Algebraic curves through commutative semigroups and its application to toric varieties
| Project/Area Number |
15540051
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Kanagawa Institute of Technology |
|---|
Principal Investigator |
KOMEDA Jiryo Kanagawa Inst.Tech., Engineering, Prof., 工学部, 教授 (90162065)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
OHBUCHI Akira Tokushima Univ., Integrated Arts and Sciences, Prof., 総合科学部, 教授 (10211111)
|
|---|
| Project Period (FY) |
2003 – 2004
|
| Project Status |
Completed (Fiscal Year 2004)
|
| Budget Amount *help |
¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2004: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2003: ¥600,000 (Direct Cost: ¥600,000)
|
|---|
| Keywords | Numerical Semigroup / Algebraic Curve / Affine Toric Variety / Weierstrass Point / Weierstrass Semigroup / Weierstrass Pair / Double Covering of a Curve / Ramification Point / Numerical Semigroup / Affine Toric Variety / Algebraic Curve / Weierstrass Point / Weierstrass semigroup / Weierstrass Pair / Double Covering of a Curve / Ramification Point / Weierstrass Semigroup / Triple Covering of a Curve |
|---|
| Research Abstract |
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.
|
Report
(3 results)
Research Products
(23 results)
