On the dimension of the Hilbert scheme of curves

TL;DR

This paper establishes lower bounds for the dimension of components of the Hilbert scheme parametrizing smooth irreducible nondegenerate curves in projective varieties, with applications to rigidity and sharpness of bounds.

Contribution

It provides new lower bounds for the dimension of Hilbert scheme components for curves in P^3, P^4, and smooth quadric threefolds, including sharpness examples.

Findings

Lower bounds for Hilbert scheme component dimensions in specific projective varieties.

Construction of examples using determinantal varieties demonstrating bound sharpness.

Application of bounds to the study of rigid curves.

Abstract

Consider a component of the Hilbert scheme whose general point corresponds to a degree d genus g smooth irreducible and nondegenerate curve in a projective variety X. We give lower bounds for the dimension of such a component when X is P^3, P^4 or a smooth quadric threefold in P^4 respectively. Those bounds make sense from the asymptotic viewpoint if we fix d and let g vary. Some examples are constructed using determinantal varieties to show the sharpness of the bounds for d and g in a certain range. The results can also be applied to study rigid curves.