# Existence of high degree rational curve in Fano which also lie in a fixed proper subvariety

Let $$X$$ be a smooth Fano variety of Picard number one. Let $$Y \subset X$$ be a closed subvariety. Let $$d$$ be an integer such that there is a dominating component $$K_d$$ of rational curves of degree $$d$$ in $$X$$. In other words, the family $$K_d$$ of curves covers $$X$$.

Question: For all such positive integer $$d$$, does there exist a rational curve $$l \in K_d$$ passing through a general point $$p \in Y$$ which lie in $$Y$$?

• Who lies in $Y$ --- point $P$ (or $p$?), or the curve? Commented May 19, 2023 at 4:28
• The curve. The point $p$ is on $Y$. There was some typo. I have corrected. Commented May 19, 2023 at 4:31
• If the curve lies in $Y$ then $X$ is irrelevant for the problem --- you are just asking if there is a curve in $Y$ passing through $p$. Of course, this depends very much on the properties of $Y$. For instance, if $Y$ is a smooth K3 surface the answer is negative. Commented May 19, 2023 at 4:45
• There's a typo in the title
– YCor
Commented May 19, 2023 at 5:02
• Assume that $Y$ is unirulled. If $K^Y_d$ is a component of rational curves of degree $d$, then my question is equivalent to asking whether $K_d^Y \cap K_d$ has dimension at least $\text{dim}(Y)$. Is that always true ? Commented May 19, 2023 at 5:03

No, just take $$Y$$ the product of $$\mathbb P^1$$ with a variety with no rational curves where the $$\mathbb P^1$$ is embedded in $$X$$ as something of degree $$2$$, and take $$d$$ odd. The only rational curves in $$Y$$ will be covers of the $$\mathbb P^1$$ which have even degree.