2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/106049Given 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.22 pagesCommutative AlgebraAlgebraic Geometry13P10; 14Q99text