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arXiv · 0901.0212

The Hilbert scheme of the diagonal in a product of projective spaces

Dustin Cartwright·Bernd Sturmfels
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Abstract

The diagonal in a product of projective spaces is cut out by the ideal of 2x2-minors of a matrix of unknowns. The multigraded Hilbert scheme which classifies its degenerations has a unique Borel-fixed ideal. This Hilbert scheme is generally reducible, and its main component is a compactification of PGL(d)^n/PGL(d). For n=2 we recover the manifold of complete collineations. For projective lines we obtain a space of trees that is irreducible but singular. All ideals in our Hilbert scheme are radical. We also explore connections to affine buildings and Deligne schemes.

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Dustin Cartwright, Bernd Sturmfels. 2009-08-27. The Hilbert scheme of the diagonal in a product of projective spaces. https://arxiv.org/abs/0901.0212

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