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
P
4
, smooth threefolds in
P
5
In the classification of smooth codimension two subvarieties and by Hartshorne’s conjecture, we stop with threefolds in
P
5
, it has typically been the case that adjunction theory or other methods have been used to narrow down the possibilities see for instance [8], and then liaison has been
used to construct examples. We give an illustration of this idea by sketching a result of Mir´o-Roig from [81]. A natural
question is to determine the degrees d for which there exists a smooth, non-arithmetically Cohen- Macaulay threefold in
P
5
. It had been shown by B˘anic˘a [5] that such threefolds exist for any odd d ≥ 7 and for any even d = 2k 8 with k = 5s + 1, 5s + 2, 5s + 3 or 5s + 4. It had been
shown by Beltrametti, Schneider and Sommese [7] that any smooth threefold in P
5
of degree 10 is arithmetically Cohen-Macaulay.
It remained to consider the case where d = 10n, n ≥ 2. Mir´o-Roig proved the existence of such threefolds using liaison. Her idea was to begin with well-known non-arithmetically Cohen-
Macaulay threefolds in P
5
and use the fact that the property of being arithmetically Cohen- Macaulay is preserved under liaison. In addition, she used the following result of Peskine and
Szpiro [91] to guarantee smoothness: T
HEOREM
15. Let X ⊂ P
n
, n ≤ 5, be a local complete intersection of codimension two. Let m be a twist such that I
X
m is globally generated. Then for every pair d
1
, d
2
≥ m there exist forms F
i
∈ H I
X
d
i
, i = 1, 2, such that the corresponding hypersurfaces V
1
and V
2
intersect properly and link X to a variety X
′
. Furthermore, X
′
is a local complete inter- section with no component in common with X , and X
′
is nonsingular outside a set of positive codimension in Sing X .
This special case of the theorem is quoted from [33], Theorem 2.1. Mir´o-Roig considered an arithmetically Buchsbaum threefold Y with locally free resolution
0 → O
P
5
⊕ O
P
5
1
3
→
1
3 → I
Y
6 → 0 see also Example 16. Since I
Y
6 is globally generated, Theorem 15 applies. Linking by two general hypersurfaces of degrees 6 and 7, respectively, she obtains a smooth residual threefold X
of degree 30, and using the mapping cone construction she obtains the locally free resolution of I
X
. Playing the same kind of game, she is able to obtain from X smooth threefolds of degrees 10n, n ≥ 5, by linking X using hypersurfaces of degree 10 and n + 3. The remaining cases,
degrees 20 and 40, are obtained by similar methods, starting with different Y .
11.4. Hilbert function questions
We have seen above that liaison is useful for showing the existence of interesting objects. In this section we will see that liaison can sometimes be used to prove non-existence results, as well as
results which reduce the possibilities. For instance, we consider the question of describing the possible Hilbert functions of sets of points in
P
3
with the Uniform Position Property. E
XAMPLE
20. Does there exist a set of points in P
3
with the Uniform Position Property and h-vector
1 3 6 5 6,
Liaison and related topics: notes 117
and if so, what can we say about it? Suppose that such a set, Z , does exist. Note that the growth in the h-vector from degree 3 to degree 4 is maximal, according to Macaulay’s growth condition
[66]. This implies, thanks to [11] Proposition 2.7, that the components [I
Z
]
3
and [I
Z
]
4
both have a GCD of degree 1, defining a plane H . It also follows using the same argument as [11]
Example 2.11 that Z consists of either 14 or 15 points on H , plus 6 or 7 points not on H of which 4 or 5 are on a line. Such a Z clearly does not have the Uniform Position Property
E
XAMPLE
21. Does there exist a set of points in P
3
with the Uniform Position Property and h-vector
1 3 6 5 5, and if so, what can we say about it? Let Z be such a set. In this case we do not have maximal
growth from degree 3 to degree 4, but we again consider the component in degree 3. This time we will not have a GCD, but we can consider the base locus of the linear system |[I
Z
]
3
|. Suppose that this base locus is zero-dimensional. Then three general elements of [I
Z
]
3
give a complete intersection, I
X
= F
1
, F
2
, F
3
. This means that Z is linked by X to a zeroscheme W , and we can make a Hilbert function h-vector calculation cf. Corollary 9 and Example 12 c:
degree 1
2 3
4 5
6 7
RI
X
1 3
6 7
6 3
1 RI
Z
1 3
6 5
5 RI
W
2 1
3 1
This means that the residual, W , has h-vector 1 3 1 2, which is impossible it violates Macaulay’s growth condition.
Thus we are naturally led to look for an example consisting of a set of 20 general points, Z , on an irreducible curve C of degree 5. We do not justify this, although similar considerations
can be found in the proof of Theorem 4.7 of [11], but we hope that it is clear that this is the natural place to look, even if it is not clear that it is the only place to look. The Hilbert function
of Z has to agree with that of Z up to degree 4. One can check that a general curve C of degree 5 and genus 1 will do the trick and no other will. Hence the desired set of points does exist.
11.5. Arithmetically Buchsbaum curves specialize to stick figures
