118 J.C. Migliore - U. Nagel
and F
2
can be chosen to be unions of planes, and for a sufficiently general choice, the curve C constructed by Liaison Addition will be a stick figure. To see that this procedure can give a
minimal element for any Buchsbaum even liaison class is somewhat more technical, but is an extension of this idea.
For the second part, recall that a basic double link is obtained by starting with a curve C and a surface F containing C, and taking the union Y of C and a general hyperplane section of
F. If C is a union of lines and F is a union of planes then clearly Y will also be a union of lines. The first problem is to show that we can always arrange that there exists a surface F which is a
union of planes. For instance, if C is a union of ≥ 3 skew lines on a quadric surface this is not arithmetically Buchsbaum, but gives the idea, and if we want deg F = 2, then F clearly cannot
be chosen to be a union of planes. So we have to show that a union of planes can always be obtained in our case. But there is a more subtle problem.
For example, suppose that C is a set of two skew lines, and suppose that we make a sequence of three basic double links using F
1
, F
2
and F
3
of degrees 20, 15 and 4 respectively, obtaining curves Y
1
, Y
2
and Y
3
of degrees 22, 37 and 41 respectively. A little thought shows that one cannot avoid that Y
3
have a triple point The key is that deg F
1
deg F
2
deg F
3
. Thus this sequence of basic double links cannot yield a stick figure.
The solution to this dilemma is to show that there is a cohomologically equivalent sequence of basic double links using surfaces G
1
, G
2
, G
3
with deg G
1
≤ deg G
2
≤ deg G
3
. Then the type of problem described in the last paragraph does not occur. Again, the details are technical,
and we refer the reader to [18] and [19].
11.6. The minimal free resolution of generic forms
An important problem, variations of which have been studied by many people, is to describe the Hilbert function or minimal free resolution of an ideal I ⊂ R = K [x
1
, . . . , x
n
] generated by a general set of forms of fixed degrees not necessarily all the same. The answer to the Hilbert
function problem has been conjectured by Fr¨oberg and we will not describe it here. It is known to hold when n ≤ 3 and when the number of generators is n + 1.
For the minimal free resolution, the answer has been conjectured by Iarrobino. At the heart of this is the idea that if the forms are general then there should be no “ghost terms” in the
minimal free resolution, i.e. there should be no summand R−t that appears in consecutive free modules in the resolution. One can see immediately that this is too optimistic, however. For
instance, if I has two generators of degree 2 and one of degree 4 then there is a term R−4 corresponding to a first syzygy and a term R−4 corresponding to a generator. So the natural
conjecture is that apart from such terms which are forced by Koszul relations, there should be no ghost terms.
This was proved to be false in [74]. A simple counterexample is the case of four generators in K [x
1
, x
2
, x
3
] of degrees 4,4,4 and 8. The minimal free resolution turns out to be 0 →
R−10 ⊕
R−11
2
→
R−8
3
⊕ R−9
2
⊕ R−10
→
R−4
3
⊕ R−8
→ R → RI → 0
The term R−8 that does not split arises from Koszul relations, as above, but the summand R−10 shared by the second and third modules also does not split and this does not arise from
Koszul relations.
Liaison and related topics: notes 119
The paper [74] made a general study of the minimal free resolution of n + 1 general forms in R. The minimal free resolution was obtained in many cases depending on the degrees of
the generators and the main tools were liaison and a technical lemma from [77] giving a bound on the graded Betti numbers for Gorenstein rings. The key to this work is Corollary 12 above,
which says that our ideal I can always be directly linked to a Gorenstein ideal.
Here is the basic idea. Knowing the Hilbert function for the n + 1 general forms leads to the Hilbert function of the linked Gorenstein ideal. The technical lemma of [77] then gives good
bounds for the graded Betti numbers of the linked Gorenstein ideal, and in fact these bounds can often be shown to be sharp. Then the mapping cone obtained from the first sequence of Lemma
10 can be used to give a free resolution of RI . One can then determine to what extent this resolution is minimal. In particular, ghost terms in the minimal free resolution of the Gorenstein
ideal translate to ghost terms in the minimal free resolution of I . Especially when n = 3, we can often arrange ghost terms for the Gorenstein ideal thanks to the Buchsbaum-Eisenbud structure
theorem [22] and the work of Diesel [34].
12. Open problems
