J. I. García-García, D. Marín-Aragón, F. Torres, A. Vigneron-Tenorio
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.
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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.
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.
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。
We investigate the number of Galois Weierstrass points whose Weierstrass semigroups are generated by two positive integers.
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。
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…
We show that three numerical semigroups <5,6,7,8>, <3,7,8 > and <3,5> are of double covering type, i.e., the Weierstrass semigroups of ramification points on double covers of curves. Combining this with the results of Oliveira-Pimentel and Komeda we can determine the Weierstrass semigroups of the ramification points on double covers of genus two curves.
Let C be a complete non-singular irreducible curve of genus 4 over an algebraically closed field of characteristic 0. We determine all possible Weierstrass semigroups of ramification points on double covers of C which have genus greater than 11. Moreover, we construct double covers with ramification points whose Weierstrass semigroups are the possible ones.
In this paper, we determine the Weierstrass semigroup $H(P_{\infty})$ and the full automorphism group of a certain family of curves $\mathcal{X}_{n,r}$, which was recently introduced by Borges and Conceição.
We define a class of numerical semigroups S, which we call Castelnuovo semigroups, and study the subvariety $M^S_{g,1}$ of $M_{g,1}$ consisting of marked smooth curves with Weierstrass semigroup S. We determine the number of irreducible components of these loci and determine their dimensions. Curves with these Weierstrass semigroups are always Castelnuovo curves, which provides the basic tool for our argument. This analysis provides examples of numerical semigroups for which $M^S_{g,1}$ is reducible and non-equidimensional.
We present an explicit method to produce upper bounds for the dimension of the moduli spaces of complete integral Gorenstein curves with prescribed symmetric Weierstrass semigroups.
This paper has been withdrawn by the authors, due a gap in the proof of the main Theorem.
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。
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.
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。
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…
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.
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。
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…
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.
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.
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.
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.
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。
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.
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。
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。
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。
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.
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…
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 …
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.
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…
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。
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.
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…
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…
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。
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…
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…
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…
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…
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}.
On the Maximality, Weierstrass Semigroups, and Automorphism Group of the Curve $Y^{q+1} = X^n(X^n + 1)$
研究有限域 $\mathbb{F}_{q^2}$ 上曲线 $y^{q+1} = x^n(x^n+1)$，证明当 $q$ 为奇数时其非奇异模型是 $\mathbb{F}_{q^2}$-最大的当且仅当 $n \mid (q+1)$，并确定了该族最大曲线的 Weierstrass 半群和自同构群。
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,…
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。
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…
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…
In the 80's D. Eisenbud and J. Harris developed the general theory of limit linear series, Invent. math. 85 (1986), in order to understand what happens to linear systems and their ramification points on families of non-singular curves degenerating to curves of compact type. They applied their theory to the study of limits of Weierstrass points, among other endeavours. In one of their articles, Invent. math. 87 (1987), they asked: "What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type?" In this eprint we address this question within a …
For applications in algebraic geometric codes, an explicit description of bases of Riemann-Roch spaces of divisors on function fields over finite fields is needed. We give an algorithm to compute such bases for one point divisors, and Weierstrass semigroups over an optimal tower of function fields. We also explicitly compute Weierstrass semigroups till level eight.
Local Complete Intersections and Weierstrass Points
本文证明了在局部完全交条件下，具有对称Weierstrass半群的完全积分Gorenstein曲线的模空间是加权射影空间，且适用于正特征域。
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.
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…
We determine de Weierstrass semigroup of a pair of certain rational points on the GK-curves. We use this semigroup to obtain two-point AG codes with better parameters than comparable one-point AG codes arising from these curves. These parameters are new records in the MinT's tables.
On Kummer extensions with one place at infinity
本文显式描述了 Kummer 扩展曲线在唯一无穷远点处的 Weierstrass 半群，确定了其 Frobenius 数与重数，讨论了对称性条件，并刻画了极大 Castle 曲线。
We present an explicit construction of a compactification of the locus of smooth curves whose symmetric Weierstrass semigroup at a marked point is odd. The construction is an extension of Stoehr's techniques using Pinkham'sequivariant deformation of monomial curves by exploring syzygies. As an application we prove the rationality of the locus for genus at most six.
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