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The structure of algebraic threefolds: An introduction to Mori’s program. (English) Zbl 0649.14022
The paper is an excellent introduction, mainly devoted to nonexperts, to the minimal model program (MMP), developed by S. Mori with relevant contributions due to many people (Benveniste, Kawamata, the author, Reid, Shokurov). The aim of the paper is to give an accessible outline of the recent progresses in MMP. It also contains many illuminating examples and comments. [See also the following review.]
Reviewer: M.Beltrametti

MSC:
 14J30 $$3$$-folds 14J10 Families, moduli, classification: algebraic theory 14E30 Minimal model program (Mori theory, extremal rays) 14C20 Divisors, linear systems, invertible sheaves
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References:
 [1] C. Herbert Clemens, A scrapbook of complex curve theory, Plenum Press, New York-London, 1980. The University Series in Mathematics. · Zbl 0456.14016 [2] William Fulton, Algebraic curves. An introduction to algebraic geometry, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes written with the collaboration of Richard Weiss; Mathematics Lecture Notes Series. · Zbl 0681.14011 [3] David Mumford, Algebraic geometry. I, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties; Grundlehren der Mathematischen Wissenschaften, No. 221. · Zbl 0356.14002 [4] Miles Reid, Undergraduate algebraic geometry, London Mathematical Society Student Texts, vol. 12, Cambridge University Press, Cambridge, 1988. · Zbl 0701.14001 [5] W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. · Zbl 0718.14023 [6] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. · Zbl 0408.14001 [7] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001 [8] Kenji Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack. · Zbl 0299.14007 [9] Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283 – 360. · Zbl 0672.14006 [10] Shigefumi Mori, Classification of higher-dimensional varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 269 – 331. · Zbl 1103.14301 [11] Miles Reid, Tendencious survey of 3-folds, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 333 – 344. · Zbl 0643.14022 [12] Miles Reid, Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345 – 414. · Zbl 0634.14003 [13] P. M. H. Wilson, Towards birational classification of algebraic varieties, Bull. London Math. Soc. 19 (1987), no. 1, 1 – 48. · Zbl 0612.14033 · doi:10.1112/blms/19.1.1 · doi.org
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