Exceptional loci of F-blowups and $G$-Hilbert schemes

TL;DR

This paper investigates the structure of exceptional loci in F-blowups of normal toric varieties, providing formulas, algorithms, and conditions related to their dimensions and connections to $G$-Hilbert schemes.

Contribution

It offers a combinatorial formula and an algorithm for the dimension of centers of prime divisors on F-blowups, linking to $G$-Hilbert schemes and singularity theory.

Findings

Derived a formula for the dimension of centers of prime divisors.

Developed an algorithm to compute the dimension using combinatorial data.

Identified conditions for positive-dimensional centers over toric varieties.

Abstract

We study exceptional loci of F-blowups of normal toric varieties. In the $\Q$-factorial case, this study amounts to studying the exceptional loci of $G$-Hilbert schemes. We give a formula for the dimension of the center of a prime divisor on the F-blowup in terms of combinatorial data, together with an algorithm for computing it. Moreover, we study the relation between F-blowups and essential divisors for three-dimensional terminal singularities and canonical singularities. Finally, we give a simple condition ensuring that a prime divisor over the given toric variety has a positive-dimensional center on the F-blowup.