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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)
##### Keywords:
Fano varieties; elementary contractions
Full Text:
##### References:
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