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

(1)设H_i:f_i=0（1【小于等于】i【小于等于】3）为P^4(C)中的非奇异超曲面，它们彼此横向相交。在P^4(C)中，由方程f=A(f_1f_2f_3)+B(f_1f_2)^2+C(f_2f_3)^2+D(f_3f_1)^2=0所定义的超曲面，其中A、B、C、D是五变量的适当次充分一般齐次多项式，除了普通的双点、普通的三点和尖点外，局部具有由方程（xy）^2+(yz)^2+(zx)^2+wxyz=0所定义的拟普通奇点（X,0）。关于（X,0）的归一化（X^*,0），我们证明了它是：(i)有理且多重4，（ii）在小变形下是刚性的，（iii）Cohen-Macaulay, （iv）指标2的Gorenstein， (v)终端，因此是正则化的。(2)用X表示(1)中f=0定义的超曲面，用D、T、C和Σq分别表示X的双点轨迹、三点轨迹、尖点轨迹和四点轨迹。利用这些符号，我们证明了X的Segre类如下：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]，其中σ:C^*→C是C的正则化类，[kc^*]是C的正则化类。由此，我们得到了给出P^4(C)中X的类数的公式。更进一步，利用由X的超线段组成的线性铅笔（Lefschetz铅笔），我们得到了关于X的归一化X^*的欧拉数_X（X^*）的公式：_X(X^*)-_X(X_0^*)=deg[k_<C0 b>]-deg[k_<C^*>]-70#[Σq]，其中X_0是P^4(C)中的一个超曲面，其各奇异轨迹的度数与X的度数相同，但没有四重点，X_0^*是X_0的归一化，_X（X_0^*）是X_0^*的欧拉数，[k_<C0>]是X_0的丘次曲线C_0（非奇异）的正则类。

(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.