Deformation formulas for parameterized hypersurfaces (Q6567977)

This paper studies deformations of non-normal hypersurfaces \(X=V(f)\) that are obtained by deforming the normalisation map \(Y\to X\). The main case is where the normalisation \(Y\) is smooth or a rational homology manifold. When \(\mathbb Q_X^\bullet\) is perverse, the kernel \(\boldsymbol N_X^\bullet\) of the natural surjection of perverse sheaves from \(\mathbb Q_X^\bullet\) to the intersection cohomology complex \(\boldsymbol I_X^\bullet\) (with constant \(\mathbb Q\)-coefficients) is a perverse sheaf, called the comparison complex. Then \(\boldsymbol N_X^\bullet\) has stalk cohomology concentrated in degree \(-n+1\) if and only if \(Y\) is a rational homology manifold. Under the assumption of isolated instability the author expresses the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the comparison complex.\N\NThe resulting formula gives for \(n=1\) Milnor's formula \(\mu=2\delta-r+1\), where the singularity degree \(\delta\) is interpreted as the number of double points in a general 1-parameter deformation. For the surface case one obtains that \(\lambda^0_{\boldsymbol N_{V(f_0)}^\bullet,\boldsymbol z}(\boldsymbol 0)- \lambda^1_{\boldsymbol N_{V(f_0)}^\bullet,\boldsymbol z}(\boldsymbol 0)= C-T-\delta-\chi(\mathbb L_{\Sigma f,\boldsymbol0})\), where \(T\) is the number of triple points, \(C\) the number of cross caps (\(D_\infty\) singularities), \(\delta\) the number of \(A_1\) singularities and \(\mathbb L_{\Sigma f,\boldsymbol0}\) the complex link of \(\Sigma f\) at \(\boldsymbol 0\).