A Characterization of Some Fano 4-folds Through Conic Fibrations

We find a characterization for Fano 4-folds X with Lefschetz defect delta(X) = 3: besides the product of two del Pezzo surfaces, they correspond to varieties admitting a conic bundle structure f : X -> Y with rho(X) - rho(Y) = 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 4-folds due to Batyrev and Sato we provide explicit examples of Fano conic bundles from toric 4-folds with delta(X) = 3.