On some components of Hilbert schemes of curves

TL;DR

This paper constructs new irreducible components of Hilbert schemes of curves with specific properties, extending previous results and revealing components that are larger than expected and dominate the moduli space of curves.

Contribution

It introduces new irreducible components of Hilbert schemes of curves that are generically smooth, higher-dimensional, and dominate the moduli space, extending prior work by Choi, Iliev, and Kim.

Findings

Constructed irreducible components beyond the distinguished component.

Components are generically smooth and of higher than expected dimension.

General points correspond to ramified covers of curves in a surface cone.

Abstract

Let $\mathcal{I}_{d,g,R}$ be the union of irreducible components of the Hilbert scheme whose general points parametrize smooth, irreducible, curves of degree $d$, genus $g$, which are non--degenerate in the projective space $\mathbb{P}^R$. Under some numerical assumptions on $d$, $g$ and $R$, we construct irreducible components of $\mathcal{I}_{d,g,R}$ other than the so-called {\em distinguished component}, dominating the moduli space $\mathcal{M}_g$ of smooth genus--$g$ curves, which are generically smooth and turn out to be of dimension higher than the expected one. The general point of any such a component corresponds to a curve $X \subset \mathbb{P}^R$ which is a suitable ramified $m$--cover of an irrational curve $Y \subset \mathbb{P}^{R-1}$, $m \geqslant 2$, lying in a surface cone over $Y$. The paper extends some of the results in previous papers of Y. Choi, H. Iliev, S. Kim (cf. [12,13] in Bibliography).