The Hilbert-Chow morphism and the incidence divisor

Abstract

For a smooth projective variety $P$ of dimension $n$, we construct a Cartier divisor supported on the incidence locus in the product of Chow varieties $\mathscr{C}_a (P) \times \mathscr{C}_{n -a - 1}(P)$. There is a natural definition of the corresponding line bundle on a product of Hilbert schemes, and we show this bundle descends to the Chow varieties. This answers a question posed by Barry Mazur.