bundles, i.e., its complement has codimension at least 2.