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

The moduli space of bundle stable pairs ${\bar{M}}_C(2,\mathrm{\Lambda })$ 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 ${\bar{M}}_C(2,\mathrm{\Lambda })$ to that of the moduli spaces of stable vector bundles ${\bar{N}}_C(2,\mathrm{\Lambda })$. In particular, Thaddeus’ moduli spaces for genus two curves are all K-stable.