| Co-Investigator(Kenkyū-buntansha) |
MIYAJIMA Kimio Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (40107850)
YOKURA Shoji Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (60182680)
AIKOU Tadashi Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (00192831)
OBITSU Kunio Kagoshima University, Faculty of Science, Associate Professor, 理学部, 助教授 (00325763)
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| Research Abstract |
(1)Let H_i:f_i=0 (1【less than or equal】i【less than or equal】3) be non-singular hypersurfaces in P^4(C), which intersect transversely each other. Then the hypresurface defined by the equation f=A(f_1f_2f_3)+B(f_1f_2)^2+C(f_2f_3)^2+D(f_3f_1)^2=0 in P^4(C), where A, B, C, and D are sufficiently general homogeneous polynomials in 5 variables of appropriate degrees, has the quasi-ordinary singularity (X,0) defined locally by the equation (xy)^2+(yz)^2+(zx)^2+wxyz=0 other than ordinary double points, ordinary triple points and cuspidal points. Regarding the normalization (X^*,0) of (X,0), we have proved that it is : (i)rational and of multiplicity 4, (ii)rigid under small deformations, (iii)Cohen-Macaulay, (iv)Gorenstein of index 2, (v)terminal, and so canonical.
(2)We denote by X the hypersurface defined by f=0 in (1), and by D, T, C and Σq the double point locus, triple point locus, cuspidal point locus and quadruple point locus of X, respectively. With these notations, we have proved that the Segre classes of X are given as follows : s(J,X)_0= [X]^2[D]-2[D]^2+5[X][T]+[K_<P4>][C]-σ^*[kc^*]-59[Σq], s(J,X)_1=-[X][D]-3[T]+2[C], s(J,X)_2=2[D], where σ:C^*→C is the normalization of C, and [kc^*] the canonical class of C^*. By this, we have obtained a formula which gives the class number of X in P^4(C). Furthermore, by use of a linear pencil consisting of hyperlane sections of X (Lefschetz pencil), we have the following formula concerning the Euler number _X(X^*) of the normalization X^* of X : _X(X^*)-_X(X_0^*)=deg[k_<C0>]-deg[k_<C^*>]-70#[Σq], where X_0 is a hypersurface in P^4(C) whose degrees of the various singular loci are the same as those of X, but without quadruple points, X_0^* the normalization of X_0, _X(X_0^*) the Euler number of X_0^*, and [k_<C0>] the canonical class of the cupidal curve C_0(non-singular) of X_0.
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