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On Fano manifolds with a birational contraction sending a divisor to a curve. (English) Zbl 1184.14072
Let \(X\) be a smooth complex \(n\)-dimensional Fano variety so that \(\omega _X^\vee\) is an ample line bundle. It is known that for fixed \(n\) such varieties can vary only in a finite number of families. It follows that the Picard numbers \(\rho _X\) (which in this case coincide with the second Betti numbers) are also bounded. If \(n=3\), the optimal bound \(\rho _X\leq 10\) is known. In higher dimensions, upper bounds are are known only in special cases. For example if \(X\) has a birational elementary contraction sending a divisor to a point, then by a result of T. Tsukioka [Geom. Dedicata 123, 179–186 (2006; Zbl 1121.14036)], \(\rho _X\leq 3\).
In this paper the author shows that if \(X\) has a birational elementary contraction sending a divisor to a curve then \(\rho _X\leq 5\). The case when \(\rho _X= 5\) is then studied in detail.

MSC:
14J45 Fano varieties
14E30 Minimal model program (Mori theory, extremal rays)
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