The Cone of Curves in the Smooth Case

  • Olivier Debarre
Part of the Universitext book series (UTX)


Let X be a smooth projective variety. We defined in Chapter 1 the cone of curves NE(X) of X as the convex cone in N 1(X) R generated by classes of effective curves. We prove here Mori’s theorem on the structure of the closure \(\overline {NE} (X)\) of this cone, more exactly of the part where K X is negative. We show that it is generated by countably many extremal rays, and that these rays can only accumulate on the hyperplane K X = 0. We will give in the next chapter (§7.9) a totally different proof of the cone theorem which works for mildly singular varieties, but only in characteristic zero, and relies heavily on cohomology calculations. The methods of this second proof will also give a very important additional piece of information: the existence of the contraction (see 1.16) of extremal rays on which K X is negative, which is at present unattainable by Mori’s geometric approach.


Normal Bundle Abelian Variety Rational Curf Effective Divisor Smooth Projective Variety 
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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Olivier Debarre
    • 1
  1. 1.IRMA—Université Louis Pasteur—CNRSStrasbourg CédexFrance

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