An analogue of Brill-Noether for hypersurfaces?

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 $$\rho(d, g, r) := (r + 1)d - r g - r(r + 1)$$ 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?)

• It is not true that Brill-Noether theory implies that when there exists at least one component $\mathcal{H}_{d,g,r}$ dominating $\mathcal{M}_g$, then this component is unique. I already addressed this in a previous MO answer: mathoverflow.net/questions/178822/… – Jason Starr Mar 17 '17 at 20:32
• corrected. Though out of curiousity - as you said in the previous question: it is true if I add "smooth, embedded, linearly nondegenerate curves", right? – Nati Mar 17 '17 at 20:43
• Yes, it is true if you add "linearly nondegenerate" to the hypotheses for a general member of $\mathcal{H}_{d,g,r}$. – Jason Starr Mar 17 '17 at 20:45
• Your question appears to be asking the following: for a general curve $C$, and for a general $\mathfrak{g}^r_d$ on that curve, say $\phi:C\to \mathbb{P}^r$ (let's assume $r\geq 3$ so that $\phi$ is an embedding), for the ideal sheaf $I$ of $\phi(C)\subset \mathbb{P}^r$, for which $e>0$ is $h^0(\mathbb{P}^r,I(e))$ positive? Of course if $e> 2(g-1)/d$, then $h^0(\mathbb{P}^r,I(e)) \geq \binom{r+e}{r} - (ed + 1 -g)$. – Jason Starr Mar 18 '17 at 22:28
• @Jason: thanks again! But if I understand correctly - your answer explains when a specific curve is contained in a hypersurface of degree $e$, but what about the deformation theory? i.e. this is probably a naive question, but why is it obvious you can choose these hypersurfaces to vary in a flat family? – Nati Mar 19 '17 at 1:12