# Existence of curves of a given degree in threefolds

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?

• If $L$ is very ample, a curve of degree 1 is a line; otherwise it is not necessarily so. Feb 25 at 18:47
• A related question has a detailed answer here. Feb 26 at 3:51

## 1 Answer

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.