K-moduli of log Fano complete intersections

TL;DR

This paper explicitly describes the K-moduli compactifications and wall crossings of certain Fano complete intersections, establishing isomorphisms with GIT quotients and introducing computational methods and a new moduli continuity technique.

Contribution

It provides explicit descriptions of K-moduli of Fano complete intersections, characterizes GIT quotients based on singularities, and introduces computational and reverse moduli continuity methods.

Findings

Explicit K-moduli compactifications for specific Fano complete intersections.

Isomorphisms between K-moduli and GIT quotients for these varieties.

Development of computational algorithms for studying VGIT quotients.

Abstract

We explicitly describe the K-moduli compactifications and wall crossings of log pairs formed by a Fano complete intersection of two quadric threefolds and a hyperplane, by constructing an isomorphism with the VGIT quotient of such complete intersections and a hyperplane. We further characterize all possible such GIT quotients based on singularities. We also explicitly describe the K-moduli of the deformation family of Fano 3-folds 2.25 in the Mori--Mukai classification, which can be viewed as blow ups of complete intersections of two quadrics in dimension three, by showing there exists an isomorphism to a GIT quotient which we also explicitly describe. Furthermore, we also construct computational algorithmic methods to study VGIT quotients of complete intersections and hyperplanes, which we use to obtain the explicit descriptions detailed above. We also introduce the reverse moduli continuity method, which allows us to relate canonical GIT compactifications to K-moduli of Fano varieties.