Non-normal affine monoid algebras

Abstract

We give a geometric description of the set of holes in a non-normal affine monoid Q. The set of holes turns out to be related to the non-trivial graded components of the local cohomology of \({\mathbb{K}[Q]}\). From this, we see how various properties of \({\mathbb{K}[Q]}\) like local normality and Serre’s conditions (R1) and (S2) are encoded in the geometry of the holes. A combinatorial upper bound for the depth the monoid algebra \({\mathbb{K}[Q]}\) is obtained which in some cases can be used to compute its depth.