Structural properties of the lattice cohomology of curve singularities
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909331480379392 |
|---|---|
| author | Kubasch, Alexander A. Némethi, András Schefler, Gergő |
| author_facet | Kubasch, Alexander A. Némethi, András Schefler, Gergő |
| contents | The lattice cohomology of a reduced curve singularity is a bigraded ${\mathbb Z}[U]$-module ${\mathbb H}^*=\oplus_{q,n}{\mathbb H}^q_{2n}$, 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 ${\mathbb H}^0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00551 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structural properties of the lattice cohomology of curve singularities Kubasch, Alexander A. Némethi, András Schefler, Gergő Algebraic Geometry 32S05, 32S10, 32S25 (Primary) 14Bxx, 57K18 (Secondary) The lattice cohomology of a reduced curve singularity is a bigraded ${\mathbb Z}[U]$-module ${\mathbb H}^*=\oplus_{q,n}{\mathbb H}^q_{2n}$, 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 ${\mathbb H}^0$ 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. |
| title | Structural properties of the lattice cohomology of curve singularities |
| topic | Algebraic Geometry 32S05, 32S10, 32S25 (Primary) 14Bxx, 57K18 (Secondary) |
| url | https://arxiv.org/abs/2410.00551 |