Schwartz $\kappa$-densities on the moduli stack of rank $2$ bundles near stable bundles

TL;DR

This paper establishes algebro-geometric conditions ensuring boundedness of Schwartz space functions on the moduli stack of rank 2 stable bundles over certain curves, advancing understanding of these functions near stable bundles.

Contribution

It formulates and proves new algebro-geometric statements that imply boundedness of Schwartz space functions on the moduli stack of rank 2 bundles, specifically for genus 2 and non-hyperelliptic genus 3 curves.

Findings

Boundedness of functions on the moduli space is established for genus 2 curves.

Results are extended to non-hyperelliptic genus 3 curves.

Functions are shown to be locally constant on the locus of very stable bundles.

Abstract

Let $C$ be a curve over a non-archimedean local field of characteristic zero. We formulate algebro-geometric statements that imply boundedness of functions on the moduli space of stable bundles of rank $2$ and fixed odd degree determinant over $C$, coming from the Schwartz space of $\kappa$-densities on the corresponding stack of bundles (earlier we proved that these functions are locally constant on the locus of very stable bundles). We prove the relevant algebro-geometric statements for curves of genus $2$ and for non-hyperelliptic curves of genus $3$.