Starting from a Fano variety, namely a smooth, complex, projective variety
whose anticanonical divisor is ample, we consider a particular kind of fiber
type contraction on it, a "Fano conic bundle" whose fibers are isomorphic to
conics in the two-dimensional projective space.
First we recall some geometric properties about such contractions.
Then we focus on the relative cone of the conic bundle, that is the real
vectorial subspace of the cone of curves in which there are all classes of
numerical equivalence of curves that are contracted by the conic bundle.
In particular, we consider the contractions in which the dimension of the
relative cone is greater than one, called " non-elementary" .
We will discuss a new result about non-elementary Fano conic bundles, that
gives us a bound for the dimension of the relative cone, and allows us to
deduce other information about the geometry of our varieties.