  Let F(X,n):= X^n-\Delta be the complementary of the union \Delta of the
diagonals of X^n and let U be a quotient of F(X,n) (possibly trivial) by a
subgroup of the symmetric group S_n. 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.
