Generic initial ideals of points and curves

Let I be the defining ideal of a smooth irreducible complete intersection space curve C with defining equations of degrees a and b. We use the partial elimination ideals introduced by Mark Green to show that the lexicographic generic initial ideal of I has Castelnuovo-Mumford regularity 1+ ab(a− 1)(b− 1)/2 with the exception of the case a = b = 2, where the regularity is 4. Note that ab(a− 1)(b− 1)/2 is exactly the number of singular points of a general projection of C to the plane. Additionally, we show that for any term ordering τ , the generic initial ideal of a generic set of points in Pr is a τ -segment ideal.