A characterization of some Fano 4-folds through conic fibrations

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

On some Fano 4-folds with Lefschetz defect 3

“…Being ψ a smooth K-negative contraction one has that for every p ∈ P 2 the fiber S p := ψ −1 (p) is a smooth del Pezzo surface. As done in the proof of [15,Proposition 4.1] notice that in case (*) all the fibers of ψ are isomorphic to P 1 × P 1 , while in the latter case (**) they are isomorphic to F 1 . This is because of the deformation invariance of the Fano index (see for instance [10, Proposition 6.2]), then the fibers of ψ must be all isomorphic each another.…”

“…The strategy followed in some parts of the proof of Theorem 3.0.1 is similar to the one used in [15, Proposition 4.1] but here we are considering all the varieties Y ∼ = P P 2 (O ⊕ O(a)) with a ∈ {0, 1, 2} and not only Y ∼ = P 1 × P 2 as done in [15,Proposition 4.1]. Namely, by means of Proposition 2.4 (a) we are considering all the possible targets of Fano conic bundles f : X → Y where dim X = 4 and ρ X − ρ Y = 3.…”

“…This makes some parts of the proof more difficult and general, and it requires a more detailed analysis. We will refer to the proof of [15,Proposition 4.1] whenever we run the same arguments.…”

“…Set E := Exc (h). In the same spirit of the proof of [15,Proposition 4.1] we divide our argument in some steps.…”

“…Remark 4.2.2). Notice that by [15,Corollary 1.3] we already know that the varieties in which we are interested in are rational.…”