Structural properties of the lattice cohomology of curve singularities

Abstract

The lattice cohomology of a reduced curve singularity is a bigraded $\mathbb{Z}[U]$-module ${\mathbb{H}}^{\ast }={\oplus }_{q,n}{\mathbb{H}}_{2n}^q$, that categorifies the $\delta$-invariant and extract key geometric information from the semigroup of values. In the present paper we prove three structure theorems for this new invariant: (a) the weight-grading of the reduced cohomology is (just as in the case of the topological lattice cohomology of normal surface singularities) nonpositive; (b) the graded $\mathbb{Z}[U]$-module structure of ${\mathbb{H}}^0$ determines whether or not a given curve is Gorenstein; and finally (c) the lattice cohomology module ${\mathbb{H}}^0$ of any plane curve singularity determines its multiplicity.