Classification of Fano 4-folds with Lefschetz defect 3 and Picard number 5

TL;DR

This paper classifies certain complex Fano 4-folds with specific Picard number and Lefschetz defect conditions, identifying six families and introducing a new one, advancing the understanding of their geometric structure.

Contribution

It provides a complete classification of Fano 4-folds with Picard number 5 under Lefschetz defect constraints, including a novel family.

Findings

Six families of Fano 4-folds with Picard number 5 identified

One new family of Fano 4-folds discovered

Classification of Fano 4-folds with Picard number >4 with specific contractions

Abstract

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. If X contains a prime divisor D with rho(X)-rho(D)>2, then either X is a product of del Pezzo surfaces, or rho(X)=5 or 6. In this setting, we completely classify the case where rho(X)=5; there are 6 families, among which one is new. We also deduce the classification of Fano 4-folds with rho(X)>4 with an elementary divisorial contraction sending a divisor to a curve.