Components of the Hilbert Scheme of smooth projective curves using ruled surfaces

TL;DR

This paper investigates the structure of the Hilbert scheme of smooth projective curves, showing the existence of new generically smooth components formed by double covers of irrational curves under certain conditions.

Contribution

It introduces a method using families of curves on cones to identify new components of the Hilbert scheme with specific geometric properties.

Findings

Existence of generically smooth components for certain degrees and genera.

Construction of a regular component different from the main component.

Explicit examples when the Brill-Noether number is non-negative.

Abstract

Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the Hilbert scheme whose general points correspond to smooth irreducible non-degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. We use families of curves on cones to show that under certain numerical assumptions for $d$, $g$ and $r$, the scheme $\mathcal{I}_{d,g,r}$ acquires generically smooth components whose general points correspond to curves that are double covers of irrational curves. In particular, in the case $\rho(d,g,r) := g-(r+1)(g-d+r) \geq 0$ we construct explicitly a regular component that is different from the distinguished component of $\mathcal{I}_{d,g,r}$ dominating the moduli space $\mathcal{M}_g$.