Algebraic Geometry of Plane Curves

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The head investigator proved that for rational cuspidal plane curves of degree d, the degree d is less than or equal to the 3× the maximal multiplicity (Math.Ann. 285, (1989), 233-247). In the project, if the maximal multiplicity is equal to d-2, rational cuspidal plane curves are completely classified and the defining equations of the curves are determined (Osaka J.Math., 37, (2000), 405-415). As a corollary, it turns out that such rational cuspidal plane curves are transformable into a line by Cremona transformations. The systematic study of the effect of degenerate quadratic transformations on germs of plane curve singularities are used.The head investigator published a book on algebra (The theory of Rings and Fileds, Kyoritsu, 1997). Based on new concept, the theory of rings and fields are written to the study of algebraic geometry.