UC San Diego

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 $(V,aB)$.

Abstract: A branchvariety of a projective k-scheme X is a geometrically reduced scheme Y equipped with a finite map to X. Alexeev and Knutson showed the existence of a proper moduli space of branchvarieties with fixed Hilbert polynomial and degree sequence, but the projectivity of this space remained an open question. In this talk, we will discuss positivity results for some line bundles related to the determinant line bundle on the moduli space of equidimensional branchvarieties. As a consequence, we establish that this moduli space is projective.

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