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On the degrees of Fano four-folds of Picard number 1. (English) Zbl 1016.14022

Summary: We show that the anti-canonical degree of a four-dimensional Fano manifold of Picard number 1 is bounded by \(5^4=625\) and when the degree is exactly 625 the Fano four-fold must be the four-dimensional projective space. The proof uses the geometry of standard rational curves on the Fano four-folds, especially the projective geometry of the variety of standard rational tangents. The most interesting case is when the variety of standard rational tangents is a space curve, in which case the bound on the anti-canonical degree of the Fano four-fold follows from a bound on the postulation of the space curve.

MSC:

14J35 \(4\)-folds
14J45 Fano varieties
14C22 Picard groups
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