Dedekind sums and parsing of Hilbert series

Abstract

Given a polarized variety $\left(X,D\right)$, we can associate a graded ring and a Hilbert series. Assume $D$ is an ample $\mathbb{Q}$ Cartier divisor, and $\left(X,D\right)$ is quasi smooth and projectively Gorenstein, we give a parsing formula for the Hilbert series according to their singularities. Here we allow the variety to have singularities of dimension $\le 1$, that is, both singularities of dimension $1$ and singular points, extending a 2013 result of Buckley, Reid and the author about varieties with only isolated singularities.