The Weierstrass semigroup of a pair of GaloisWeierstrass points with prime degree on a curve

We describe the Weierstrass semigroup of a Galois Weierstrass point with prime degree and the Weierstrass semigroup of a pair of Galois Weierstrass points with prime degree, where a Galois Weierstrass point with degree n means a total ramification point of a cyclic covering of the projective line of degree n.*Supported by Korea Research Foundation Grant (KRF-2003-041-C20010).**Partially supported by Grant-in-Aid for Scientific Research (15540051), JSPS.     

The Weierstrass semigroup of a pair and moduli inM 3

We classify all the Weierstrass semigroups of a pair of points on a curve of genus 3, by using its canonical model in the plane. Moreover, we count the dimension of the moduli of curves which have a pair of points with a specified Weierstrass semigroup.This work has been supported by the Japan Society for the Promotion of Science and the Korea Science and Engineering Foundation (Project No. 976-0100-001-2). Also the first author is partially supported by Korea Research Foundation Grant (KRF-99-005-D00003).     

Special Pairs in the Generating Subset of the Weierstrass Semigroup at a Pair

We discuss the structure of the Weierstrass semigroup at a pair of points on an algebraic curve. It is known that the Weierstrass semigroup at a pair (P, Q) contains the unique generating subset (P, Q). We find some characterizations of the elements of (P, Q) and prove that, for any point P on a curve, (P, Q) consists of only maximal elements for all except for finitely many points QP on the given curve. Also we obtain more results concerning special and nonspecial pairs.     

Two-point AG codes from the Beelen-Montanucci maximal curve

In this paper we investigate two-point algebraic-geometry codes (AG codes) coming from the Beelen-Montanucci (BM) maximal curve. We study properties of certain two-point Weierstrass semigroups of the curve and use them for determining a lower bound on the minimum distance of such codes. AG codes with better parameters with respect to comparable two-point codes from the Garcia-Güneri-Stichtenoth (GGS) curve are discovered.     

Existence of the non-primitive Weierstrass gap sequences on curves of genus 8

We show that for any possible Weierstrass gap sequence L on a non-singular curve of genus 8 with twice the smallest positive non-gap is less than the largest gap there exists a pointed non-singular curve (C, P) over an algebraically closed field of characteristic 0 such that the Weierstrass gap sequence at P is L. Combining this with the result in [6] we see that every possible Weierstrass gap sequence of genus 8 is attained by some pointed non-singular curve. *Partially supported by Grant-in-Aid for Scientific Research (17540046), Japan Society for the Promotion of Science. **Partially supported by Grant-in-Aid for Scientific Research (17540030), Japan Society for the Promotion of Science.     

Codes from curves with total inflection points

The concept of pure gaps of a Weierstrass semigroup at several points of an algebraic curve has been used lately to obtain codes that have a lower bound for the minimum distance which is greater than the Goppa bound. In this work, we show that the existence of total inflection points on a smooth plane curve determines the existence of pure gaps in certain Weierstrass semigroups. We then apply our results to the Hermitian curve and construct codes supported on several points that compare better to one-point codes from that same curve.      

Gorenstein Curves with Quasi-Symmetric Weierstrass Semigroups

We consider a canonical Gorenstein curve C of arithmetic genus g in P g-1 (K), that admits a non-singular point P, whose Weierstrass semigroup is quasi-symmetric in the sense that the last gap is equal to 2g-2. By making local considerations at the point P and the second point of the curve C on its osculating hyperplane at P we construct monomial bases for the spaces of higher order regular differentials. We give an irreducibility criterion for the canonical curve in terms of the coefficients of the quadratic relations. We also realize each quasi-symmetric numerical semigroup as the Weierstrass semigroup of a reducible canonical Gorenstein curve, but we give examples of such semigroups that cannot be realized as Weierstrass semigroups of smooth curves.     

Intersection Divisors of a Canonically Embedded Curve with its Osculating Spaces

We present a new result on the geometry of nonhyperelliptic curves; namely, the intersection divisors of a canonically embedded curve C with its osculating spaces at a point P, not considering the intersection at P, can only vary in dimensions given by the Weierstrass semigroup of the curve C at P. We obtain, under a reasonable geometrical hypothesis, monomial bases for the spaces of higher-order regular differentials. We also give a sufficient condition on the Weierstrass semigroup of C at P in order for this geometrical hypothesis to be true. Finally, we give examples of Weierstrass semigroups satisfying this condition.     

On Weierstrass semigroups and sets: a review with new results

In this work we present a survey of the main results in the theory of Weierstrass semigroups at several points, with special attention to the determination of bounds for the cardinality of its set of gaps. We also review results on applications to the theory of error correcting codes. We then recall a generalization of the concept of Weierstrass semigroup, which is the Weierstrass set associated to a linear system and several points. We finish by presenting new results on this Weierstrass set, including some on the cardinality of its set of gaps.      

Curves covered by the Hermitian curve

A family of maximal curves is investigated that are all quotients of the Hermitian curve. These curves provide examples of curves with the same genus, the same automorphism group and the same Weierstrass semigroup at a generic point, but that are not isomorphic.     

Pure Weierstrass gaps from a quotient of the Hermitian curve

In this paper, by employing some results on Kummer extensions, we give an arithmetic characterization of pure Weierstrass gaps at many totally ramified places on a quotient of the Hermitian curve, including the well-studied Hermitian curve as a special case. The cardinality of these pure gaps is explicitly investigated. In particular, the numbers of gaps and pure gaps at a pair of distinct places are determined precisely, which can be regarded as an extension of the previous work by Matthews (2001) considered Hermitian curves. Additionally, some concrete examples are provided to illustrate our results.     

Weierstrass Semigroups and Codes from a Quotient of the Hermitian Curve

We consider the quotient of the Hermitian curve defined by the equation y^q + y = x^m over where m > 2 is a divisor of q+1. For 2≤ r ≤ q+1, we determine the Weierstrass semigroup of any r-tuple of -rational points on this curve. Using these semigroups, we construct algebraic geometry codes with minimum distance exceeding the designed distance. In addition, we prove that there are r-point codes, that is codes of the form where r ≥ 2, with better parameters than any comparable one-point code on the same curve. Some of these codes have better parameters than comparable one-point Hermitian codes over the same field. All of our results apply to the Hermitian curve itself which is obtained by taking m=q +1 in the above equation Communicated by: J.W.P. Hirschfeld     

Weierstrass Pairs and Minimum Distance of Goppa Codes

We prove that elements of the Weierstrassgap set of a pair of points may be used to define a geometricGoppa code which has minimum distance greater than the usuallower bound. We determine the Weierstrass gap set of a pair ofany two Weierstrass points on a Hermitian curve and use thisto increase the lower bound on the minimum distance of particularcodes defined using a linear combination of the two points.     

On Weierstrass semigroups of double covering of genus two curves

In this work we prove that are Weierstrass semigroups all numerical semigroups whose three first positive non-gaps are 6, 8 and 10, resolving the problem of the numerical semigroups that appear as Weierstrass semigroups in double coverings of genus two curves.     

The arithmetic extensions of a numerical semigroup

Abstract

In this paper we introduce the notion of extension of a numerical semigroup. We provide a characterization of the numerical semigroups whose extensions are all arithmetic and we give an algorithm for the computation of the whole set of arithmetic extension of a given numerical semigroup. As by-product, new explicit formulas for the Frobenius number and the genus of proportionally modular semigroups are obtained.     

Weierstrass Multiplicative Points on a Compact Riemann Surface

We introduce Weierstrass multiplicative points and develop the theory of Weierstrass multiplicative points for multiplicative meromorphic functions and Prym differentials on a compact Riemann surface. We prove some analogs of the Weierstrass and Noether theorems on the gaps of multiplicative functions. We obtain two-sided estimates for the number of Weierstrass multiplicative points and q-points. We propose a method for studying the Weierstrass and Noether gaps and Weierstrass multiplicative points by means of filtrations in the Jacobi variety of a compact Riemann surface.     

On numerical semigroups related to covering of curves

We investigate arithmetical properties of a class of semigroups that includesthose appearing as Weierstrass semigroups at totally ramified points of coveringof curves.     

Weierstrass semigroups whose minimum positive integers are even

We consider three subsets of the set of 2n-semigroups, where for a positive integer n a 2n-semigroup means a numerical semigroup whose minimum positive integer is 2n. These three subsets are obtained by the Weierstrass semigroups of total ramification points on a cyclic covering of the projective line, the Weierstrass semigroups of ramification points on a double covering of a non-singular curve and the Weierstrass semigroups of points on a non-singular curve. We show that the three subsets are different for n ≧ 3. Partially supported by Grant-in-Aid for Scientific Research (17540046), Japan Society for the Promotion of Science. Received: 19 June 2006     

Semigroup closures of finite rank symmetric inverse semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations I _λ ⁿ of the rank ≤ n is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.