Abstract
Let π : X → Y be the blow-up of a four-dimensional complex manifold Y along a smooth curve C. Assume that X is a Fano manifold and has another (3,1)-type extremal contraction \({\varphi : X \to Z}\) whose exceptional divisor meet that of the blow-up π : X → Y. We show that if the exceptional divisor of \({\varphi}\) is smooth, then Y is isomorphic to four-dimensional projective space and C is an elliptic curve of degree 4.
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Tsukioka, T. A remark on Fano 4-folds having (3,1)-type extremal contractions. Math. Ann. 348, 737–747 (2010). https://doi.org/10.1007/s00208-010-0497-3
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DOI: https://doi.org/10.1007/s00208-010-0497-3