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

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