Hilbert scheme of smooth projective curves of unexpected dimension \& existence of a component with less than the expected number of moduli

Abstract

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. Denoting by $\mathcal{M}_g$ the moduli space of smooth curves of genus $g$, let $\mu: \mathcal{H}_{d,g,r}\dasharrow \mathcal{M}_g$ be the natural map sending $X\in\mathcal{H}_{d,g,r}$ to its isomorphism class $\mu (X)=[X]\in\mathcal{M}_g$. It has been conjectured that a component $\mathcal{H}\subset\mathcal{H}_{d,g,r}$ has the minimal possible dimension $$\Xx(d,g,r):=3g-3+\rho(d,g,r)+\dim\operatorname{Aut}(\mathbb{P}^r)$$ \noindent if $\codim_{\mathcal{M}_g}\mu(\mathcal{H})\le g-5$ provided $\Xx(d,g,r)\ge 0$, where $\rho(d,g,r):=g-(r+1)(g-d+r)$ is the Brill-Noether number. In this article, we exhibit examples against the conjecture discuss further for the study of the functorial map $\mu: \\mathcal{H}{d,g,r}\dasharrow\mathcal{M}_g$ along this line. A component $\mathcal{H}\subset \mathcal{H}_{d,g,r}$ is said to have the {\it expected number of moduli} if $$\dim\mu({\Hh})=\min\{3g-3, 3g-3+\rho(d,g,r)\},$$provided $3g-3+\rho(d,g,r)\ge 0$. The existence of a component with strictly less than the expected number of moduli has not been known. In this paper, we show the existence of components with less than the expected number of moduli.