Families of nodal curves in P^r with the expected number of moduli

TL;DR

This paper constructs irreducible components of the Hilbert scheme of nodal curves in projective space with specified degree, genus, and number of nodes, ensuring they have the expected number of moduli under certain conditions.

Contribution

It provides a method to explicitly construct irreducible components of the Hilbert scheme with the expected number of moduli for various nodal curves in projective space.

Findings

Constructed irreducible components with the expected number of moduli.

Applicable under suitable numerical assumptions on degree and genus.

Ensures the existence of such components for all node counts up to the genus.

Abstract

Let V^{r}_{d,g, \delta} be the Hilbert scheme of nodal curves in P^r of degree d and arithmetic genus g with \delta nodes. Under suitable numerical assumptions on d and g, for every 0 \le \delta \le g we construct an irreducible component of V^{r}_{d,g, \delta} having the expected number of moduli.