The Connectedness of Space Curve Invariants

Let S = C[x1, . . . , xn+1] be the set of polynomials in n+1 variables over C (usually corresponding to the homogeneous coordinate ring of P). Let ≻ be the reverse lexicographical order on the monomials of S. Given a homogeneous ideal I in S, gin(I), the generic initial ideal of I, is the Borel-fixed monomial ideal associated to I. (For details see [B], [BM], [BS] or [Gr].) The generic initial ideal reflects much of the structure of the original ideal, for example, it has the same Hilbert function and the same regularity. Also, many geometric problems can be reduced to combinatorial ones which may be phrased in terms of generic initial ideals. Thus, we would like to have some rules governing the kinds of Borel-fixed monomial ideals which can occur from geometry. The first set of rules are supplied by a result of Gruson and Peskine ([GP]):