K-stability of Casagrande–Druel varieties

Abstract We introduce a new subclass of Fano varieties (Casagrande–Druel varieties) that are 𝑛-dimensional varieties constructed from Fano double covers of dimension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> \mathbb{P}^{n-1} ramified over smooth hypersurfaces of degree <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>d</m:mi> </m:mrow> </m:math> 2d with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>&gt;</m:mo> <m:mi>d</m:mi> <m:mo>&gt;</m:mo> <m:mfrac> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:mfrac> <m:mo>&gt;</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> n&gt;d&gt;\frac{n}{2}&gt;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.