2001 Fiscal Year Final Research Report Summary

Research for Morse theory and module maps of degenerate familios of curves

Research Project

Project/Area Number	12640037
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Single-year Grants
Section	一般
Research Field	Algebra
Research Institution	Tohoku-Gakuin University
Principal Investigator	ASHIKAGA Tadashi Faculty of Engineering, Tohoku-Gakuin University, Professor, 工学部, 教授 (90125203)
Project Period (FY)	2000 – 2001
Keywords	degeneration / morsification / family of curves / atomic fiber / signature / monodromy / hyperlliptic curre / semi-stable reduction
Research Abstract	We continued the study of the Modification problem of degenerate families of hyperelliptic curves with Tatsuya Arakawa. We obtained the results about the certain obstruction for the existence of hyperelliptic splitting families, the classification of hyperelliptic atomic fibers of genus 3 and so on. We wrote the paper with him entitled "Local splitting families of hyperelliptic pencils, II". (The part I of this series was already published, see "REFERENCES".) Based on the recent studies of myself and Kazuhiro Konno, we wrote the survey paper with him about the results and open problems in the field of global and local studies of pencils of algebraic curves, see "REFERENCES". We obtained the certain formula of the signature defect of degeneration of curves, which comes from the semi-stable reduction of the family. This is an application of the G-signature theorem of Atiyah-Singer. We are now trying to develop this study, and will publish it in near future.

(6 results)