Local or global characteristic numbers of complex projective hypersurfaces and the resolution or improvement of their singularities

点击翻译按钮获取中文摘要

(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.