Nodal curves and components of the Hilbert scheme of curves in $\mathbb {P}^r$ with the expected number of moduli

TL;DR

This paper investigates the existence of specific components within the Hilbert scheme of nodal curves in projective space, focusing on those with the expected number of moduli based on given geometric parameters.

Contribution

It establishes conditions for the existence of Hilbert scheme components with the expected number of moduli for integral nodal curves with prescribed invariants.

Findings

Identifies conditions for components with expected moduli in the Hilbert scheme.

Provides constructions of nodal curves with specified degree, genus, and singularities.

Enhances understanding of the structure of the Hilbert scheme of curves.

Abstract

We study the existence of components with the expected number of moduli of the Hilbert scheme of integral nodal curves $C \subset \mathbb {P}^r$ with prescribed degree, arithmetic genus and number of singular points.