On tautness of two-dimensional $F$-regular and $F$-pure rational singularities

TL;DR

This paper proves that $F$-regular singularities in two dimensions are uniquely determined by their weighted dual graphs, extending the concept of tautness from complex to positive characteristic cases.

Contribution

It establishes the tautness of $F$-regular singularities and explores tautness properties of $F$-pure rational singularities in positive characteristic.

Findings

$F$-regular singularities are taut, uniquely determined by their dual graphs.

Tautness of $F$-pure rational singularities is discussed.

Extends tautness results from complex to positive characteristic cases.

Abstract

The weighted dual graph of a two-dimensional normal singularity $(X, x)$ represents the topological nature of the exceptional locus of its minimal log resolution. $(X, x)$ and its graph are said to be taut if the singularity can be uniquely determined by the graph. Laufer gave a complete list of taut singularities over $\mathbb{C}$. In positive characteristics, taut graphs over $\mathbb{C}$ are not necessarily taut and tautness have been studied only for special cases. In this paper, we prove the tautness of $F$-regular singularities. We also discuss the tautness of $F$-pure rational singularities.