Diane Maclagan
Smooth irreducible multigraded Hilbert schemes
Abstract: The multigraded Hilbert scheme parameterizes all ideals in a
polynomial ring with a given multigraded Hilbert function. It
generalizes
many known examples of Hilbert schemes, such as the classic Hilbert
scheme
of subschemes of projective space, the Hilbert scheme of points in
affine
space, and the toric Hilbert scheme. While multigraded Hilbert
schemes
are typically highly singular, Haiman and Sturmfels conjectured that
they
are smooth and irreducible when the polynomial ring is k[x,y]. I will
discuss joint work with Greg Smith in which we prove this conjecture.