Description:

Title: K-stability and K-moduli of Casagrande-Druel Varieties

Abstract: Arising in the construction of ramified double-covers of Fano varieties, as well as the investigation of Fano four-folds, Casagrande-Druel varieties are conic bundles $Y$ realized as certain blow-ups of projective bundles over their base $V$ with discriminant locus $B$. These varieties are Fano under minor additional assumptions.
        We show that a Casagrande-Druel variety $Y$ satisfying an additional proportionality condition, is K-poly/semistable if and only if the pair $(V,aB)$ is K-poly/semistable for an explicit coefficient $a$, and that Casagrande-Druel variety failing this proportionality condition is K-unstable. Using this we then construct the K-moduli spaces of Casagrande-Druel varieties by directly relating the K-moduli of such Fano varieties to the K-moduli spaces of the base pairs $(V,aB)$.