2006 Fiscal Year Final Research Report Summary
Research on coverings of curves and toric varieties through Weierstrass points
| Project/Area Number |
17540046
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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 Institute of Technology, Center for Basic Education and Integrated Learning, Prof., 基礎・教養教育センター, 教授 (90162065)
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| Co-Investigator(Kenkyū-buntansha) |
OHBUCHI Akira Tokushima University, Integrated Arts and Sciences, Prof., 総合科学部, 教授 (10211111)
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| Project Period (FY) |
2005 – 2006
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| Keywords | 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 |
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| Research Abstract |
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.
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