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

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.