5

Let $\mathfrak X\to \mathbb P^1$ be a projective family, where $\mathfrak X\subset \mathbb P^n\times\mathbb P^1$. Then we have the relative Fano variety of lines $F(\mathfrak X/\mathbb P^1)$; besides, using Segre embedding, $\mathfrak X\subset \mathbb P^n\times \mathbb P^1\subset\mathbb P^{2n+1}$, so we can talk about $F(\mathfrak X)$, the Fano variety of lines in the usual sense.

My question is:

Is $F(\mathfrak X/\mathbb P^1)$ isomorphic to $F(\mathfrak X)$?

(Because the lines in Segre embedding only lies in one fibre, I think it is reasonable to guess they are the same)

Akatsuki
  • 3,250
  • I think a necessary and sufficient condition is that the family has no fixed point in $\mathbb P^n$. I'm too busy to work this out now, but I hope someone else will find the time. – Ben Jun 25 '18 at 16:29
  • @Ben Yes, we need assume there is no fixed point, otherwise there are lines in the other fibres. – Akatsuki Jun 26 '18 at 17:01
  • Actually, it should work in any case if we just discard the components of $F(\mathfrak X\subset\mathbb P^{2n+1})$ corresponding to the fixed locus of $\mathfrak X$ in $\mathbb P^n$ – Ben Jun 26 '18 at 18:14

0 Answers0