The Rational Quotient

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


Let X be a variety. We define an equivalence relation ℛ on X by saying that two points are ℛ-equivalent if they can be connected by a chain of rational curves (so that on a rationally chain-connected variety, two general points are ℛ-equivalent). The set of ℛ-equivalence classes is not in general an algebraic variety (there exist, for example, nonruled complex projective surfaces that contain countably many rational curves!). However, Campana realized in [Cl] and [C4] that it is nevertheless possible to construct a very good substitute for the quotient if one throws away a countable union of proper subvarieties.


Irreducible Component Rational Curf Hilbert Scheme Dense Open Subset General Fiber 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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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