On Casagrande-Druel Fano varieties with Lefschetz defect 2

TL;DR

This paper classifies Fano varieties with Lefschetz defect 2, focusing on those arising from Casagrande-Druel constructions, and extends the classification to Fano 4-folds and generalized constructions.

Contribution

It provides a complete classification of Casagrande-Druel Fano 4-folds with delta(X)=2 and links most Fano 3-folds with delta(X)=2 to these constructions.

Findings

15 of 19 Fano 3-fold families with delta=2 arise from Construction A

147 Fano 4-fold families with delta=2 and Picard number >3 are classified

Most Fano 3-folds with delta=2 are related to the generalized Construction B

Abstract

The larger the Lefschetz defect delta(X) of a smooth complex Fano variety X, the more information we can deduce about the geometry of X. The structure of varieties with delta(X) greater than 2 is known. In this paper, we study the case delta(X)=2. In particular, we focus on Fano varieties with delta(X)=2 arising from the so called Casagrande-Druel construction, which we refer to as Construction A. We show that among the 19 families of Fano 3-folds with delta(X)=2 classified by Mori and Mukai, 15 arise from such construction. Moreover, we construct all Fano 4-folds with Picard number greater than 3 and delta(X)=2 admitting such a structure, obtaining 147 distinct families in total. This completes the classification of all Casagrande-Druel Fano 4-folds with delta(X)=2. To broaden the scope, we also study a generalized version of Construction A, which we call Construction B, and we show that 18 out of the 19 families of Fano 3-folds with delta(X)=2 arise from it.