Some observations on the dimension of Fano K-moduli

TL;DR

This paper demonstrates that the dimension of the K-moduli space of n-dimensional Fano varieties can be unbounded, with the stack dimension also unbounded, even when the coarse space dimension stays bounded.

Contribution

It reveals the unboundedness of the dimension of K-moduli spaces and stacks for Fano varieties, highlighting nuanced differences between stack and coarse space dimensions.

Findings

Dimension of K-moduli space can be unbounded.

Stack dimension can be unbounded while coarse space dimension is bounded.

Provides new insights into the structure of Fano K-moduli spaces.

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.