  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.
