Abstract

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 4-folds due to Batyrev and Sato we provide explicit examples of Fano conic bundles from toric 4-folds with δX = 3.