Generic initial ideals of points and curves

Let I be the defining ideal of a smooth complete intersection space curve C with defining equations of degrees a andb. 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 is a τ-segment ideal.