On an extension of Galligo's theorem concerning the Borel-fixed points on the Hilbert scheme

Given an ideal I and a weight vector w which partially orders monomials we can consider the initial ideal inw(I) which has the same Hilbert function. A well known construction carries this out via a one-parameter subgroup of a GLn+1 which can then be viewed as a curve on the corresponding Hilbert scheme. Galligo [A. Galligo, Théorème de division et stabilité en géométrie analytique locale, Ann. Inst. Fourier (Grenoble) 29 (2) (1979) 107–184, vii] proved that if I is in generic coordinates, and if w induces a monomial order up to a large enough degree, then inw(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.