The Hilbert Scheme of the Diagonal in a Product of Projective Spaces

The diagonal in a product of projective spaces is cut out by the ideal of 2×2-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...

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Bibliographic Details
Published in:International mathematics research notices 2010-01, Vol.2010 (9), p.1741-1771
Main Authors: Cartwright, Dustin, Sturmfels, Bernd
Format: Article
Language:English
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Summary:The diagonal in a product of projective spaces is cut out by the ideal of 2×2-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 novel 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.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnp201