subvarieties of uniruled variety

Question

I was wondering about the following: suppose that $X$ is a smooth projective uniruled variety and $Y\subset X$ a smooth subvariety. If we assume some hypothesis on $X$ and $Y$ (I'm thinking of about degree and codimension mainly) are there results that guarantee the uniruledness of $Y$ itself? I know that the question is quite general, but any reference or suggestion will be very appreciated.
Thanks!

Answer 1

It's hard to answer such a general question, but here's something along the lines you are suggesting, in the case where the ambient variety is projective space itself.

Any subvariety $X \subset \mathbf P^n$ not contained in a hyperplane must satisfy the inequality

$$ \operatorname{deg} X \geq 1 + \operatorname{codim} X.$$

The subvarieties for which this is an equality are called varieties of minimal degree in $\mathbf P^n$. Perhaps surprisingly, they can be classified completely:

Theorem: Let $X \subset \mathbf P^n$ be a variety of minimal degree, of codimension greater than 1. Then $X$ is either the Veronese surface $\mathbf P^2 \subset \mathbf P^5$, a rational normal scroll, or the cone over one these.

In particular, varieties of minimal degree are all rational. So this is sort of a strengthened version of your question.

For more general uniruled varieties, I don't know of any such results, and I would not really expect them. For example think about products $\mathbf P^n \times C$ where $C$ is an elliptic curve; this contains lots of copies of $C$ which are of high codimension; you have to specify what you mean by degree in this context by choosing an ample line bundle, but for most choices I think these curves will be of low degree. Of course, maybe there is some different kind of bound one could come up with.

A great reference for the claims about varieties of minimal degree above is this paper by Eisenbud and Harris.