Group completions via Hilbert schemes

الأصل · EN

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 Hₓ of the Hilbert scheme of X× X. Moreover, Hₓ is isomorphic to the ``wonderful completion'' of the connected automorphism group of X; in particular, Hₓ is non-singular. We describe explicitly the degenerations of the diagonal in X× X, that is, the points of Hₓ; these subschemes of X× X are reduced and Cohen-Macaulay.