Liaison and related topics: notes 115
d We use our knowledge of the schemes X to prove that an arithmetically Cohen-Macaulay scheme V
1
⊂ X , as described in c, in fact does exist. This is the most technical part of the proof.
11.2. Smooth curves in
P
3
A. A long-standing problem, with many subtle variations, was to determine the possible pairs d, g of degree and genus of smooth curves in
P
3
or P
n
. This was solved by Gruson and Peskine [45] for curves in
P
3
and by Rathmann [97] for curves in P
4
and P
5
. Substantial progress has been made by Chiantini, Ciliberto and Di Gennaro [28] in higher projective
spaces. One variation of this problem is to determine a bound for the arithmetic genus of a
non-degenerate, integral, degree d curve C ⊂ P
3
lying on an irreducible surface S of degree k, and to describe the extremal curves. This problem was solved by Harris [47],
who gave a specific bound. Furthermore, he showed that the curves which are extremal with respect to this bound are precisely the curves residual to a plane curve via certain
complete intersections. Note that they are thus arithmetically Cohen-Macaulay. A deeper problem is to bound the genus of a smooth curve in
P
3
not lying on any surface of degree k. There is much progress on this problem, beginning with work of Hartshorne and
Hirschowitz [54]. B. Harris’ work mentioned above used the Hilbert function of the general hyperplane section
of the curve C. He showed that the general hyperplane section must have the Uniform Position Property see Definition 13. Note that Harris’ proof of the uniform position
property for a general hyperplane section required characteristic zero. It has been proved in characteristic p for
P
n
, n ≥ 4, by Rathmann [97]. This led to natural questions: Q1. What are all the possible Hilbert functions for the general hyperplane section of an
integral curve in P
3
? Same question for P
n
. Q2. What are all the possible Hilbert functions for the general hyperplane section of an
integral arithmetically Cohen-Macaulay curve in P
3
? Same question for P
n
. Q3. What are all the possible Hilbert function of sets of points in
P
2
with the Uniform Position Property? Same question for
P
n−1
. Q4. Do the questions above for fixed n have the same answer?
The answer to these questions is known for n = 3, but open otherwise see also Section 11.4. The answer to Q4 is “yes” when n = 3, and the Hilbert functions that arise are
those of so-called decreasing type. This means the following. Let Z be the set of points either the hyperplane section of an integral curve or a set of points with the Uniform
Position Property. Then the Hilbert function of the Artinian reduction, A, of RI
Z
looks as follows. Let d
1
be the degree of the first minimal generator of I
Z
, and d
2
the degree of the second. Note that d
1
≤ d
2
. Let r be the Castelnuovo-Mumford regularity of I
Z
. Then
h
A
t =
t + 1 if t d
1
d
1
if d
1
≤ t ≤ d
2
− 1 strictly decreasing
if d
2
− 1 ≤ t ≤ r if t ≥ r
Work on this problem was carried out in [45], [67], [98]. The interesting part is to con- struct an integral arithmetically Cohen-Macaulay curve with the desired h-vector, and this
116 J.C. Migliore - U. Nagel
was done in [67] by a nice application of liaison. A completely different approach, using lifting techniques, was carried out in [29].
11.3. Smooth surfaces in
