The Hilbert-Chow morphism and the incidence divisor

TL;DR

This paper constructs a Cartier divisor on the incidence locus of certain Chow varieties and demonstrates its descent to the Chow variety, addressing a question posed by Mazur.

Contribution

It introduces a new Cartier divisor supported on the incidence locus and proves its descent to Chow varieties, linking Hilbert schemes and Chow varieties.

Findings

Constructed a Cartier divisor on the incidence locus.

Proved the line bundle descends to Chow varieties.

Answered Mazur's question regarding this descent.

Abstract

For a smooth projective variety $P$, we construct a Cartier divisor supported on the incidence locus in $\mathscr{C}_a (P) \times \mathscr{C}_{\dim(P)-a-1}(P)$. 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 Mazur.