  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.
