Weierstrass Semigroup: Structure & Applications

Updated 11 December 2025

Weierstrass semigroup is the set of possible pole orders at a point on an algebraic curve, defined by meromorphic functions with gaps reflecting the curve’s genus.
It connects algebraic geometry, number theory, and singularity theory by analyzing properties like multiplicity, Frobenius number, and effective weight.
Applications include constructing algebraic–geometry codes and studying moduli spaces, with explicit descriptions found in Kummer and cyclic covers.

A Weierstrass semigroup encodes the pole order structure of meromorphic functions at one or several points on an algebraic curve, providing an invariant connecting algebraic geometry, number theory, and singularity theory. Its paper illuminates the structure of algebraic curves, moduli of pointed curves, and coding theory via algebraic–geometry codes, as well as deep connections to the realizability and distribution of numerical semigroups.

1. Definition and Foundational Properties

Let $C$ be a smooth, projective curve of genus $g$ over a field $k$ , and $P\in C$ a $k$ -rational point. The (single-point) Weierstrass semigroup at $P$ is

$H(P) = \{ n\in\mathbb{N}_0 ~|~ \exists f\in k(C)^\times,~ (f)_\infty = nP \},$

the set of possible pole orders at $P$ of functions regular away from $P$ (Kaplan et al., 2012). The set of positive integers not in $H(P)$ is called the set of gaps at $P$ , and is always of cardinality $g$ . A numerical semigroup $S$ is an additive submonoid of $\mathbb{N}_0$ with finite complement; $S$ is Weierstrass if $S = H(P)$ for some $(C,P)$ .

For $m$ distinct points $(P_1,\ldots,P_m)$ , the (generalized) multipoint Weierstrass semigroup is

$H(P_1,\ldots,P_m) = \left\{ (n_1,...,n_m)\in\mathbb{N}_0^m ~|~ \exists f\in k(C)^\times\;\text{with}\; (f)_\infty = \sum_{j=1}^m n_j P_j \right\} .$

The set of gaps and pure gaps are defined analogously, via vanishing of Riemann–Roch dimensions (Castellanos et al., 16 Apr 2025, Moyano-Fernández et al., 2017).

Combinatorial Structure

The set $H(P_1,\ldots,P_m)$ is a subsemigroup of $(\mathbb{N}_0^m, +)$ , closed under coordinatewise maximum (lub) (Montanucci et al., 2021).
For one-point semigroups, the multiplicity $m$ is the minimal positive generator, the Frobenius number $F$ is the largest gap, and the genus is the number of gaps (Kaplan et al., 2012).

2. Realizability and Distribution of Weierstrass Semigroups

Necessary and Sufficient Criteria

Buchweitz's necessary criterion: If $S$ is a numerical semigroup of genus $g$ and the $n$ -fold sumset of its gaps satisfies $|n\cdot H(S)| > (2n-1)(g-1)$ for some $n>1$ , then $S$ is not Weierstrass (Kaplan et al., 2012).
Eisenbud–Harris sufficient criterion: If $S$ has $F<2m$ and weight $W(S)=\sum_{a\in H(S)} a - \sum_{i=0}^{g-1} i < g-1$ , then $S$ is Weierstrass (Kaplan et al., 2012).

Proportion and Asymptotics

Both Buchweitz's and Eisenbud–Harris's criteria cover $o(1)$ of all numerical semigroups as $g\to\infty$ , i.e., the density of semigroups excluded/realized by these criteria tends to zero (Kaplan et al., 2012).
The total density of Weierstrass semigroups among all numerical semigroups remains undetermined: it is sandwiched between $0$ and $1$, but not known to be positive or zero (Kaplan et al., 2012).

Typical Structure

For large genus, almost all numerical semigroups have Frobenius number $F$ close to $2m$, and multiplicity $m$ large (Kaplan et al., 2012).
Typical gaps and Apéry sets display sharp limit laws (local limit theorems) (Kaplan et al., 2012).

3. Explicit Descriptions in Key Geometric Contexts

Kummer Extensions and Maximal Curves

Let $F/\mathbb{F}_q(x)$ be a Kummer extension $y^m = f(x)$ , with ramified rational places $Q_1, \ldots, Q_n$ .

The minimal generating set of $H(Q_1,\ldots,Q_n)$ is given by explicit formulas involving the ramification indices and arithmetic of the Kummer or superelliptic cover (Yang et al., 2016, Castellanos et al., 16 Apr 2025).
Maximal (absolute and relative) elements are characterized by combinatorial discrepancy conditions, and all elements are generated by lub's of minimals (Castellanos et al., 16 Apr 2025).
For maximal curves $X_{a,b,n,1}$ or $Y_{n,1}$ admitting many totally ramified rational points, the structure of $H(P_\infty, P_1, ..., P_m)$ is described by a finite minimal set and period lattice (Montanucci et al., 2021, Castellanos et al., 2021).

Generalized/Multipoint Semigroups

The classical (nonnegative) semigroup is obtained as $H(Q_1,\ldots,Q_m) = \widehat{H}(Q_1,\ldots,Q_m)\cap \mathbb{N}^m$ .
All elements are generated as lub's of finitely many minimals modulo a period lattice (Montanucci et al., 2021, Moyano-Fernández et al., 2017).

Points	Minimal Generator Formula	Range Restrictions
2	$(mk_1+j,\;mk_2+j)$	$1\le j \le m-1-\lfloor m/r\rfloor$ , $k_1+k_2\! =\! r-2-\lfloor\frac{rj}m\rfloor$
$l$	$(mk_1+j,\dots,mk_l+j)$	$1 \le j \le m-1-\lfloor m/r \rfloor$ , $\sum k_i = r-l-\lfloor rj/m \rfloor$
$\infty$ , $l$	$(mk_0 - rj,\;mk_1+j, \ldots)$	$k_0 \ge \lceil rj/m \rceil$ , $\sum k_i = r-l$

4. Realization in Geometric and Group-Theoretic Constructions

Double Covers and Cyclic Covers

Classification results identify all numerical semigroups arising at ramification points of double covers of genus 2 curves, using a combinatorial criterion involving the composition of even gaps and additional elements (Harui et al., 2013).
More generally, for cyclic covers $\pi: C \to B$ of a hyperelliptic curve $B$ (ramified over reduced divisors), semigroup realizability is governed by normed residue classes, a cohomological criterion for line bundles, and translation to multiplication profiles in $\mathrm{Jac}(B)$ (Cotterill et al., 2021).
For symmetric non-hyperelliptic semigroups, realization via deformation of monomial curves or as ramification in Kummer/cyclic extensions is characterized (Komeda et al., 2016, Contiero et al., 2013).

Castelnuovo Curves

For Castelnuovo semigroups (“interval-generated” $S_{r,d}$ ), the strata in the moduli space of pointed curves are described combinatorially and are reducible in some cases (Pflueger, 2016).
The semigroup encodes the embedding type, contact orders, and determines the arithmetic genus, effective weight, and geometry of the corresponding embeddings.

5. Applications and Invariants

Moduli and Effective Weight

The moduli space $\mathcal{M}_{g,1}(S)$ of pointed curves with fixed Weierstrass semigroup $S$ has codimension bounded above by the effective weight $w_{\mathrm{eff}}(S)$ , which refines the classical weight by only counting generator–gap pairs (Pflueger, 2016):

$w_{\mathrm{eff}}(S) = \sum_{b\in \text{gaps}} |\{a\in \text{generators} : a < b\}|$

There are sharp bounds on the dimension of $\mathcal{M}_{g,1}(S)$ given by deformation theory (Deligne–Greuel, Pinkham), and the minimum is attained for infinite families of symmetric multiplicity six semigroups (Contiero et al., 2021).

Coding Theory

Pure gaps at several points are used to construct algebraic–geometry codes with parameters exceeding classical Goppa bounds, notably over maximal curves not covered by Hermitian (Castellanos et al., 2021, Yang et al., 2016).
Explicit Apéry sets (minimal representatives of residue classes modulo the multiplicity) are key for computing code parameters (Beelen et al., 2020).

Divisors, Riemann–Roch, and Poincaré Series

The structure of $H(Q_1, ..., Q_m)$ governs the growth of Riemann–Roch spaces $L(D)$ for divisors supported at given points (Moyano-Fernández et al., 2017).
The Poincaré series associated to the multi-filtration by pole order vectors completely determines the semigroup (Moyano-Fernández et al., 2017).

6. Open Problems and Future Directions

The density of Weierstrass semigroups in the set of numerical semigroups is unknown; neither a positive nor zero limit has been rejected by current evidence (Kaplan et al., 2012).
New families of semigroups realized by geometric constructions—beyond classical (hyperelliptic, trigonal, Castelnuovo, AG-codes from Hermitian or maximal curves)—are being explored via cyclic covers, particularly with prescribed multiplication profiles in Jacobians (Cotterill et al., 2021).
The characterization of gap and pure gap sets, explicit minimal sets, and the arithmetic of Apéry-like invariants in classical and generalized settings remain areas of active research (Montanucci et al., 2021, Castellanos et al., 16 Apr 2025).
Interaction with automorphism groups, modular forms, and the computation of theta constants continues to yield new structural and computational insights (Beelen et al., 29 Apr 2024, Komeda et al., 2016).

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