Singular schemes of hypersurfaces
Let \(M\) be a smooth algebraic variety over a field \(k\), algebraically closed and of characteristic zero. Let \(X\subset M\) be a hypersurface, i.e., a closed subscheme locally defined by one equation. Then the singular locus \(Y\) of \(X\) has a natural scheme structure defined by the jacobian ideal. The author defines a class \(\mu_{\mathcal L}(Y)\) in the Chow group of \(Y\), depending on the data \((M,X)\): he then shows that the class only depends on \(Y\) and on the line bundle \({\mathcal L}:={\mathcal O}_M(X)|_Y\), and gives explicit methods for computing it in the case where \(Y\) is itself smooth. In \S 2 it is proven that \(\deg(\mu_{\mathcal L}(Y))\) is Parusiński's generalized Milnor number; the class is then used to study local properties near \([X]\) of the dual variety of \((M,{\mathcal O}_M(X))\), i.e., the variety of sections of \({\mathcal O}_M(X)\) defining singular hypersurfaces, recovering and partially extending previous results of Ein, Holme, Landman, Parusiński and Zak. In \S 3 it is shown that the \(\mu\) class imposes strong restrictions on which schemes \(Y\) can appear as singular subschemes of a hypersurface.
scientific article; zbMATH DE number 878417
scientific article