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.     

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     

Guarded and Banded Semigroups

The variety of guarded semigroups consists of all (S,·, ˉ) where (S,·) is a semigroup and x ↦ \overline{x} is a unary operation subject to four additional equations relating it to multiplication. The semigroup Pfn(X) of all partial transformations on X is a guarded semigroup if x \overline{f} = x when xf is defined and is undefined otherwise. Every guarded semigroup is a subalgebra of Pfn(X) for some X. A covering theorem of McAlister type is obtained. Free guarded semigroups are constructed paralleling Scheiblich's construction of free inverse semigroups. The variety of banded semigroups has the same signature but different equations. There is a canonical forgetful functor from guarded semigroups to banded semigroups. A semigroup underlies a banded semigroup if and only if it is a split strong semilattice of right zero semigroups. Each banded semigroup S contains a canonical subsemilattice g_⋆(S). For any given semilattice L, a construction to synthesize the general banded semigroup S with g_⋆ ≅ L is obtained.     

The ordinarization transform of a numerical semigroup and semigroups with a large number of intervals

A numerical semigroup is said to be ordinary if it has all its gaps in a row. Indeed, it contains zero and all integers from a given positive one. One can define a simple operation on a non-ordinary semigroup, which we call here the ordinarization transform, by removing its smallest non-zero non-gap (the multiplicity) and adding its largest gap (the Frobenius number). This gives another numerical semigroup and by repeating this transform several times we end up with an ordinary semigroup. The genus, that is, the number of gaps, is kept constant in all the transforms.This procedure allows the construction of a tree for each given genus containing all semigroups of that genus and rooted in the unique ordinary semigroup of that genus. We study here the regularity of these trees and the number of semigroups at each depth. For some depths it is proved that the number of semigroups increases with the genus and it is conjectured that this happens at all given depths. This may give some light to a former conjecture saying that the number of semigroups of a given genus increases with the genus.We finally give an identification between semigroups at a given depth in the ordinarization tree and semigroups with a given (large) number of gap intervals and we give an explicit characterization of those semigroups.     

Unary Semigroups with an Associate Subgroup

A subgroup H of a regular semigroup S is said to be an associate subgroup of S if for every s ∈ S, there is a unique associate of s in H. An idempotent z of S is said to be medial if czc = c, for every c product of idempotents of S. Blyth and Martins established a structure theorem for semigroups with an associate subgroup whose identity is a medial idempotent, in terms of an idempotent generated semigroup, a group and a single homomorphism. Here, we construct a system of axioms which characterize these semigroups in terms of a unary operation satisfying those axioms. As a generalization of this class of semigroups, we characterize regular semigroups S having a subgroup which is a transversal of a congruence on S.     

On Weierstrass semigroups of double coverings of genus three curves

Let C be a complete non-singular curve of genus 3 over an algebraically closed field of characteristic 0. We determine all possible Wierstrass semigroups of ramification points on double coverings of C whose covering curves have genus greater than 8. Moreover, we construct double coverings with ramification points whose Weierstrass semigroups are the possible ones.     

ON THE STRUCTURE OF REGULAR CRYPTO SEMIGROUPS

Superabundant semigroups are generalizations of completely regular semigroups written the class of abundant semigroups. It has been shown by Fountain that an abundant semigroup is superabundant if and only if it is a semilattice of completely J ^*-simple semigroups. Reilly and Petrich called a semigroup S cryptic if the Green's relation H is a congruence on S. In this paper, we call a superabundant semigroup S a regular crypto semigroup if H ^* is a congruence on S such that S/H ^* is a regular band. It will be proved that a superabundant semigroup S is a regular crypto semigroup if and only if S is a refined semilattice of completely J ^*-simple semigroups. Thus, regular crypto semigroups are generalization of the cryptic semigroups as well as abundant semigroups.     

Further results in the theory of generalised inflations of semigroups

We determine the structure of semigroups that satisfy xyzw∈{xy,xw,zy,zw}. These semigroups are precisely those whose power semigroup is a generalised inflation of a band. The structure of generalised inflations of the following types of semigroups is determined: the direct product of a group and a band, a completely simple semigroup and a free semigroup F(X) on a set X. In the latter case the semigroup must be an inflation of F(X). We also prove that in any semigroup that equals its square, the power semigroup is a generalised inflation of a band if and only if it is an inflation of a band.     

Weierstrass points on X 0(p ℓ) and arithmetic properties of Fourier coefficients of cusp forms

Generally it is unknown, whether or not ∞ is a Weierstrass point on the modular curve X ₀(N) if N is squarefree. A classical result of Atkin and Ogg states that ∞ is not a Weierstrass point on X ₀(N), if N=pM with p prime, p ∤ M and the genus of X ₀(M) zero. We use results of Kohnen and Weissauer to show that there is a connection between this question and the p-adic valuation of cusp forms under the Atkin–Lehner involution. This gives, in a sense, a generalization of Ogg’s Theorem in some cases.      

Split Left GC-Lpp Semigroups

A left GC-lpp semigroup S is called split if the natural homomorphism γb of S onto S/γ induced by γ is split.It is proved that a left GC-lpp semigroup is split if and only if it has a left adequate transversal.In particular,a construction theorem for split left GC-lpp semigroups is established.     

Ramification points of double coverings of curves and Weierstrass points

Summary Let π: X→C be a double covering with X smooth curve and C elliptic curve. Let R(π)⊂X be the ramification locus of π. Every P∈R(π) is a Weierstrass point of X and we study the triples (C, π, X) for which the set of corresponding Weierstrass points have certain semigroups of non-gaps. We study the same problem also for triple cyclic coverings of C. Entrata in Redazione il 17 luglio 1998. The authors were partially supported by MURST and GNSAGA of CNR (Italy).     

The direct sum of semigroups is of inner type

A semigroup S is said to be of left inner type if the monoid of left translations of S is canonically isomorphic with S¹. The semigroups of left inner type form a special class of semigroups whose simple characterization is not known. The main purpose of this paper is to show that the direct sum of at least three semigroups is of left inner type. A brief discussion is given on the exceptional case, when the semigroup is the direct sum of two summands.     

A generalization of the Weierstrass semigroup

The Weierstrass semigroup H(P) is well known and has been studied. Recently there has been a renewed interest in these semigroups because of applications in coding theory. Generalizations of the Weierstrass semigroup H(P) to n-tuples P₁,…,P_n have been made and studied. We will state and study another possible generalization.     

Permutability of normal extensions of a semilattice

LetS be a semigroup;S is said to bepermutable if, for some integern, every product ofn elements ofS can be re-ordered. We prove that every normal extension of a semilattice by an inverse permutable semigroupsis permutable. Also, some properties of permutable groups are extended to inverse semigroups.     

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.     

Equidistribution of numerical semigroup gaps modulo m

For a positive integer m, a finite set of integers is said to be equidistributed modulo m if the set contains an equal number of elements in each congruence class modulo m. In this paper, we consider the problem of determining when the set of gaps of a numerical semigroup S is equidistributed modulo m. Of particular interest is the case when the nonzero elements of an Apéry set of S form an arithmetic sequence. We explicitly describe such numerical semigroups S and determine conditions for which the sets of gaps of these numerical semigroups are equidistributed modulo m.     

Numerical Semigroups with a Monotonic Apery Set

We study numerical semigroups S with the property that if m is the multiplicity of S and w(i) is the least element of S congruent with i modulo m, then 0 < w(1) < ... < w(m − 1). The set of numerical semigroups with this property and fixed multiplicity is bijective with an affine semigroup and consequently it can be described by a finite set of parameters. Invariants like the gender, type, embedding dimension and Frobenius number are computed for several families of this kind of numerical semigroups. This paper was supported by the project BFM2000-1469. The fourth author wishes to acknowledge support from the Universidade de Evora and the CIMA-UE.     

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).