  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 tau, the
generic initial ideal of a generic set of points in P^r is a tau-segment ideal.
