Let $\pi\colon X\to Y$ be a proper morphism of smooth complex algebraic varieties with $\dim X = 2n$ and general fibers of dimension $<n$. Assume that $F := \pi^{-1}(p)$ is a an irreducible and reduced $n$-dimensional fiber, is it true that $F^2<0$?
3
-
$\begingroup$ How do you interpret $F^2$ as a number? $\endgroup$– HenriCommented Oct 17, 2019 at 13:11
-
1$\begingroup$ For simplicity asume $X$ to be complete. Then $F$ defines a class in $H^{2n}(X,\mathbb Z)$ and $F^2$ corresponds to a number via $H^{2n}(X,\mathbb Z)\times H^{2n}(X,\mathbb Z)\to H^{4n}(X,\mathbb Z)\cong\mathbb Z$. $\endgroup$– bogCommented Oct 17, 2019 at 14:26
-
$\begingroup$ thanks. I had read that $\pi$ was of relative dimension $n$ which is why I got confused $\endgroup$– HenriCommented Oct 17, 2019 at 16:30
Add a comment
|