Picard groups of Hilbert schemes of curves

TL;DR

This paper computes the Picard group of the Hilbert scheme of certain smooth, irreducible curves in projective space, providing explicit descriptions of its generators in terms of natural divisor classes.

Contribution

It explicitly determines the Picard group of the Hilbert scheme of smooth curves with specified degree and genus under certain conditions, with generators expressed via natural divisor classes.

Findings

Picard group computed over integers

Generators expressed in natural divisor classes

Applicable to curves with degree ≥ 2g+1 and r ≤ d-g

Abstract

We calculate the Picard group, over the integers, of the Hilbert scheme of smooth, irreducible, non-degenerate curves of degree $d$and genus $g \geq 4$ in ${\Bbb P}^r$, in the case when $d \geq 2g+1 $ and $r \leq d-g$. We express the classes of the generators in terms of some ``natural'' divisor classes.