Asymptotic existence of nodal curve components with expected moduli

For integers $r,d,g,t$ with $0\le t\le g$, let $E'(r,d,g,t)$ be the locus of integral, non-degenerate degree-$d$ curves in $\mathbb{P}^r$ of arithmetic genus $g$ having exactly $t$ ordinary nodes as their only singularities and satisfying $h^1(C,\mathcal{O}_C(2))=0$. An irreducible component $\Gamma$ of this locus has expected dimension when its Hilbert-scheme dimension is $(r+1)d-(r-3)(g-1)$, and has the expected number of moduli when

$$\dim\bigl(u_{r,d,g,t}(\Gamma)\bigr)=\min\{3g-3-t,3g-3+\rho(d,g,r)-t\}.$$

Asymptotic existence conjecture. Fix an integer $r\geq 3$. There is a function $\mathbb{O}_r:\mathbb{N}\to\mathbb{N}$ such that $\lim_{g\to+\infty}\mathbb{O}_r(g)/g=0$ and an integer $g_0$ such that, for all integers $g,t,d$ with $g\geq g_0$, $0\leq t\leq g$, and $-\rho(g,r,d)+t\leq 2g-\mathbb{O}_r(g)$, the locus $E'(r,d,g,t)$ has an irreducible component with expected dimension and the expected number of moduli.

This predicts asymptotic existence of components of the Hilbert scheme of curves in projective space having both unobstructed deformation theory and the smallest dimension permitted by the moduli map. The statement concerns all sufficiently large genera in the specified numerical range; no resolution status is supplied in the source.