Commutative ring theory: Regular rings

We generalize results of Homma and Kim [J. Pure Appl. Algebra Vol. 162, (2001), pp. 273--290] concerning an improvement on the Goppa bound on the minimum distance of certain Goppa codes.

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On Goppa Codes and Weierstrass Gaps at Several Points

http://users.math.yale.edu/~nk354/papers/kaplancountingsemigroups.pdf

Counting numerical semigroups by genus and some cases of a question of Wilf

Algebraic geometry : an introduction to birational geometry of algebraic varieties

Introduction to algebraic and abelian functions

Non-Weierstrass Numerical Semigroups

Let X be a smooth, proper 1-dimensional algebraic variety (of genus^2) over an algebraically closed field k of characteristic 0, and let P be a point of X. Then a positive integer v is called a gap at P if h\X, Oxikv―l)P)) = h\X, Ox(vP)), and GP denotes the set of gaps at P. If we denote by N and HP respectively the additive semigroup of non-negative integers and the complement of GP in N, then HP is a semigroup. A subsemigroup // of N whose complement is finite is called a numerical semigroup. The following problem is fundamental and is a long-standing problem. Is there a pair {X, P) with X a smooth, proper 1-dimensional algebraic variety over k and P its point, such that H=HP? Using the deformation theory on algebraic varieties with Gm-action, Pinkham [7] constructed a moduli space 3tH which classifies the set of isomorphic classes of pairs (X, P) consisting of a smooth, proper 1-dimensional algebraic variety X together with its point P such that HP―H. But he did not claim that 3AH is non-empty. Using the Pinkham's construction of JMH, some mathematicians showed that for some H, 3AH is non-empty. To state their results we prepare some notation. Let M(H)={au ・・・, an] be the minimal set of generators for the semigroup H, which is uniquely determined by H. IH denotes the kernel of the ^-algebra homomorphism <p: k\_X^\―k{,Xu ■■・ , X^\-≫k[f] defined by (p(Xi)=tai where k\_X~]and k[f\ are polynomial rings over k, and fi{H) denotes the least number of generators for the ideal IHWhen we set C# = Spec k[_X~]/IH, we denote by T£ = c jTL(/) the ^-vector space of first order deformations of CH h llEZ u with a natural graded structure. Moreover, g{H) and C{H) denote the cardinal number of the set N―H and the least integer c with c+NQH, respectively. Then 3lH is non-empty in the following cases: 1) H is a complete intersection, i.e., fi(H) = n ―l, 2) H is a special almost complete intersection (Waldi [10]),

/pdf/on-the-existence-of-weierstrass-points-with-a-certain-xoy0zakedw.pdf

On the existence of Weierstrass points with a certain semigroup generated by 4 elements

/pdf/on-primitive-schubert-indices-of-genus-g-and-weight-g-1-4gt154muci.pdf

On primitive Schubert indices of genus g and weight g-1

On the existence of Weierstrass gap sequences on curves of genus ≤8

Compact Riemann surfaces and their abelian functions are instrumental to solve integrable equations; more recently the representation theory of the Monster and related modular forms have pointed to the relevance of τ-functions, which are in turn connected with a specific type of abelian function, the (Kleinian) σ-function. Klein originally generalized Weierstrass' σ-function to hyperelliptic curves in a geometric way, then from a modular point of view to trigonal curves. Recently a modular generalization for all curves was given, as well as the geometric one for certain affine planar curves, known as (n, s) curves, and their generalizations known as "telescopic" curves. 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 for plane affine models of the Riemann surface. Our examples are not telescopic. Because our definition is algebraic, in the sense of being related to the field of meromorphic functions on the curve, we are able to consider properties of the σ-functions such as Jacobi inversion formulae, and to observe relationships between their properties and those of a Norton basis for replicable functions, in turn relevant to the Monstrous Moonshine.

Jiryo Komeda

Papers

Non-Weierstrass Numerical Semigroups

On the existence of Weierstrass points with a certain semigroup generated by 4 elements

On primitive Schubert indices of genus g and weight g-1

On the existence of Weierstrass gap sequences on curves of genus ≤8

The sigma function for weierstrass semigroups ?3, 7, 8? and ?6, 13, 14, 15, 16?