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Complex manifolds whose blow-up at a point is Fano. (Variétés complexes dont l’éclatée en un point est de Fano.) (French. Abridged English version) Zbl 1036.14020
Summary: We classify complex projective manifolds $$X$$ for which there exists a point $$a$$ such that the blow-up of $$X$$ at $$a$$ is Fano. As a consequence, we get that, in dimension greater or equal than three, the quadric is the only complex manifold $$X$$ for which there exists two distinct points $$a$$ and $$b$$ such that the blow-up of $$X$$ with center $$\{a,b\}$$ is Fano.

##### MSC:
 14J45 Fano varieties 14E30 Minimal model program (Mori theory, extremal rays)
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##### References:
 [1] D. Abramovich, K. Karu, K. Matsuki, J. Włodarczyk, Torification and factorization of birational maps, Preprint math.AG/9904135 · Zbl 1032.14003 [2] Ando, T., On extremal rays of the higher-dimensional varieties, Invent. math., 81, 2, 347-357, (1985) · Zbl 0554.14001 [3] Andreatta, M.; Ballico, E.; Wiśniewski, J.A., Two theorems on elementary contractions, Math. ann., 297, 2, 191-198, (1993) · Zbl 0789.14011 [4] Beltrametti, M.C.; Sommese, A.J., The adjunction theory of complex projective varieties, De gruyter exp. math., (1995) [5] L. Bonavero, Toric varieties whose blow-up at a point is Fano, Tohoku Math. J. (accepté) · Zbl 1021.14014 [6] Casagrande, C., On the birational geometry of toric Fano 4-folds, C. R. acad. sci. Paris, Série I, 332, 1093-1098, (2001) · Zbl 1024.14023 [7] Lazarsfeld, R., Some applications of the theory of positive vector bundles, (), 29-61 [8] Maruyama, M., Elementary transformations in the theory of algebraic vector bundles, (), 241-266 [9] Sato, H., Toward the classification of higher-dimensional toric Fano varieties, Tohoku math. J. (2), 52, 3, 383-413, (2000) · Zbl 1028.14015 [10] Wiśniewski, J.A., On contractions of extremal rays of Fano manifolds, J. reine angew. math., 417, 141-157, (1991) · Zbl 0721.14023
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