  We show several new inequalities found recently that the basic sequence of
the saturated homogeneous ideal I of an integral curve in P^3 must satisfy.
Then we compare our results with Cook's assertions on the generic initial ideal
of I, carrying out numerical computations with the use of a computer. The
outcome is that we have obtained new restrictions on the generic initial ideal
of I. When the characteristic of k is zero, the basic sequence of a homogeneous
ideal I in a polynomial ring over an infinite field k is a sequence of the
degrees of the minimal generators of the generic initial ideal of I arranged in
a suitable order.
