%0 Journal Article
%T On an extension of Galligo's theorem concerning the Borel-fixed points on the Hilbert scheme
%A Morgan Sherman
%J Mathematics
%D 2005
%I arXiv
%X Given an ideal $I$ and a weight vector $w$ which partially orders monomials we can consider the initial ideal $\init_w (I)$ which has the same Hilbert function. A well known construction carries this out via a one-parameter subgroup of a $\GL_{n+1}$ which can then be viewed as a curve on the corresponding Hilbert scheme. Galligo \cite{galligo} proved that if $I$ is in generic coordinates, and if $w$ induces a monomial order up to a large enough degree, then $\init_w(I)$ is fixed by the action of the Borel subgroup of upper-triangular matrices. We prove that the direction the path approaches this Borel-fixed point on the Hilbert scheme is also Borel-fixed.
%U http://arxiv.org/abs/math/0512023v1