Abstract
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.
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© 2001 Springer Science+Business Media New York
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Debarre, O. (2001). The Rational Quotient. In: Higher-Dimensional Algebraic Geometry. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-5406-3_5
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DOI: https://doi.org/10.1007/978-1-4757-5406-3_5
Publisher Name: Springer, New York, NY
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