Paolo Aluffi
Let Y be the singular locus of a hypersurface X in a smooth variety M, with the scheme structure defined by the Jacobian ideal of X (we will say then that Y is the singular {\it scheme\/} of X, to emphasize that the scheme structure of Y is important for our considerations). In this note we consider a class in the Chow group of~Y which arises naturally in this setup, and which captures much intersection-theoretic information about the situation. We produce constraints for a given scheme to be a singular scheme of a hypersurface, and we obtain applications to duality recovering, and sometime strengthening, results of Holme, Landman, Ein, and Zak.
The degree of the dimension-0 component of our class agrees with a known generalization, due to Parusinski, of the usual Milnor number.
The version in this archive corrects an oversight in a definition in the version appeared on Duke.