Non Weierstrass Numerical Semigroups of Genus g >= 10

On the structure of numerical sparse semigroups and applications to Weierstrass points

In this work, we are concerned with the structure of sparse semigroups and some applications of them to Weierstrass points. We manage to describe, classify and find an upper bound for the genus of sparse semigroups. We also study the realization of some sparse semigroups as Weierstrass semigroups. The smoothness property of monomial curves associated to (hyper)ordinary semigroups presented by Pinkham and Rim-Vitulli, and the results on double covering of curves by Torres are crucial in this.

一类非Weierstrass半群

Minimal Non-Weierstrass Semigroups

本文利用合冲理论提出新方法，证明某些数值半群不是Weierstrass半群，包括第一个重数6（可能最小）且亏格13（已知最小）的例子，并给出其他适用例子。

Weierstrass semigroups on the Skabelund maximal curve

In 2017, D. Skabelund constructed a maximal curve over $\mathbb{F}_{q^4}$ as a cyclic cover of the Suzuki curve. In this paper we explicitly determine the structure of the Weierstrass semigroup at any point $P$ of the Skabelund curve. We show that its Weierstrass points are precisely the $\mathbb{F}_{q^4}$-rational points. Also we show that among the Weierstrass points, two types of Weierstrass semigroup occur: one for the $\mathbb{F}_q$-rational points, one for the remaining $\mathbb{F}_{q^4}$-rational points. For each of these two types its Apéry set is computed as well as a set of generator…

Generalized Weierstrass semigroups at several points on certain maximal curves which cannot be covered by the Hermitian curve

In this paper we determine the generalized Weierstrass semigroup $ \widehat{H}(P_{\infty}, P_1, \ldots , P_{m})$, and consequently the Weierstrass semigroup $H(P_{\infty}, P_1, \ldots , P_{m})$, at $m+1$ points on the curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$. These curves has been introduced in Tafazolian et al. as new examples of curves which cannot be covered by the Hermitian curve.

广义Weierstrass半群及其Poincaré级数

Generalized Weierstrass semigroups and their Poincaré series

该研究针对有限域上曲线多点处的广义Weierstrass半群展开，给出半群结构描述以推导除子与Riemann-Roch空间的算术性质，并证明其Poincaré级数可完整描述对应半群。

The uniqueness of Weierstrass points with semigroup <a;b> and related subgroups

Assume $a$ and $b=na+r$ with $n \geq 1$ and $0<r<a$ are relatively prime integers. In case $C$ is a smooth curve and $P$ is a point on $C$ with Weierstrass semigroup equal to $<a;b>$ then $C$ is called a $C_{a;b}$-curve. In case $r \neq a-1$ and $b \neq a+1$ we prove $C$ has no other point $Q \neq P$ having Weierstrass semigroup equal to $<a;b>$. We say the Weierstrass semigroup $<a;b>$ occurs at most once. The curve $C_{a;b}$ has genus $(a-1)(b-1)/2$ and the result is generalized to genus $g<(a-1)(b-1)/2$. We obtain a lower bound on $g$ (sharp in many cases) such that all Weierstrass semigrou…

Weierstrass Semigroup and Pure Gaps at several points on the $GK$ curve

We determine the Weierstrass semigroup $H(P_{\infty}, P_{1}, \ldots , P_{m})$ at several points on the $GK$ curve. In addition, we present conditions to find pure gaps on the set of gaps $G(P_{\infty}, P_{1}, \ldots , P_{m})$. Finally, we apply the results to obtain AG codes with good relative parameters.

Triples of rational points on the Hermitian curve and their Weierstrass semigroups

In this paper, we study configurations of three rational points on the Hermitian curve over $\mathbb{F}_{q^2}$ and classify them according to their Weierstrass semigroups. For $q>3$, we show that the number of distinct semigroups of this form is equal to the number of positive divisors of $q+1$ and give an explicit description of the Weierstrass semigroup for each triple of points studied. To do so, we make use of two-point discrepancies and derive a criterion which applies to arbitrary curves over a finite field.

Irreducibility and stable rationality of the loci of Weierstrass points on curves of genus at most six

We show that for any numerical semigroup H of genus g at most 6, the locus of Weierstrass points on curves of genus g with Weierstrass semigroup H is irreducible and that for all but possibly two semigroups it is stably rational.

On multi-index filtrations associated to Weierstrass semigroups

The aim of this paper is to review the main techniques in the computation of Weierstraßsemigroup at several points of curves defined over perfect fields, with special emphasis on the case of two points. Some hints about the usage of some packages of the computer algebra software \textsc{Singular} are also given.

On reducible non-Weierstrass semigroups

Weierstrass semigroups are well-known along the literature. We present a new family of non-Weierstrass semigroups which can be written as an intersection of Weierstrass semigroups. In addition, we provide methods for calculating non-Weierstrass semigroups with genus as large as desired.

Kummer扩展中一阶完全分歧点的Weierstrass半群

Weierstrass semigroups at totally ramified places of degree one on Kummer extensions

本文研究Kummer扩展中一阶完全分歧点的Weierstrass半群，给出间隙集的显式描述，并探讨其对称性条件及最小生成集，最后应用于GGS曲线和BM曲线子覆盖。

Gap sequences of 1-Weierstrass points on non-hyperelliptic curves of genus 10

In this paper, we compute the 1-gap sequences of 1-Weierstrass points on non-hyperelliptic smooth projective curves of genus 10. Furthermore, the geometry of such points is classified as flexes, sextactic and tentactic points. Also, an upper bounds for their numbers are estimated.

The 2-Weierstrass points of genus 3 hyperelliptic curves with extra automorphisms

For each group $G$, $(|G| > 2)$ \, which acts as a full automorphism group on a genus 3 hyperelliptic curve, we determine the family of curves which have 2-Weierstrass points. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. The 1-dimensional families that we discover have the property that they contain only genus 0 components.

关于函数域$\mathbb{F}_q(x)$的任意Kummer扩张中的广义Weierstrass半群

On generalized Weierstrass Semigroups in arbitrary Kummer extensions of $\mathbb{F}_q(x)$

本文研究函数域$\mathbb{F}_q(x)$任意Kummer扩张中广义Weierstrass半群的结构、性质及极大元，给出极大元集合的明确描述并应用于特定极大曲线，推广统一了早期成果，为半群与函数域关系提供新视角。

On Weierstrass semigroup at $m$ points on curves of the type f(y) = g(x)

In this work we determine the so-called minimal generating set of the Weierstrass semigroup of certain $m$ points on curves $\mathcal{X}$ with plane model of the type $f(y) = g(x)$ over $\mathbb{F}_{q}$, where $f(T),g(T)\in \mathbb{F}_q[T]$. Our results were obtained using the concept of discrepancy, for given points $P$ and $Q$ on $\mathcal{X}$. This concept was introduced by Duursma and Park, in \cite{duursma}, and allows us to make a different and more general approach than that used to certain specific curves studied earlier.

Weierstrass semigroups on the Giulietti-Korchmáros curve

In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Giulietti-Korchmáros curve $\mathcal{X}$. We show that as the point varies, exactly three possibilities arise: One for the $\mathbb{F}_{q^2}$-rational points (already known in the literature), one for the $\mathbb{F}_{q^6} \setminus \mathbb{F}_{q^2}$-rational points, and one for all remaining points. As a result, we prove a conjecture concerning the structure of $H(P)$ in case $P$ is an $\mathbb{F}_{q^6} \setminus \mathbb{F}_{q^2}$-rational point. As a corollary we also obtain th…

Generalized Weierstrass semigroups and Riemann-Roch spaces for certain curves with separated variables

In this work we study the generalized Weierstrass semigroup $\widehat{H} (\mathbf{P}_m)$ at an $m$-tuple $\mathbf{P}_m = (P_{1}, \ldots , P_{m})$ of rational points on certain curves admitting a plane model of the form $f(y) = g(x)$ over $\mathbb{F}_{q}$, where ${f(T),g(T)\in \mathbb{F}_q[T]}$. In particular, we compute the generating set $\widehatΓ(\mathbf{P}_m)$ of $\widehat{H} (\mathbf{P}_m)$ and, as a consequence, we explicit a basis for Riemann-Roch spaces of divisors with support in $\{P_{1}, \ldots , P_{m}\}$ on these curves, generalizing results of Maharaj, Matthews, and Pirsic.

Automorphisms of Curves and Weierstrass semigroups for Harbater-Katz-Gabber covers

We study $p$-group Galois covers $X \rightarrow \mathbb{P}^1$ with only one fully ramified point. These covers are important because of the Katz-Gabber compactification of Galois actions on complete local rings. The sequence of ramification jumps is related to the Weierstrass semigroup of the global cover at the stabilized point. We determine explicitly the jumps of the ramification filtrations in terms of pole numbers. We give applications for curves with zero $p$--rank: we focus on maximal curves and curves that admit a big action. Moreover the Galois module structure of polydifferentials is…

线性化函数域中的广义Weierstrass半群

On generalized Weierstrass semigroups in linearized function fields

利用差异概念，研究线性化函数域中由n个一次全分歧点构成的n元组Q的广义Weierstrass半群，刻画并显式确定绝对最大元素集和相对最大元素集，推广了现有结果。

Minimal Weierstrass equations for genus 2 curves

We study the minimal Weierstrass equations for genus 2 curves defined over a ring of integers $\mathcal O_{\mathbb F}$. This is done via reduction theory and Julia invariant of binary sextics. We show that when the binary sextics has extra automorphisms this is usually easier to compute. Moreover, we show that when the curve is given in the standard form $y^2=f(x^2)$, where $f(x)$ is a monic polynomial, $f(0)=1$ which is defined over $\mathcal O_{\mathbb F}$ then this form is reduced.

On the dimension of the stratum of the moduli of pointed curves by Weierstrass gaps

The dimension of the moduli space of smooth pointed curves with prescribed Weierstrass semigroup at the market point is computed for three families of symmetric semigroups of multiplicity six. We also collect the dimensions of such moduli spaces for all semigroups of genus not greater than seven. A question related to an improvement of a Deligne--Pinkham's bound is also formulated, suggesting that the positive graded part of the first module of the cotangent complex associated to a semigroup algebra is a missing invariant. The answer for this question is positive for all these moduli that we k…

函数域上的 Weierstrass 半群、纯间隙与码

Weierstrass Semigroup, Pure Gaps and Codes on Function Fields

该研究确定了 Kummer 扩张中一个和两个完全分歧位处的 Weierstrass 半群，刻画任意函数域两有理位的纯间隙，并用于构造多有理位函数域上的代数几何码。

New examples of Weierstrass semigroups associated with a double covering of a curve on a Hirzebruch surface of degree one

Let $φ:Σ_1\longrightarrow \mathbb{P}^2$ be a blow up at a point on $\mathbb{P}^2$. Let $C$ be the proper transform of a smooth plane curve of degree $d\geq 4$ by $φ$, and let $P$ be a point on $C$. Let $π:\tilde{C}\longrightarrow C$ be a double covering branched along the reduced divisor on $C$ obtained as the intersection of $C$ and a reduced divisor in $|-2K_{Σ_1}|$ containing $P$. In this paper, we investigate the Weierstrass semigroup $H(\tilde{P})$ at the ramification point $\tilde{P}$ of $π$ over $P$, in the case where the intersection multiplicity at $φ(P)$ of $φ(C)$ and the tangent lin…

The Proportion of Weierstrass Semigroups

We solve a problem of Komeda concerning the proportion of numerical semigroups which do not satisfy Buchweitz' necessary criterion for a semigroup to occur as the Weierstrass semigroup of a point on an algebraic curve. We also show that the family of semigroups known to be Weierstrass semigroups using a result of Eisenbud and Harris, has zero density in the set of all semigroups. In the process, we prove several more general results about the structure of a typical numerical semigroup.

Representations of cyclic groups in positive characteristic and Weierstrass semigroups

We study the $k[G]$-module structure of the space of holomorphic differentials of a curve defined over an algebraically closed field of positive characteristic, for a cyclic group $G$ of order $p^\ell n$. We also study the relation to the Weierstrass semigroup for the case of Galois Weierstrass points.

Classes of Weierstrass points on genus 2 curves

We study the codimension n locus of curves of genus 2 with n distinct marked Weierstrass points inside the moduli space of genus 2, n-pointed curves, for n <= 6. We give a recursive description of the classes of the closure of these loci inside the moduli space of stable curves. For n<= 4, we express these classes using a generating function over stable graphs indexing the boundary strata of moduli spaces of pointed stable curves. Similarly, we express the closure of these classes inside the moduli space of curves of compact type for all n. This is a first step in the study of the structure of…

On non-primitive Weierstrass points

We give an upper bound on the codimension in $M_{g,1}$ of the variety $M^S_{g,1}$ of marked curves $(C,p)$ with a given Weierstrass semigroup. The bound is a combinatorial quantity which we call the effective weight of the semigroup; it is a refinement of the weight of the semigroup, and differs from it precisely when the semigroup is not primitive. We prove that whenever the effective weight is less than g, the variety $M^S_{g,1}$ is nonempty and has a component of the predicted codimension. These results extend previous results of Eisenbud, Harris, and Komeda to the case of non-primitive sem…

Automorphisms of Curves and Weierstrass semigroups

The relation of the Weierstrass semigroup with several invariants of a curve is studied. For Galois covers of curves with group $G$ we introduce a new filtration of the group decomposition subgroup of $G$. The relation to the ramification filtration is investigated in the case of cyclic covers. We relate our results to invariants defined by Boseck and we study the one point ramification case. We also give applications to Hasse-Witt invariant and symmetric semigroups.

关于最大费马函数域的魏尔斯特拉斯半群

On Weierstrass semigroups of maximal Fermat function fields

本文研究了最大费马函数域在不同位置处的魏尔斯特拉斯半群，揭示了其复杂性和丰富的结构特性。

Two remarks on the Weierstrass flag

We show that the locally closed strata of the Weierstrass flags on the moduli spaces of curves of genus g and on the moduli space of curves of genus g with one marked point are almost never affine.

On the minimum distance of AG codes, on Weierstrass semigroups and the smoothability of certain monomial curves in 4-Space

In this paper we treat several topics regarding numerical Weierstrass semigroups and the theory of Algebraic Geometric Codes associated to a pair $(X, P)$, where $X$ is a projective curve defined over the algebraic closure of the finite field $F_q$ and P is a $F_q$-rational point of $X$. First we show how to evaluate the Feng-Rao Order Bound, which is a good estimation for the minimum distance of such codes. This bound is related to the classical Weierstrass semigroup of the curve $X$ at $P$. Further we focus our attention on the question to recognize the Weierstrass semigroups over fields of …

Extremality of loci of hyperelliptic curves with marked Weierstrass points

The locus of genus-two curves with n marked Weierstrass points has codimension n inside the moduli space of genus-two curves with n marked points, for n<=6. It is well known that the class of the closure of the divisor obtained for n=1 spans an extremal ray of the cone of effective divisor classes. We generalize this result for all n: we show that the class of the closure of the locus of genus-two curves with n marked Weierstrass points spans an extremal ray of the cone of effective classes of codimension n, for n<=6. A related construction produces extremal nef curve classes in moduli spaces …

具有第三大亏格的极大曲线的Weierstrass半群与自同构群

Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus

本文确定具有第三大亏格的极大曲线$\mathcal{X}_3$在所有点处的Weierstrass半群与完整自同构群，揭示其Weierstrass点异常丰富，并证明除小$q$外其自同构群完全继承自Hermitian曲线，为唯一性公开问题提供关键线索。

On certain N--sheeted coverings and numerical semigroups which cannot be realized as Weierstrass semigroups

A curve $X$ is said to be of type $(N,γ)$ if it is an $N$--sheeted covering of a curve of genus $γ$ with at least one totally ramified point. A numerical semigroup $H$ is said to be of type $(N,γ)$ if it has $γ$ positive multiples of $N$ in $[N,2Nγ]$ such that its $γ^{th}$ element is $2Nγ$ and $(2γ+1)N \in H$. If the genus of $X$ is large enough and $N$ is prime, $X$ is of type $(N,γ)$ if and only if there is a point $P \in X$ such that the Weierstrass semigroup at $P$ is of type $(N,γ)$ (this generalizes the case of double coverings of curves). Using the proof of this result and the Buchweitz…

Weierstrass semigroup at $m+1$ rational points in maximal curves which cannot be covered by the Hermitian curve

We determine the Weierstrass semigroup $H(P_\infty,P_1,\ldots,P_m)$ at several rational points on the maximal curves which cannot be covered by the Hermitian curve introduced by Tafazolian, Teherán-Herrera, and Torres. Furthermore, we present some conditions to find pure gaps. We use this semigroup to obtain AG codes with better relative parameters than comparable one-point AG codes arising from these curves.

Bounding the number of points on a curve using a generalization of Weierstrass semigroups

In this article we use techniques from coding theory to derive upper bounds for the number of rational places of the function field of an algebraic curve defined over a finite field. The used techniques yield upper bounds if the (generalized) Weierstrass semigroup [P. Beelen, N. Tutaş: A generalization of the Weierstrass semigroup, J. Pure Appl. Algebra, 207(2), 2006] for an $n$-tuple of places is known, even if the exact defining equation of the curve is not known. As shown in examples, this sometimes enables one to get an upper bound for the number of rational places for families of function…

Weierstrass semigroups at every point of the Suzuki curve

In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Suzuki curve $\mathcal{S}_q$. As the point $P$ varies, exactly two possibilities arise for $H(P)$: one for the $\mathbb{F}_q$-rational points (already known in the literature), and one for all remaining points. For this last case a minimal set of generators of $H(P)$ is also provided. As an application, we construct dual one-point codes from an $\mathbb{F}_{q^4}\setminus\fq$-point whose parameters are better in some cases than the ones constructed in a similar way from an $\fq$-r…

Remarks on numerical semigroups

We extend results on Weierstrass semigroups at ramified points of double covering of curves to any numerical semigroup whose genus is large enough. As an application we strengthen the properties concerning Weierstrass weights in \cited{To}.

Bounding the number of rational places using Weierstrass semigroups

Let Lambda be a numerical semigroup. Assume there exists an algebraic function field over GF(q) in one variable which possesses a rational place that has Lambda as its Weierstrass semigroup. We ask the question as to how many rational places such a function field can possibly have and we derive an upper bound in terms of the generators of Lambda and q. Our bound is an improvement to a bound by Lewittes which takes into account only the multiplicity of Lambda and q. From the new bound we derive significant improvements to Serre's upper bound in the cases q=2, 3 and 4. We finally show that Lewit…

超椭圆曲线循环覆盖的 Weierstrass 半群

Weierstrass semigroups from cyclic covers of hyperelliptic curves

本文研究了超椭圆曲线循环覆盖的 Weierstrass 半群可实现性问题，证明了其可实现性由超椭圆 Jacobi 簇中相关除子类在 $j$ 倍乘法下的行为决定。

第三大可能亏格的极大函数域在 $q \equiv 1 \pmod 3$ 时的 Weierstrass 半群与自同构群

Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 1 \pmod 3$

针对 q≡1 mod 3 时具有第三大亏格的极大函数域 Y_3，本文显式确定其所有位的 Weierstrass 半群与完整自同构群，发现其 Weierstrass 位结构异常丰富，但自同构群除小 q 外均继承自 Hermitian 函数域。

On nonnegatively graded Weierstrass points

We provide a new lower bound for the dimension of the moduli space of smooth pointed curves with prescribed Weierstrass semigroup at the marked point, derived from the Deligne-Greuel formula and Pinkham's equivariant deformation theory. Using Buchweitz's description of the first cohomology module of the cotangent complex for monomial curves, we show that our lower bound improves a recently one given by Pflueger. By allowing semigroups running over suitable families of symmetric semigroups of multiplicity six, we show that this new lower bound is attained, and that the corresponding moduli spac…

The Riemann constant for a non-symmetric Weierstrass semigroup

The zero divisor of the theta function of a compact Riemann surface $X$ of genus $g$ is the canonical theta divisor of Pic${}^{(g-1)}$ up to translation by the Riemann constant $Δ$ for a base point $P$ of $X$. The complement of the Weierstrass gaps at the base point $P$ given as a numerical semigroup plays an important role, which is called the Weierstrass semigroup. It is classically known that the Riemann constant $Δ$ is a half period $\frac{1}{2}Γ_τ$ for the Jacobi variety $\mathcal{J}(X)=\mathbb{C}^g/Γ_τ$ of $X$ if and only if the Weierstrass semigroup at $P$ is symmetric. In this article,…

度量图的几何 Weierstrass 点的枚举

Enumeration of geometric Weierstrass points of metric graphs

本文引入度量图的几何 Weierstrass 点概念，证明亏格为 $g$ 的一般度量图按重数计算有 $g^3-g$ 个此类点，并为亏格大于等于 $2$ 的度量图上 Weierstrass 点的存在性提供了新证明。

计算Weierstrass半群及奇异平面模型上的Feng-Rao距离

Computing Weierstrass semigroups and the Feng-Rao distance from singular plane models

本文提出一种算法，在P为曲线奇异平面模型唯一无穷远分支时，计算Weierstrass半群及函数，并导出L(mP)基、Feng-Rao距离，应用于解码。

The sigma function for Weierstrass semigroups <3,7,8> and <6,13,14,15,16>

Compact Riemann surfaces and their abelian functions are instrumental to solve integrable equations; more recently the representation theory of the Monster and related modular form have pointed to the relevance of $τ$-functions, which are in turn connected with a specific type of abelian function, the (Kleinian) $σ$-function. This paper proposes a construction of $σ$-functions based on the nature of the Weierstrass semigroup at one point of the Riemann surface as a generalization of the construction of plane affine models of the Riemann surface. Because our definition is algebraic, we are able…

Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups

We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painlevé irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields. Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem which have interesting consequences…

Separating semigroup of hyperelliptic curves and of genus 3 curves

A rational function on a real algebraic curve $C$ is called separating if it takes real values only at real points. Such a function defines a covering $\Bbb R C\to\Bbb{RP}^1$. Let $A_1,\dots,A_n$ be connected components of $C$. In a recent paper M. Kummer and K. Shaw defined the separating semigroup of $C$ as the set of all sequences $(d_1(f),\dots,d_n(f))$ where $f$ is a separating function and $d_i(f)$ is the degree of the restriction of $f$ to $A_i$. We describe the separating semigroup for hyperelliptic curves and for genus 3 curves.

具有第三大可能亏格的极大函数域的 Weierstrass 半群与自同构群，$q \equiv 0 \pmod 3$

Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 0 \pmod 3$

本文针对 $q\equiv0\pmod3$，显式确定第三大亏格的 $\mathbb{F}_{q^2}$-极大函数域 $Z_3$ 的各位 Weierstrass 半群与完整自同构群。

One- and Two-Point Codes over Kummer Extensions

We compute the Weierstrass semigroup at one totally ramified place for Kummer extensions defined by $y^m=f(x)^λ$ where $f(x)$ is a separable polynomial over $\mathbb{F}_q$. In addition, we compute the Weierstrass semigroup at two certain totally ramified places. We then apply our results to construct one- and two-point algebraic geometric codes with good parameters.

低亏格带点代数曲线模空间的维数

The Dimension of the Moduli Space of Pointed Algebraic Curves of Low Genus

本文针对亏格不超过7的多种情形，显式计算了以给定数值半群为Weierstrass半群的带点代数曲线的模空间，并确定了所有亏格7半群的维数。

On the Geil-Matsumoto Bound and the Length of AG codes

The Geil-Matsumoto bound conditions the number of rational places of a function field in terms of the Weierstrass semigroup of any of the places. Lewittes' bound preceded the Geil-Matsumoto bound and it only considers the smallest generator of the numerical semigroup. It can be derived from the Geil-Matsumoto bound and so it is weaker. However, for general semigroups the Geil-Matsumoto bound does not have a closed formula and it may be hard to compute, while Lewittes' bound is very simple. We give a closed formula for the Geil-Matsumoto bound for the case when the Weierstrass semigroup has two…

The Weierstrass subgroup of a curve has maximal rank

We show that the Weierstrass points of the generic curve of genus $g$ over an algebraically closed field of characteristic 0 generate a group of maximal rank in the Jacobian.