Theodoros Papazachariou - K-moduli for log Fano complete intersections

An important category of geometric objects in algebraic geometry is smooth Fano varieties, which have positive curvature. These have been classified in 1, 8 and 105 families in dimensions 1, 2 and 3 respectively, while in higher dimensions the number of Fano families is yet unknown, although we know that their number is bounded. An important problem is compactifying these families into moduli spaces via K-stability. In this talk, I will describe the compactification of the family of Fano threefolds, which is obtained by blowing up the projective space along a complete intersection of two quadrics which is an elliptic curve, into a K-moduli space using Geometric Invariant Theory (GIT). A more interesting setting occurs in the case of pairs of varieties and a hyperplane section where the K-moduli compactifications tessellate depending on a parameter. In this case it has been shown recently that the K-moduli decompose into a wall-chamber decomposition depending on a parameter, but wall-crossing phenomena are still difficult to describe explicitly.  Using GIT, I will describe an explicit example of wall-crossing in the K-moduli spaces, where both variety and divisor differ in the deformation families before and after the wall, given by log pairs of Fano surfaces of degree 4 and a hyperplane section.