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

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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.