Research on coverings of curves and toric varieties through Weierstrass points

基于Weierstrass点的曲线和复曲面簇覆盖研究

基本信息

批准号：
17540046
负责人：
KOMEDA Jiryo
金额：
$ 0.83万
依托单位：
Kanagawa Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540046/
关键词：
Weierstrass point Weierstrass semigroup Double covering of a curve Affine toric variety Numerical semigroup Non-singular plane curve Non-singular curve of genus 9 Rational ruled surface 非特異平面代数曲線種数10の数値半群アフィン・トーリック多様体神奈川工科大学研究報告半群

项目摘要

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.

这项研究致力于以下内容：（1）对曲线双重覆盖及其存在的脉冲分离点的描述。（2）对仿射旋转品种的研究，其中包含与低属的数值半群相关的单次曲线。在任何点上构建了曲线的双重覆盖曲线，并描述了分支点的Weierstrass Semigroup。 M-序列是指最小正整数为m的数值半群。我们表明，有一个Weierstrass 2n-序列，它不是任何n> 2的曲线双重覆盖的任何后流点的Weierstrass半群。但是我们还证明，任何四个序列都是曲线双重覆盖的某个后流的Weierstrass Semigroup。在这种情况下，如果分支点的数量很少，我们使用某些有理统治的表面的爆炸获得了这样的覆盖。（2）我们找到了一个仿生的圆磨品种，其中包含由5或6个元素产生的非重要性7个属的属9，除两种情况外。此外，我们还发现了一个仿生的复曲面品种，其中包含一个属9的6个序列，除了半群，这些半群是双重覆盖物上的RAMIFIENT点的Weierstrass半群。由于结果，只有两个属9的数值半群，这些属尚未决定是否是Weierstrass。