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

Tomohiro Okuma; Kei-ichi Watanabe; Ken-ichi Yoshida

Paper

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

Abstract

Let

be an excellent two-dimensional normal local ring containing an algebraically closed field and let

be a resolution of singularity. We prove a theorem giving a condition under which the dimension of the cohomology group of invertible sheaves on

coincides with a natural lower bound. Applying this theorem, we establish upper bounds for the normal reduction number

. For example, we prove the inequality

, where

denotes the arithmetic genus, a fundamental combinatorial (topological) invariant. We introduce the notion of almost cone singularities and give a sharper inequality

for such singularities, where

denotes the fundamental genus. We also show that

is not a combinatorial invariant in general.