On a Conjecture of Drinfeld

TL;DR

This paper proves Drinfeld's conjecture that the wobbly locus forms a divisor in the moduli space of semi-stable vector bundles on a high-genus curve, specifically when the rank and degree are coprime.

Contribution

The paper provides a proof of Drinfeld's conjecture regarding the codimension of the wobbly locus in the moduli space for coprime rank and degree.

Findings

Wobbly locus is a divisor in the moduli space for coprime n and d.

Confirmed the purity and codimension one property of the wobbly locus.

Established the conjecture for a broad class of vector bundles on high-genus curves.

Abstract

Let $C$ be a smooth irreducible irreducible projective curve of genus $g \ge 2$. Let $\mathcal{M}_C(n, \delta)$ be the moduli space of semi-stable vector bundles on $C$ of rank $n$ and fixed determinant $\delta$ of degree $d$. Then the locus of wobbly bundles is known to be closed in $\mathcal{M}_C(n, \delta)$. It was announced by Laumon and attributed to Drinfeld that the wobbly locus is pure of co-dimension one, i.e., they form a divisor in $\mathcal{M}_C(n, \delta)$. This is now known as Drinfeld's conjecture. In this article, we will give a proof of the conjecture when $n$ and $d$ are coprime.