The Rational Quotient
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.
KeywordsIrreducible Component Rational Curf Hilbert Scheme Dense Open Subset General Fiber
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