Unstable singular del Pezzo hypersurfaces with lower index

Abstract

We prove the existence of singular del Pezzo surfaces that are neither K-semistable nor contain any anticanonical polar cylinder.

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

“…As a consequence we prove the K‐polystability of quasi‐smooth hypersurfaces in $$\mathbb {P}(1,1,a,a)$$ of degree $$2a$$, hence partially confirming [27, Conjecture 1.3]. Corollary Let $$a \geqslant 2$$ be an integer and let $$X$$ be a degree $$2a$$ quasi‐smooth hypersurface in $$\mathbb {P}(1,1,a,a)$$.…”

“…The second author wishes to thank Anne‐Sophie Kaloghiros for many fruitful conversations and Yuji Odaka for helpful e‐mail exchanges; he is grateful also to Ivan Cheltsov and Jihun Park for useful remarks on an earlier draft of this manuscript and for sharing a preliminary version of [27]. The first author is partially supported by the NSF grant DMS‐2148266.…”

“…The (non‐)existence of KE metrics on many del Pezzo surfaces which are quasi‐smooth hypersurfaces in weighted projective 3‐spaces with Fano index $$\ge$$2 has been studied in [14–16, 27].…”

“…To the authors' knowledge it is not known if they are K‐polystable for an integer $$a=3$$ or $$a \geqslant 5$$. In [27], Kim and Won conjecture that these surfaces are K‐polystable and not K‐stable.…”

K-stability of log del Pezzo hypersurfaces with index 2

“…As a consequence, it was conjectured in [9,Conjecture 1.10], that for I " 2, all S d admit an orbifold Kähler-Einstein metric. But this was disproved by Kim and Won in [19,Theorem 1.2].…”