Elliptic curves in moduli space of stable bundles

TL;DR

This paper investigates elliptic curves within the moduli space of stable rank 2 bundles on a curve, establishing degree bounds and classifying minimal degree elliptic curves, with implications for higher genus cases.

Contribution

It provides degree bounds for elliptic curves in the moduli space and classifies those with minimal degree, advancing understanding of the geometry of these moduli spaces.

Findings

Elliptic curves on the moduli space have degree at least 6 when the curve is generic.

Complete classification of elliptic curves of degree 6 in the moduli space.

Elliptic curves passing through a generic point have degree at least 12 when genus > 4.

Abstract

Let $M$ be the moduli space of rank $2$ stable bundles with fixed determinant of degree $1$ on a smooth projective curve $C$ of genus $g\ge 2$. When $C$ is generic, we show that any elliptic curve on $M$ has degree (respect to anti-canonical divisor $-K_M$) at least 6, and we give a complete classification for elliptic curves of degree $6$. Moreover, if $g>4$, we show that any elliptic curve passing through the generic point of $M$ has degree at least $12$. We also formulate a conjecture for higher rank.