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?