Research on coverings of curves and toric varieties through Weierstrass points

本文主要研究：(1)曲线的双覆盖上分支点的Weierstrass半群的描述及其存在性。(2)含单项式曲线与低属半群关联的仿射托木品种的研究。(3)低次非奇异平面曲线上点的Weierstrass半群候选点的确定。对于(1)，我们构造了具有分支点的曲线在任意点上的双覆盖，并描述了分支点的Weierstrass半群。m-半群是指最小正整数为m的数值半群。我们证明了存在一个Weierstrass 2n-半群，它不是曲线上任何分支点对任意n bbbb2的Weierstrass半群。但我们也证明了任何4-半群都是曲线上某分支点的Weierstrass半群。在这种情况下，如果分支点的数量很小，我们使用一些有理直纹曲面的放大来得到这样的覆盖。对于(2)，我们发现了一个仿射环面品种，除了两种情况外，它包含由5个或6个元素产生的非原始7-半群9属。此外，我们还发现了一个仿射环面变异，除了双复盖上分枝点的Weierstrass半群外，还包含一个非原始的9属6半群。由结果可知，只有两个9属的数值半群不能确定是否为Weierstrass。对于(3)，我们给出了7次非奇异平面曲线上一点的Weierstrass半群的候选点的完整描述。

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.