A variant of R{\"o}hr's vanishing theorem with an application to the normal reduction number for normal surface singularities

TL;DR

This paper extends R{"o}hr's vanishing theorem to provide bounds on the normal reduction number of two-dimensional normal surface singularities, introducing almost cone singularities and exploring their properties.

Contribution

It proves a new vanishing theorem for invertible sheaves on resolutions of surface singularities and establishes bounds on the normal reduction number, including for almost cone singularities.

Findings

Proved a vanishing theorem linking cohomology dimensions to lower bounds.

Established upper bounds for the normal reduction number in terms of invariants.

Introduced the concept of almost cone singularities and derived sharper bounds.

Abstract

Let $A$ be an excellent two-dimensional normal local ring containing an algebraically closed field and let $X\to \mathrm{Spec} (A)$ be a resolution of singularity. We prove a theorem giving a condition under which the dimension of the cohomology group of invertible sheaves on $X$ coincides with a natural lower bound. Applying this theorem, we establish upper bounds for the normal reduction number $\bar{\mathrm{r}}(A)$ of $A$. For example, we prove the inequality $\bar{\mathrm{r}}(A) \le p_a(A)+1$, where $p_a(A)$ denotes the arithmetic genus, a fundamental combinatorial (topological) invariant. We introduce the notion of almost cone singularities and give a sharper inequality $\bar{\mathrm{r}}(A) \le p_f(A)+1$ for such singularities, where $p_f(A)$ denotes the fundamental genus. We also show that $\bar{\mathrm{r}}(A)$ is not a combinatorial invariant in general.