Explicit K-stability of Fano Varieties - Etxabarri Alberdi, Erroxe (2024)
This thesis completes the classification of local stability thresholds (δ-invariant) for smooth del Pezzo surfaces of degree 2 and explores the compactification of K-moduli for Fano 3-folds. In the first part, we show that this invariant is irrational if and only if there is a unique (-1)-curve passing through the point where we are computing the local invariant. This work can be useful for future verification of K-stability in higher dimensions, this is because the computations of δ-invariants of higher dimensional varieties are often reduced to the computations of δ-invariants of del Pezzo surfaces. The irrationality of the local stability threshold also implies the existence of infinitely many local degenerations of the variety, which can lead to interesting further studies. In the second part, we work on the compactification of one-dimensional components of the moduli spaces of Fano 3-folds by studying degenerate objects. The result on K-moduli gives some of the few existing examples of compactifications of components of the K-moduli space for Fano varieties. There are a total of 6 families with one-dimensional moduli. In this thesis, we focus on 3 of those families, we explain the parameterization of each family, the proof of K-polystability of singular elements and the compactification of the K-moduli component by explicitly describing each K-polystable member of the family.