Structural properties of the lattice cohomology of curve singularities

2024/10/01 by Alexander A. Kubasch, András Némethi, Kubasch, Alexander A. +3 · 1 citation
Mathematics · #32S05 #32S10 #32S25 (Primary) 14Bxx #57K18 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2410.00551

openalex publication_date 2024/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The lattice cohomology of a reduced curve singularity is a bigraded \mathbb Z[U]-module \mathbb H^*=⊕_q,n\mathbb H^q_2n, that categorifies the δ-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 H⁰ determines whether or not a given curve is Gorenstein; and finally (c) the lattice cohomology module \mathbb H⁰ of any plane curve singularity determines its multiplicity.

Cited by

• Cohen-Macaulay Type via Lattice Homology and the Motivic Poincaré Series

Related

• Analytic lattice cohomology of isolated curve singularities
• Lattice cohomology and the embedded topological type of plane curve singularities
• Filtered lattice homology of curve singularities
• Filtered lattice homology of surface singularities
• Lattice cohomology of normal surface singularities