Non-normal affine monoid algebras

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 K[Q]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{K}[Q]}$$\end{document}. From this, we see how various properties of K[Q]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{K}[Q]}$$\end{document} 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 K[Q]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{K}[Q]}$$\end{document} is obtained which in some cases can be used to compute its depth.