Algebraic curves through commutative semigroups and its application to toric varieties

我们对以下两个半群感兴趣。一种是这样的半群，它的元素是有理函数的极点的阶，在代数曲线上的某一不动点上只有一个极点。这个半群称为点的魏尔斯特拉斯半群。这个半群变成一个数值半群。另一类是两点对的魏尔斯特拉斯半群，其定义类似于一个点的情形。本文主要研究内容如下：1.研究了一条魏尔斯特拉斯半群为给定数值半群的点曲线的存在性及其构造。射影直线覆盖的分支点对的魏尔斯特拉斯半群的刻画。研究了由点的魏尔斯特拉斯半群构成的仿射环簇。对于第一种情况，我们构造了一个超椭圆曲线在魏尔斯特拉斯点上分支的双重覆盖，并研究了它的分支点的魏尔斯特拉斯半群。证明了每一个4-半群，即一个最小正元为4的数值半群，都是由这种构造得到的。对于第二种情况，我们刻画了素数次投影线的循环覆盖上的两个全分支点的对的魏尔斯特拉斯半群。第三种情况对应于环簇的应用。研究了由4元生成的6-半群或7-半群构成的仿射环簇.在未来，我们将研究由4元生成的6-半群或7-半群构成的点曲线的存在性和构造.对于第一种情况，我们将研究其魏尔斯特拉斯半群为亏格8或9的点曲线的构造；对于第二种情况，我们将研究超椭圆曲线的双覆盖；对于第三种情况，我们将研究由5元2维仿射环簇生成的6-半群、7-半群.

We are interested in the following two semigroups. One is the semigroup whose element is the order of a pole of a rational function with only one pole at a given fixed point on an algebraic curve. This semigroup is called the Weierstrass semigroup of the point. This semigroup becomes a numerical semigroup. Another is the Weierstrass semigroup of the pair of two points which is defined in a similar way to the case of one point. This research is devoted to the following :1. Study on the existence or the construction of a pointed curve whose Weierstrass semigroup is a given numerical semigroup.2. The description of the Weierstrass semigroup of the pair of ramification points of a covering of the projective line.3. Research into affine toric varieties constructed from the Weierstrass semigroup of a point.For the first case we construct a double covering of a hyperelliptic curve which is ramified over a Weierstrass point and investigate the Weierstrass semigroup of the ramification point. We can show that every 4-semigroup, i.e., a numerical semigroup whose minimum positive element is four, is derived from such a construction.For the second case we describe the Weierstrass semigroup of the pair of two total ramification points on a cyclic covering of the projective line with prime degree.The third case corresponds to the application to toric varieties. We study the affine toric varieties constructed from a 6-semigroup or a 7-semigroup generated by 4 elements.In the future we will study the existence and the construction of a pointed curve whose Weierstrass semigroup is of genus 8 or 9 for the first case and a double covering of a hyperelliptic curve for the second case and 6-semigroups, 7-semigroups generated by 5 elements, 2-dimensional affine toric varieties for the third case.