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Delta invariant of $\mathbb{Q}$-Cartier curve germs and the genus of representable numerical semigroups

Zsolt Baja, Tamás László, András Némethi

TL;DR

The paper develops two delta-invariant formulae for a curve germ embedded as a $ obreak \mathbb{Q}$-Cartier divisor in a normal surface singularity with a $ obreak \mathbb{Q}$-homology sphere link, linking analytic and topological data through equivariant genera. It then relates representable numerical semigroups to the value semigroups of generic $ obreak \mathbb{C}^*$-orbits on weighted homogeneous surface singularities, producing a combinatorially computable genus formula in terms of resolution data. A duality for equivariant geometric genus is established, and the genus formula is used to characterize when representable semigroups are symmetric, expressed via Seifert invariants and delta-type invariants. The results unify singularity theory methods with semigroup invariants, enabling topological computation of the genus and providing insight into when symmetry occurs, with explicit examples. Overall, the work advances the program connecting numerical semigroups to surface singularities and offers practical tools for computing genus and assessing symmetry from purely topological data.

Abstract

In this article, first we give two formulae for the delta invariant of a complex curve singularity that can be embedded as a ${\mathbb Q}$-Cartier divisor in a normal surface singularity with rational homology sphere link. Next, we consider representable numerical semigroups, they are semigroups associated with normal weighted homogeneous surface singularities with rational homology sphere links (via the degrees of the homogeneous functions). We then prove that such a semigroup can be interpreted as the value semigroup of a generic orbit (as a curve singularity) given by the $\mathbb{C}^*$-action on the weighted homogeneous germ. Furthermore, we use the delta invariant formula to derive a combinatorially computable formula for the genus of representable semigroups. Finally, we characterize topologically those representable semigroups which are symmetric.

Delta invariant of $\mathbb{Q}$-Cartier curve germs and the genus of representable numerical semigroups

TL;DR

The paper develops two delta-invariant formulae for a curve germ embedded as a -Cartier divisor in a normal surface singularity with a -homology sphere link, linking analytic and topological data through equivariant genera. It then relates representable numerical semigroups to the value semigroups of generic -orbits on weighted homogeneous surface singularities, producing a combinatorially computable genus formula in terms of resolution data. A duality for equivariant geometric genus is established, and the genus formula is used to characterize when representable semigroups are symmetric, expressed via Seifert invariants and delta-type invariants. The results unify singularity theory methods with semigroup invariants, enabling topological computation of the genus and providing insight into when symmetry occurs, with explicit examples. Overall, the work advances the program connecting numerical semigroups to surface singularities and offers practical tools for computing genus and assessing symmetry from purely topological data.

Abstract

In this article, first we give two formulae for the delta invariant of a complex curve singularity that can be embedded as a -Cartier divisor in a normal surface singularity with rational homology sphere link. Next, we consider representable numerical semigroups, they are semigroups associated with normal weighted homogeneous surface singularities with rational homology sphere links (via the degrees of the homogeneous functions). We then prove that such a semigroup can be interpreted as the value semigroup of a generic orbit (as a curve singularity) given by the -action on the weighted homogeneous germ. Furthermore, we use the delta invariant formula to derive a combinatorially computable formula for the genus of representable semigroups. Finally, we characterize topologically those representable semigroups which are symmetric.

Paper Structure

This paper contains 17 sections, 9 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices associated with a resolution
  4. Minimal H--representatives and the generalized Laufer algorithm
  5. Minimal H--representatives and the generalized Laufer algorithm
  6. The Riemann-Roch function
  7. Universal abelian cover and the equivariant geometric genus
  8. Exact sequences and vanishing theorems
  9. The delta invariant of a $\mathbb{Q}$-Cartier curve
  10. The main formulae
  11. Example
  12. Generic orbits and representable semigroups of weighted homogeneous germs
  13. Preliminaries on representable semigroups
  14. Notations and some preliminary facts
  15. The generic orbit curve and its semigroup
  16. ...and 2 more sections

Key Result

Theorem 2.1

For any $(X,0)$ (even if $\Sigma$ is not a $\mathbb{Q}HS^3$) and any $\mathcal{L}\in {\rm Pic}(\tilde{X})$ with $-c_1({\mathcal{L}})\in Z_K+ {\mathcal{S}}'$ one has $h^1({\tilde{X}},{\mathcal{L}})=0$.

Theorems & Definitions (18)

  • Theorem 2.1: Grauert--Riemenschneider vanishing GrRieLaufer-rationalRam
  • Theorem 2.2: Generalized Grauert--Riemenschneider vanishing EFBook
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.1
  • ...and 8 more