K-moduli of log Fano complete intersections

We explicitly describe the K-moduli compactifications of log pairs formed by a Fano complete intersection of hypersurfaces of fixed degree and a hyperplane, by constructing an isomorphism with certain GIT quotients. We characterize all possible GIT quotients of complete intersections of these hypersurfaces and hyperplanes. Our main result is the first example of wall crossing for the K-moduli of log pairs, where both variety and divisor admit deformations before and after the wall. 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. 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 GIT compactifications to K-moduli of Fano varieties. We use explicit GIT methods to classify in detail the GIT stability of a complete intersection of two quadrics in dimension four together with a hyperplane section. We also classify the GIT stability of a complete intersection of two quadrics in dimension three. Furthermore, we explicitly compactify the moduli space of log Fano pairs of complete intersections and hyperplane sections, by establishing a direct link with GIT. We use the above methods to prove our main result, namely the explicit description of the first wall crossing for the K-moduli of log Fano pairs, of complete intersections of two quadrics in dimension four, and a hyperplane section. The reverse moduli continuity method, alongside our computational GIT techniques, allow us to compactify the K-moduli of the deformation family of Fano 3-folds 2.25, which is one of the only three known explicit compactifications of K-moduli of Fano threefolds.