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