2006 Fiscal Year Final Research Report Summary
Geometry of plane algebraic curves
| Project/Area Number |
15540007
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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 | Saitama University |
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Principal Investigator |
SAKAI Fumio Saitama University, Graduate School of Engineering and Science, Professor, 大学院理工学研究科, 教授 (40036596)
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| Co-Investigator(Kenkyū-buntansha) |
KOJIMA Hisashi Saitama University, Graduate School of Engineering and Science, Professor, 大学院理工学研究科, 教授 (90146118)
FUKUI Toshizumi Saitama University, Graduate School of Engineering and Science, Professor, 大学院理工学研究科, 教授 (90218892)
EBIHARA Madoka Saitama University, Graduate School of Engineering and Science, Lecturer, 大学院理工学研究科, 講師 (80213578)
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| Project Period (FY) |
2003 – 2006
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| Keywords | plane curve / singularity / gonality / Cremona transformation |
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| Research Abstract |
Let C be an irreducible plane curve of degree d over the complex number field. To a non-constant rational function Φ on C, we can associate a morphism Φfrom the non-singular model of C to P^1. The gonality of C, denoted by G (or Gon(C)), is defined to be the minimum of the degrees of such morphisms. Let ν denote the maximal multiplicity of the singular points of C. In this situation, we say that C is of type (d,ν). We then easily see that G≦d-ν. The head investigator and his student M. Ohkouchi proved two kinds of criteria for the equality : G=d-ν (Tokyo J.Math.J.2004). However, for many plane curves, the equality is not the case. The head investigator also proved two lower bounds for G for the case in which G<d-ν and discussed various kinds of examples (Preprint, under submission). More recently, he obtained a relation between the genus g of C and the gonality G. More precisely, if g≦B (d,ν), then the equality G=d-ν holds.
The head investigator and his student M. Saleem classified rational plane curves C of type (d, d-2) (Saitama Math.J.27,2005). In particular, they provide an inductive algorithm to construct such curves and proved that any such curve C is transformable into a line by a Cremona transformation. Previously, rational plane curves of type (d, d-2) with only cusps were classified. To describe multi-branched plane curve singularities, the notion of the system of multiplicity sequences were introduced. These results are generalized to type (d, d-2) plane curves with arbitrary genus, which are elliptic and hyperelliptic curves (Preprint).
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