Connected monomial invariants

We define the monomial invariants of a projective variety Z; they are invariants coming from the generic initial ideal of Z. We prove that, under suitable hypotheses, a variety of codimension at least two has connected monomial invariants; as a corollary, we generalize a result of Cook: if Z is an integral variety of codimension two, satisfying some assumptions on the minimal degree of a hypersurface containing it, then its invariants are connected.