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

@article{Casagrande2020ClassificationOF, title={Classification of Fano 4-folds with Lefschetz defect 3 and Picard number 5.}, author={C. Casagrande and E. Romano}, journal={arXiv: Algebraic Geometry}, year={2020} }

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.

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