The virtual number of \(D_ \infty\) points. II (Q1813891)

Let \(X\) be a divisor in a compact complex manifold \(Y\) of dimension \(n+1\). Suppose \(X\) has a one-dimensional singular locus \(\Sigma\), endowed with the reduced structure, such that \(X\) has only \(A_ \infty\) singularities on \(\Sigma-\{0\}\). Let \((X,\sigma)\) be a germ of the nonisolated singularity of \(X\) at \(\sigma\in \Sigma\). Then the virtual number \(VD_ \infty(X,\sigma)\) of \(D_ \infty\) points of the hypersurface germ \((X,\sigma)\) is defined [see Part I, the second author, ibid., 175-184 (1990; see the preceding review)]. The authors prove the following formula for the total virtual number of \(D_ \infty\) points of \(X:\sum_{\sigma\in\Sigma}VD_ \infty(X,\sigma)=\langle 2K_ Y+nX,\Sigma\rangle+4\chi(\Sigma,{\mathcal O}_ \Sigma)\), where \(K_ Y\) denotes the canonical divisor on \(Y\) and \(\langle , \rangle\) the intersection product on \(Y\). As a good example the classical case of a surface in \(\mathbb{P}^ 3\) with only ordinary singularities is considered.