Compactification of configuration spaces via Hilbert schemes

Original · EN

Let F(X,n):= Xⁿ-Δbe the complementary of the union Δof 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.