eo m / 9 50 30 24 v 1 2 9 M ar 1 99 5 Degrees of Generators of Ideals Defining Subschemes of Projective Space

In this paper, we are interested in giving an upper bound for the degrees of the generators for an ideal defining a curve in projective space, and in investigating the properties of curves for which this bound is realized. While our bound works for subschemes of any dimension in any projective space, our characterizations for when the bound is realized only work for curves in P. Using the known structure theory for liaison classes of curves in P allows us to give a reasonably complete picture of the liaison classes containing the extremal curves, mainly in terms of the dimensions of the components of the deficiency module for the liaison class. We are also able to give some similar conditions for when a liaison class contains an integral curve satisfying our upper bound for degrees of generators. In contrast to the minimal degree of a generator, in most cases, the maximal degree is not readily computed in terms of other invariants, for example, the Hilbert function. There is, of course, Castelnuovo–Mumford regularity, [Mu], bounding the degrees of generators in terms of the cohomology modules (and more generally, bounding the degrees of generators of the syzygies). Another result which gives information about a certain class of curves is in [DGM], where they show that the defining ideals of arithmetically Cohen–Macaulay curves have generators whose degree is bounded above by one plus the last integer for which the second difference of the Hilbert function is non-zero. Furthermore, in [CGO], most arithmetically Cohen–Macaulay curves achieving this bound are classified; generally speaking, with some exceptions, the arithmetically Cohen–Macaulay curves whose ideals have a generator of maximal degree are linked in the minimal number of steps to plane curves. In that paper, however, they note that little is known about the non-arithmetically Cohen–Macaulay case, or the non-codimension two case. This is what sparked our interest in the