K-moduli of log Fano_complete_intersections

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. Our main result is the first example of wall-crossing for the K-moduli of log pairs, where both the variety and divisor admit deformations before and after the wall-crossing. Furthermore, we 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. This is one of the only three known explicit compactifications of K-moduli of Fano threefolds. Our work uses the moduli continuity method for log pairs by relating the K-moduli to certain GIT compactifications. In addition, we introduce the reverse moduli continuity method, which allows us to relate canonical GIT compactifications to K-moduli of Fano varieties. We also compute the CM line bundle for complete intersections of hypersurfaces of fixed degree with a hyperplane section, and we show it isomorphic to an ample line bundle in the Picard group of the canonical GIT quotient of complete intersections and a hyperplane. We use explicit GIT methods to classify in detail the GIT stability of a complete intersection of two quadrics in dimension four and three, together with a hyperplane section. Furthermore, we explicitly compactify the moduli space of log Fano pairs of complete intersections and hyperplane sections, by establishing a direct link with the GIT compactification.