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References
Ando, T.: On extremal rays of the higher dimensional varieties. Invent. Math.81, 347–357 (1985)
Beltrametti, M.: Ond-folds whose canonical bundle is not numerically effective according to Mori and Kawamata. Ann. Mat. Pura Appl.147, 151–172 (1987)
Beltrametti, M.: Contractions of non numerically effective extremal rays in dimension 4, Proc. Alg. Geom. Teubner-Texte Math. 92, 24–37, Berlin: Teubner 1986
Cutkosky, S.: Elementary contractions of Govenstein threefolds, Math. Ann.280, 521–525 (1988)
Friedman, R.: Simultaneous resolution of threefold double points. Math. Ann.274, 671–689 (1986)
Ionescu, P.: Generalized adjunction and applications. Math. Proc. Camb. Phil. Soc.99, 452–472 (1986)
Kawamata, Y.: On the finiteness of generators of a pluri-canonical ring for a 3-fold of general type. Am. J. Math.106, 1503–1512 (1984)
Kawamata, Y.: Crepant blowing-up of 3-dimensional canonical singularities and its application to degeneration of surfaces. Ann. Math.127, 93–163 (1988)
Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. Adv. St. Pure Math.10, 283–360 (1987)
Miyaoka, Y., Mori, S.: A numerical criterion for uniruledness. Ann. Math.124, 65–69 (1986)
Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math.116, 133–176 (1982)
Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. Progress in Math. 3. Basel: Birkhäuser 1980
Pinkham, H.: Factorization of birational maps in dimension 3. Proc. Symp. Pure Math.40, 343–371 (1983)
Reid, M.: Minimal models of canonical 3-folds. Adv. St. Pure Math.1, 131–180 (1983)
Van de Ven, A.: On uniform vector bundles. Math. Ann.195, 245–248 (1972)
Wilson, P.M.H.: Fano fourfolds of index greater than one. J. reine angew. Math.379, 172–181 (1987)
Wisniewski, J.: Length of extremal rays and a characterization of projective space, preprint, Notre Dame, 1987
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Kawamata, Y. Small contractions of four dimensional algebraic manifolds. Math. Ann. 284, 595–600 (1989). https://doi.org/10.1007/BF01443353
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DOI: https://doi.org/10.1007/BF01443353