| Co-Investigator(Kenkyū-buntansha) |
OHMOTO Toru Kagoshima Univ., Faculty of Sciences, Associate Professor, 理学部, 助教授 (20264400)
YOKURA Shoji Kagoshima Univ., Faculty of Sciences, Professor, 理学部, 教授 (60182680)
MIYAJIMA Kimio Kagoshima Univ., Faculty of Sciences, Professor, 理学部, 教授 (40107850)
NAKASHIMA Masaharu Kagoshima Univ., Faculty of Sciences, Professor, 理学部, 教授 (40041230)
AIKOU Tadashi Kagoshima Univ., Faculty of Sciences, Associate Professor, 理学部, 助教授 (00192831)
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| Research Abstract |
1. We have formulated the infinitesimal mixed Torelli problem for a locally trivial analytic family of complex projective surfaces with ordinary singularities, parametrized by a manifold, relativizing the notion of cubic hyper-resolution due to V.Navarro Aznar, F.Guillen et al., and have gave cohomological sufficient conditions for this problem to be affirmatively solved. Furthermore we have constructed a few examples for which these sufficient conditions are satisfied.
2. We have also considered the infinitesimal mixed Torelli problem for complex projective threefolds of so-called type (n, r_1, r_2, r_3, r_4). In this procedure we have found a certain weakly normal, non-isolated singularity which is a degenerate one of an ordinary triple point, and is described as (xy)^2+(yz)^2+(zx)^2+wxyz=0 by use of affine coordinates. It has turned out that singularity is a cone over the Steiner surface which is a rational surface with ordinary singularities in P^3 (C). The normalization of it is a cone over P^2 (C) embedded in P^5 (C) by the Veronese map of degree 2, a rational isolated singularity of multiplicity 4, and is rigid under deformation.
3. Besides the above results, we have obtained a formula which gives the Euler number of the non-singular normalization of a complex hypersurface with ordinary singularities in P^4 (C), generalizing the classical one for a complex hypersurface with ordinary singularities in P^3 (C) due to Enriques. This work is in preparation to be published.
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