Moduli space of stable maps into very ample hypersurfaces!

Let $X$ be a smooth complex projective variety and $L$ be some ample divisor. For a holomorphic map $u:\Sigma \to X$, we define its degree to be $deg(u^*L)$.

Question: For a given positive integer $M$, is there a positive integer $N$ such that for a generic smooth hypesurface $D$ in the linear system $|NL|$, there are no nontrivial holomorphic maps of both degree and genus less than $M$ into $D$?

Remark1: for a given degree $d$ and genus $g$, virtual dimension of moduli spaces of genus $g$ degree $d$ maps into a hypersurface $D\in |NL|$ is $$c_1^D(d)+(dim(D)-3)(1-g) = c_1^X(d)-Nd +(1-g)(dim(D)-3).$$ If $g$ and $d$ are bounded, then for $N$ big enough, virtual dimension will be negative.

Remark2: It seems that the answer to some similar question in symplectic setting is positive.

• I think you left out the bound on the degree in the formulation of your question. Commented Feb 5, 2014 at 21:53
• "degree and genus less than M". I am adding a "both" now. Commented Feb 5, 2014 at 22:23

Without loss of generality $L$ is very ample. This gives an embedding $X \to \mathbb P^n$. The genus $g\leq M$ degree $d\leq M$ curves in $X$ live in finitely many finite-dimensional moduli spaces (it's a subspace of finitely many components of the Hilbert scheme of $\mathbb P^n$). For each curve $C$, the codimension of the space of hypersurfaces of degree $N$ containing $C$ in the space of hypersurfaces of degree $N$ is an increasing function of $N$ - in fact for $N$ sufficiently large it is is $H^0(C, \mathcal O_C(N))$. So for $N$ large enough, the codimension will be greater than the dimension of the moduli space. This will lead to a generic hypersurface not containing any curve.
For $D$ a surface, there is work of Bogomolov. The results are weaker for higher dimensional varieties, but they do rule out genus $0$ and genus $1$ curves, cf. the work of G. Xu, Lawrence Ein, Claire Voisin and Gianluca Pacienza.