Elliptic Curves in Moduli Space of Stable Bundles of Rank 3

TL;DR

This paper studies elliptic curves within the moduli space of rank 3 stable bundles on a curve, establishing degree bounds and classifying certain elliptic curves, challenging existing conjectures.

Contribution

It provides degree bounds for elliptic curves in the moduli space and classifies degree 6 elliptic curves, offering new insights into their structure.

Findings

Elliptic curves on the moduli space have degree at least 6 for generic curves.

Complete classification of degree 6 elliptic curves is provided.

Elliptic curves passing through generic points have degree at least 18 when genus > 12.

Abstract

Let $M$ be the moduli space of rank 3 stable bundles with fixed determinant of degree 1 on a smooth projective curve of genus $g\geq 2$. When $C$ is generic, we show that any essential 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, which is not in conformity with Sun's Conjecture. Moreover, if $g>12$, we show that any elliptic curve passing through the generic point of $M$ has degree at least 18.