Group completions via Hilbert schemes

Details Group completions via Hilbert schemes Group completions via Hilbert schemes

2000-10-23

arxiv.org/abs/math/0010215v1

Mathematics

Algebraic Geometry

LaTeX2e, 20 pages

Scientific paper

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.

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