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

Abstract

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]),