Let X be a complex projective manifold, L an ample line bundle on X, and assume that we have a ℂ* action on (X;L). We classify such triples (X; L;ℂ*) for which the closure of a general orbit of the ℂ* action is of degree ≤ 3 with respect to L and, in addition, the source and the sink of the action are isolated fixed points, and the ℂ* action on the normal bundle of every fixed point component has weights ±1. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension 11 and 13 are homogeneous if their group of automorphisms is reductive of rank ≥ 2.

Let X be a complex, projective, smooth and Fano variety. We study Fano conic bundles f : X → Y . Denoting by ρX the Picard number of X, we investigate such contractions when ρX − ρY > 1, called non-elementary. We prove that ρX − ρY ≤ 8, and we deduce new geometric information about our varieties X and Y , depending on ρX −ρY . Using our results, we show that some known examples of Fano conic bundles are elementary. Moreover, when we allow that X is locally factorial with canonical singularities and with at most finitely many non-terminal points, and f : X → Y is a fiber type KX -negative contraction with one-dimensional fibers, we show that ρX − ρY ≤ 9.

We find a characterization for Fano 4-folds X with Lefschetz defect δX = 3: besides the product of two del Pezzo surfaces, they correspond to varieties admitting a conic bundle structure f : X → Y with ρX − ρY = 3. Moreover, we observe that all of these varieties are rational. We give the list of all possible targets of such contractions. Combining our results with the classification of toric Fano 4folds due to Batyrev and Sato we provide explicit examples of Fano conic bundles from toric 4-folds with δX = 3.