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

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.

accepted version; 36 pages, with an appendix joint with Benjamin Church; a computational error fixed