The Lefschetz defect of a smooth Fano veriety

The Lefschetz defect is an invariant of a smooth complex Fano variety $X$ introduced by C. Casagrande. Informally, it measures the failure of Lefschetz hyperplane theorem for non-ample prime divisors on $X$. The main property of the Lefschetz defect is that in a certain sense the classification by the Lefschetz defect generalizes the classification of smooth del Pezzo surfaces. Using the Lefschetz defect, it is possible to recover the Mori–Mukai classification of smooth Fano 3-folds of Picard number $\rho(X)>4$.
In the talk we will discuss the connection of the Lefschetz defect with birational geometry, and its applications to the classification of smooth Fano 4-folds.