Casagrande, Cinzia; Romano, Eleonora A. Classification of Fano 4-folds with Lefschetz defect 3 and Picard number 5. (English) Zbl 1469.14086 J. Pure Appl. Algebra 226, No. 3, Article ID 106864, 13 p. (2022). Summary: 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 \(\rho_X=5,6\). 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 \(\rho_X\geq 5\) with an elementary divisorial contraction sending a divisor to a curve. Cited in 2 ReviewsCited in 5 Documents MSC: 14J45 Fano varieties 14J35 \(4\)-folds 14E30 Minimal model program (Mori theory, extremal rays) PDFBibTeX XMLCite \textit{C. Casagrande} and \textit{E. A. Romano}, J. Pure Appl. Algebra 226, No. 3, Article ID 106864, 13 p. (2022; Zbl 1469.14086) Full Text: DOI arXiv References: [1] Batyrev, V. V., On the classification of toric Fano 4-folds, J. Math. Sci. (N.Y.), 94, 1021-1050 (1999) · Zbl 0929.14024 [2] Casagrande, C., On the Picard number of divisors in Fano manifolds, Ann. Sci. Éc. Norm. Supér., 45, 363-403 (2012) · Zbl 1267.14050 [3] Casagrande, C., Numerical invariants of Fano 4-folds, Math. Nachr., 286, 1107-1113 (2013) · Zbl 1286.14057 [4] Casagrande, C.; Codogni, G.; Fanelli, A., The blow-up of \(\mathbb{P}^4\) at 8 points and its Fano model, via vector bundles on a del Pezzo surface, Rev. Mat. Complut., 32, 475-529 (2019) · Zbl 1435.14041 [5] Casagrande, C.; Druel, S., Locally unsplit families of large anticanonical degree on Fano manifolds, Int. Math. Res. Not., 2015, 10756-10800 (2015) · Zbl 1342.14088 [6] Fulton, W., Intersection Theory (1998), Springer · Zbl 0885.14002 [7] Mori, S.; Mukai, S., Classification of Fano 3-folds with \(b_2 \geq 2\), Manuscr. Math., 110, 407-162 (2003), Erratum: · Zbl 0478.14033 [8] Montero, P.; Romano, E. A., A characterization of some Fano 4-folds through conic fibrations, Int. Math. Res. Not. (2019), published online [9] Romano, E. A., Non-elementary Fano conic bundles, Collect. Math., 70, 33-50 (2019) · Zbl 1441.14142 [10] Sato, H., Toward the classification of higher-dimensional toric Fano varieties, Tohoku Math. J., 52, 383-413 (2000) · Zbl 1028.14015 [11] Tsukioka, T., Fano manifolds obtained by blowing up along curves with maximal Picard number, Manuscr. Math., 132, 247-255 (2010) · Zbl 1203.14044 [12] Wiśniewski, J. A., On deformation of nef values, Duke Math. J., 64, 325-332 (1991) · Zbl 0773.14003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.