Some observations on the dimension of Fano K-moduli

Abstract

In this short note we show the unboundedness of the dimension of the K-moduli space of n-dimensional Fano varieties, and that the dimension of the stack can also be unbounded while, simultaneously, the dimension of the corresponding coarse space remains bounded.

On K‐stability of some del Pezzo surfaces of Fano index 2

“…Let us now compute the dimension of Γ by analysing the deformation theory of the K-polystable toric del Pezzo surface 𝑌 introduced in Proposition 3. Note that a similar study was discussed in [38].…”

A 1-dimensional component of K-moduli of del Pezzo surfaces

“…The goal of this note is to show how toric geometry and deformation theory can help understanding the geometry of explicit components of K-moduli. Similar ideas were used in [15] to construct examples of reducible or non-reduced K-moduli of Fano 3-folds, in [19] to study the K-stability of certain del Pezzo surfaces with Fano index 2, and in [23] to study the dimension of K-moduli. In this note we analyse a specific example of K-polystable toric del Pezzo surface and we prove the following: Theorem 1.1.…”