A remark on a theorem of Narasimhan and Ramanan

TL;DR

This paper offers an alternative proof of a theorem linking the moduli space of certain vector bundles over a genus 2 curve to projective 3-space, using Seshadri constants and Fano variety criteria.

Contribution

It introduces a new proof method for the theorem by applying a criterion based on Seshadri constants, differing from previous approaches.

Findings

The moduli space is isomorphic to .

The proof utilizes Seshadri constants to characterize projective space.

The approach connects vector bundle moduli with Fano variety properties.

Abstract

In this short note, we provide an alternative proof of a notable theorem by Narasimhan and Ramanan. The theorem states that the moduli space of $S$-equivalence classes of semistable rank $2$ vector bundles over a curve $X$ of genus $2$ with trivial determinant is isomorphic to $\mathbb{P}^3$. Our proof relies on a criterion by Bauer and Szemberg, which characterizes projective spaces among smooth Fano varieties using Seshadri constants.