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
