Families of curves with "almost-general" moduli

Question

The Brill-Noether theorem says that, if $\rho(d, g, r) := (r + 1)d - rg - r(r + 1) \geq 0$, then there exists a unique component of the Hilbert scheme of curves of degree $d$ and genus $g$ in $\mathbb{P}^r$ which dominates $M_g$.

Is it true that all the other components of the Hilbert scheme have "small" image in $M_g$?

For example (under the hypothesis $\rho(d, g, r) \geq 0$), is it true that there is no other component of the Hilbert scheme whose image in moduli is, say, of codimension 1? Of codimension 2? Of codimension less than r?

Answer 1

I realize this is an old question, but I only now noticed it. I do not know why you say that the Brill-Noether theorem implies that there is a unique component of the Hilbert scheme that dominates the moduli space. For instance, when $g>4$, when $d=2g-2$, and when $r=g-1$, then there is, indeed, a unique irreducible component of $\text{Hilb}^{dt+1-g}_{\mathbb{P}^r_k/k}$ that dominates moduli space and whose generic point parameterizes a smooth, canonically embedded, genus $g$ curve. However, there is another irreducible component that dominates moduli, whose generic point parameterizes an embedded, genus $g$ curve of degree $d=2g-2$ such that the restriction of $\mathcal{O}_{\mathbb{P}^r}(1)$ is not the canonical bundle, and such that the linear span of the curve is a hyperplane in $\mathbb{P}^r_k$.

The Hilbert scheme parameterizes all manner of closed subschemes, including smooth, embedded curves that are linearly degenerate. Perhaps you want to add a hypothesis that you consider only the open subset of the Hilbert scheme that parameterizes smooth, embedded, linearly nondegenerate curves.