Dedekind sums and parsing of Hilbert series

Abstract

Given a polarized variety (X,D), we can associate a graded ring and a Hilbert series. Assume D is an ample ℚ Cartier divisor, and (X,D) 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 ≤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.