Michel Brion

"Wonderful completions via Hilbert schemes"

Let X be a projective variety, homogeneous under a linear algebraic group. We show that the diagonal of X belongs to a unique irreducible component H_X of the Hilbert scheme of X times X. Moreover, H_X is isomorphic to the ``wonderful completion'' of the connected automorphism group of X; in particular, H_X is non-singular. We describe explicitly the degenerations of the diagonal in X times X, that is, the points of H_X; these subschemes of X times X are reduced and Cohen-Macaulay.