K-stability of Thaddeus' moduli of stable bundle pairs on genus two curves

TL;DR

This paper investigates the K-stability of Thaddeus' moduli space of stable bundle pairs on genus two curves, establishing its isomorphism with a GIT moduli space and confirming its K-stability.

Contribution

It demonstrates the K-stability of Thaddeus' moduli spaces for genus two curves and relates them to GIT moduli of lines in quartic del Pezzo threefolds.

Findings

K-moduli space is isomorphic to GIT moduli of lines in quartic del Pezzo threefolds

Thaddeus' moduli spaces for genus two are all K-stable

Constructs a natural forgetful morphism between K-moduli spaces

Abstract

The moduli space of bundle stable pairs $\overline{M}_C(2,\Lambda)$ on a smooth projective curve $C$, introduced by Thaddeus, is a smooth Fano variety of Picard rank two. Focusing on the genus two case, we show that its K-moduli space is isomorphic to a GIT moduli of lines in quartic del Pezzo threefolds. Additionally, we construct a natural forgetful morphism from the K-moduli of $\overline{M}_C(2,\Lambda)$ to that of the moduli spaces of stable vector bundles $\overline{N}_C(2,\Lambda)$. In particular, Thaddeus' moduli spaces for genus two curves are all K-stable.