Compactification of configuration spaces via Hilbert schemes
Abstract
Let F(X,n):= Xn- be the complementary of the union of the diagonals of Xn and let U be a quotient of F(X,n) (possibly trivial) by a subgroup of the symmetric group Sn. We construct compactifications of U in products of Hilbert schemes. Our approach generalizes and unifies classical constructions by Schubert-Semple, Le Barz-Keel, Kleiman and Cheah. An extensive study is done in the case n<4. This includes in particular a complete classification and a description of the quotients by the natural actions.
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