# Curves in conic bundles

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.

• The components of the reducible fibers are parameterized by an irreducible curve, hence they are all algebraically equivalent.
– 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.