Let $d,g,r$ be natural numbers such that $d \geq 1$, $g \geq 2$, $r \geq 3$. Denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme classifying subschemes of $\mathbb{P}^r$ with Hilbert polynomial $P(x) = d x + (1 - g)$. There is a natural rational map from any component of $\mathcal{H}_{d,g,r}$ whose general member is a smooth curve to the moduli of curves $\mathcal{M}_g$.

The fundamental result of Brill-Noether theory asserts that there exists such a component that dominates $\mathcal{M}_g$ if and only if the Brill-Noether number, defined by the formula \begin{equation} \rho(d, g, r) := (r + 1)d - r g - r(r + 1) \end{equation} is non-negative.

**My question is:** are there any reasonable conditions one can be imposed on the curve, the line bundle, the $\mathfrak{g}^r_d$ etc to insure that the natural forgetful map from the flag-Hilbert scheme parametrizing curves embedded in degree $e$ hypersurfaces $C \subset X \subset \mathbb{P}^r$ to the Brill-Noether component in the Hilbert scheme of curves is dominant? (more generally, what about complete intersections?)