Consider a smooth minimal $3$-fold conic bundle $f:X\rightarrow\mathbb{P}^2$. Then $X$ has Picard rank two and consequently also the vector space of $1$-cycles is $2$-dimensional. Then the cone of effective curves of $X$ has two generators. On the other hand if $F = F_1 + F_2$ is a reducible fibers it seems that $F_1,F_2$ should be generators, and there should be another generator coming from a curve $C\subset X$ which is not contracted by $f$. What am I missing here? What are the generators of the cone of curves of $X$?

Thank you.

  • 1
    $\begingroup$ The components of the reducible fibers are parameterized by an irreducible curve, hence they are all algebraically equivalent. $\endgroup$
    – abx
    Jul 20 '21 at 18:36

Both $F_1$ and $F_2$ are annihilated by the pullback of the line class of $\mathbb{P}^2$. Moreover, both intersect the canonical class of the conic bundle by $-1$. Therefore, they are numerically equivalent.


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