[Paper Review] Picard groups of Hilbert schemes of curves

Alexis Kouvidakis|ArXiv.org|Jun 2, 1993

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper computes the integral Picard group of the Hilbert scheme of smooth, irreducible, non-degenerate curves of degree $d$ and genus $g \geq 4$ in $\mathbb{P}^r$, under the conditions $d \geq 2g+1$ and $r \leq d-g$. It identifies the generators of the Picard group in terms of natural divisor classes, providing a complete description of the group structure in this range of parameters.

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.

Motivation & Objective

• To determine the structure of the Picard group of the Hilbert scheme of smooth, irreducible, non-degenerate curves in $\mathbb{P}^r$ over the integers.
• To understand the divisor class group of Hilbert schemes parametrizing curves of degree $d$ and genus $g \geq 4$ under specific geometric constraints.
• To express the generators of the Picard group in terms of geometrically meaningful, 'natural' divisor classes.
• To establish a complete description of the Picard group in the range $d \geq 2g+1$ and $r \leq d-g$, where the geometry of the curves is sufficiently rigid.

Proposed method

• The method relies on deformation theory and the study of linear systems on curves embedded in $\mathbb{P}^r$.
• It uses the fact that for $d \geq 2g+1$, the line bundle $\mathcal{O}(1)$ on the curve is very ample and the embedding is projectively normal.
• The analysis involves computing the Néron-Severi group of the Hilbert scheme via the action of the automorphism group and the geometry of the universal family.
• It identifies natural divisor classes arising from the determinant of cohomology and the Plücker embedding of the Grassmannian of linear systems.
• The proof uses the fact that the Hilbert scheme is smooth and irreducible in the given range, allowing the application of standard Picard group techniques.
• The key step is showing that the Picard group is generated by the classes of the determinant of the relative dualizing sheaf and the pullback of the hyperplane class from the Grassmannian.

Experimental results

Research questions

• RQ1 What is the structure of the Picard group of the Hilbert scheme of smooth, irreducible, non-degenerate curves of degree $d$ and genus $g \geq 4$ in $\mathbb{P}^r$?
• RQ2 How do the natural divisor classes—such as the determinant of cohomology and the hyperplane class—generate the Picard group in the specified range?
• RQ3 Under what geometric conditions on $d$, $g$, and $r$ does the Picard group become torsion-free and generated by explicit classes?
• RQ4 Can the generators of the Picard group be expressed in terms of geometrically meaningful divisors rather than abstract cohomological classes?

Key findings

• The Picard group of the Hilbert scheme of smooth, irreducible, non-degenerate curves of degree $d$ and genus $g \geq 4$ in $\mathbb{P}^r$ is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$ under the conditions $d \geq 2g+1$ and $r \leq d-g$.
• The generators of the Picard group are explicitly described as the class of the determinant of the relative dualizing sheaf and the pullback of the hyperplane class from the Grassmannian of linear systems.
• The Picard group is torsion-free and generated by two geometrically meaningful divisor classes in the given range.
• The result holds over the integers, meaning the Picard group is freely generated by these two classes without torsion.
• The computation relies on the fact that the Hilbert scheme is smooth and irreducible in this range, and that the curves are projectively normal.
• The paper establishes that no additional independent divisor classes exist beyond these two generators in the specified parameter regime.