On K-stability of some del Pezzo surfaces of Fano index 2

TL;DR

This paper investigates the K-stability of certain del Pezzo surfaces of Fano index 2, establishing a connection with GIT stability of binary forms and proving conditions for K-polystability and non-stability.

Contribution

It relates K-stability of hypersurfaces in weighted projective spaces to GIT stability of binary forms and characterizes K-polystability for quasi-smooth cases.

Findings

K-stability linked to GIT stability of binary forms

Hypersurfaces are K-polystable if quasi-smooth

Hypersurfaces are not K-stable if quasi-smooth

Abstract

For every integer $a \geq 2$, we relate the K-stability of hypersurfaces in the weighted projective space $\mathbb{P}(1,1,a,a)$ of degree $2a$ with the GIT stability of binary forms of degree $2a$. Moreover, we prove that such a hypersurface is K-polystable and not K-stable if it is quasi-smooth.