Existence of curves of a given degree in threefolds

Question

Let $X$ be a projective complex smooth threefold such that its Picard group is generated by an ample line bundle $L$. I have the following question:

For each given integer $d\geq 1$, does there exist a one-dimensional closed subscheme $Y\subset X$ such that $\int_X c_1(L)\cap Y=d$?

Note that here I do not assume anything on the genus of $Y$. When $L$ is very ample and $d=1$, it seems to me that this is equivalent to the existence of a line on $X$, but I can not find a good reference even in this simplest case...

It seems that I only need to construct $Y$ such that $\int_X c_1(L)\cap Y=d$ for each $\int_X c_1(L)^3\cap X\geq d\geq 1$. Is there any reference for such kind of results?

Answer 1

If $X$ is a general sextic hypersurface in $\mathbb{P}^4$, it сontains no lines. Indeed, the Hilbert scheme of lines on $X$ is the zero locus of the global section of the vector bundle $S^6\mathcal{U}^\vee$ on the Grassmannian $\mathrm{Gr}(2,5)$ (where $\mathcal{U}$ is the tautological bundle) corresponding to the equation of $X$, and since $$ \mathrm{rank}(S^6\mathcal{U}^\vee) = 7 > 6 = \dim(\mathrm{Gr}(2,5)), $$ this zero locus is empty when $X$ is general.