K-stability of Casagrande-Druel varieties

TL;DR

This paper introduces Casagrande-Druel varieties, a new class of Fano varieties constructed from double covers, and investigates their K-stability, providing partial proofs and applications to moduli spaces.

Contribution

It defines Casagrande-Druel varieties, conjectures their K-polystability under certain conditions, and proves this for specific cases, advancing understanding of their moduli.

Findings

Proved K-polystability for smoothable Casagrande-Druel threefolds.

Established K-polystability for varieties from double covers of projective space ramified over smooth hypersurfaces.

Described connected components of K-moduli spaces for certain Fano threefold families.

Abstract

We introduce a new subclass of Fano varieties (Casagrande-Druel varieties), that are $n$-dimensional varieties constructed from Fano double covers of dimension $n-1$. We conjecture that a Casagrande-Druel variety is K-polystable if the double cover and its base space are K-polystable. We prove this for smoothable Casagrande-Druel threefolds, and for Casagrande-Druel varieties constructed from double covers of $\mathbb{P}^{n-1}$ ramified over smooth hypersurfaces of degree $2d$ with $n>d>\frac{n}{2}>1$. As an application, we describe the connected components of the K-moduli space parametrizing smoothable K-polystable Fano threefolds in the families 3.9 and 4.2 in the Mori-Mukai classification.