Singular schemes of hypersurfaces

Abstract

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 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. The guiding question we have in mind is: which schemes Y can arise as singular schemes of hypersurfaces? We will obtain strong constraints showing for instance that the only hypersurfaces in P whose singular schemes are positive dimensional linear subspaces of P are quadrics, and that no (reduced) nodal curve can be the singular scheme of a hypersurface in a non-singular variety. Many more statements of this sort can be found in section 3. A different type of application is in section 2: we show how our class relates to other invariants of the singularity of a hypersurface; the class can be used to recover results of Holme and Parusiǹski on degree and multiplicity of dual varieties, and leads naturally to a generalization of the notion of ‘ranks’ of a (smooth) projective variety. Also, we obtain a strengthening of Landman’s parity result, and a new proof of a result of Zak on the dimension of the dual of a smooth variety. The duality results follow by applying the framework to hyperplane sections of M : the singular scheme of a section is supported on the locus of contact of the hyperplane with M , and the class can be used to measure this contact. For example, the class measures how ‘general’ a given section is: we show (Corollary 2.6) that if the contact scheme is a linear subspace Pr−1, then the corresponding hyperplane is a smooth point of the dual variety of M , and the dual variety has codimension r. The main general results are in section 1, where we prove (Corollary 1.7) that the class we introduce depends in fact only on Y and on the line bundle L = O(X)|Y , and not on the ambient variety M (provided that Y is the singular scheme of a