Research on coverings of curves and toric varieties through Weierstrass points

本文主要研究了以下几个问题：（1）曲线的双覆盖上分歧点的Weierstrass半群的刻画及其存在性。（2）研究了含有与低亏格数值半群相关的单项曲线的仿射环面簇。（3）低次非奇异平面曲线上点的Weierstrass半群候选的确定对于（1），构造了曲线在任意点上具有分歧点的双重覆盖，并刻画了分歧点的Weierstrass半群. m-半群是指最小正整数为m的数值半群。证明了对于任意n>2，存在一个Weierstrass 2n-半群，它不是曲线的二重覆盖上的任何分歧点的Weierstrass半群.但我们也证明了任何4-半群都是曲线的二重覆盖上的某个分歧点的Weierstrass半群。对于（2），我们找到了一个仿射复曲面簇，它包含亏格为9的5元或6元生成的非本原7-半群，除了两种情况外.此外，我们还找到了一个仿射环面簇，它包含亏格为9的非本原6-半群，但不包含双覆盖上分歧点的Weierstrass半群。对于（3）给出了7次非奇异平面曲线上点的Weierstrass半群的候选者的完整描述.

This research is devoted to the following:(1)The description of the Weierstrass semigroup of a ramification point on a double covering of a curve and its existence.(2)Study on affine toric varieties which contain a monomial curve associated with a numerical semigroup of low genus.(3)The determination of the candidates of the Weierstrass semigroup of a point on a non-singular plane curve of low degree.For (1) we constructed a double covering of a curve with a ramification point over any point and describe the Weierstrass semigroup of the ramification point. An m-semigroup means a numerical semigroup whose minimum positive integer is m. We showed that there is a Weierstrass 2n-semigroup which is not the Weierstrass semigroup of any ramification point on a double covering of a curve for any n>2. But we also proved that any 4-semigroup is the Weierstrass semigroup of some ramification point on a double covering of a curve. In this case, if the number of the ramification points is small, we got such a covering using blow-ups of some rational ruled surface.For (2) we found an affine toric variety which contains a non-primitive 7-semigroup of genus 9 generated by 5 or 6 elements except two cases. Moreover, we also found an affine toric variety which contains a non-primitive 6-semigroup of genus 9 except the semigroups which are the Weierstrass semigroups of ramification points on double coverings. By virtue of the results there are only two numerical semigroups of genus 9 which are not decided whether it is Weierstrass or not.For (3) we gave the complete description of the candidates for the Weierstrass semigroup of a point on a non-singular plane curve of degree 7.