Tower sets and other configurations with the Cohen-Macaulay property

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by  Giuseppe Favacchio

TOWER SETS AND OTHER CONFIGURATIONS WITH
THE COHEN-MACAULAY PROPERTY
GIUSEPPEFAVACCHIO, ALFIORAGUSAANDGIUSEPPE ZAPPALA`
Abstract. Some well-known arithmetically Cohen-Macaulay configu-
4 rations oflinear varieties inPr ask-configurations, partial intersections
1 and star configurations are generalized by introducing tower schemes.
0 Tower schemes are reduced schemes that are finiteunion of linear vari-
2 etieswhosesupportsetisasuitablefinitesubsetofZc calledtowerset.
+
n We prove that the tower schemes are arithmetically Cohen-Macaulay
a and we compute their Hilbert function in terms of their support. Af-
J terwards, since even in codimension 2 not every arithmetically Cohen-

5 Macaulay squarefree monomial ideal is the ideal of a tower scheme, we
1 slightly extend this notion by defining generalized tower schemes (in
codimension 2) and we show that the support of these configurations

] (thegeneralized towerset)givesacombinatorial characterization ofthe
C
primarydecompositionofthearithmeticallyCohen-Macaulaysquarefree
A
monomial ideals.
.
h
t
a
m
Introduction
[

In the last few years a large number of researchers in algebraic geometry
1
in order to produce projective schemes with suitable Hilbert functions and
v
5 graded Betti numbers constructed special configurations of linear varieties
3 relatedtosomesubsetsofZc .Amongtheseshouldbecitedthepartialinter-
5 +
section schemes introducedfirstin [MR]andgeneralized inany codimension
3
. in [RZ] and the k-configurations defined in [GS] and [GHS] to obtain maxi-
1
mal graded Betti numbers with respect to a fixed Hilbert function. On the
0
4 other hand, to study the extremal Hilbert functions for fat point schemes
1 in the plane, secant varieties of some classical algebraic varieties and some
:
v properties of the symbolic powers of ideals, the star configurations were
Xi defined and deeply investigated (see for instance [AS], [GHM]). All these
configurations lead to aCM ideals, mostly monomial and squarefree. Look-
r
a ing at what all these configurations have in common, in this paper we define
the tower sets (Definition 2.1), suitable finite subsets of Zc , on which are
+
supported the tower schemes (Definition 2.3), which generalize all the pre-
vious mentioned configurations. These tower sets enclose the combinatorial
aspects of such configurations.
Also for these schemes we are able to prove that they have the aCM
property (Theorem 2.6). Moreover, we compute the Hilbert function of
the tower schemes in terms of its tower set support. At this point one
can believe that, at least for monomial squarefree ideals, all aCM ideals
can be constructed in this way. Unfortunately, already in codimension 2,
2010 Mathematics Subject Classification. 13 H 10, 14 N 20, 13 D 40.
Key words and phrases. Cohen-Macaulay, Monomial ideals, Configurations.
1
2 Introduction
this is false as we show in Example 3.4. So the question which arises is to
find the right configuration which could characterize all the aCM monomial
squarefree ideals in a polynomial ring. Here we give a complete answer in
codimension 2 (Theorems 3.19, 3.32 and 3.35) defining a slight modification
of the tower schemes (generalized tower sets and schemes, see Definitions
3.12 and 3.13). The codimension bigger than 2 case remains open.
After preliminaries and basic facts, in section 2 we introduce tower sets
and tower schemes and we prove that all these schemes are aCM (Theorem
2.6). Then we show that every tower scheme has the same Hilbert func-
tion as a corresponding tower scheme supported on a left segment whose
Hilbert function was computed in [RZ] (see Proposition 2.11 and Corollary
2.12). Section 3 is devoted to give a combinatorial characterization for aCM
squarefree monomial ideals of codimension 2. To do that we give a slight
generalization of tower sets and tower schemes (Definitions 3.12 and 3.13).
Then we prove numerous preparatory results about these sets and schemes
and finally in Theorems 3.19 and 3.35 we prove the stated characterization.
1. Notation and preliminaries
Throughout the paper k will be a field and R := k[x ,...,x ] = ⊕ R
1 n d d
will be the standard graded polynomial k-algebra.
We will denote by Z := {r ∈ Z | r > 0}. If r ∈ Z we will set [r] :=
+ +
{1,...,r}. If c,r ∈ Z we will denote by C the set of the subsets of [r] of
+ c,r
cardinality c.
Moreover, wewillsetπ : Zc → Z theprojection on thei-th component.
i + +
Onthe set Zc wewill usethe following standardpartial order. If α,β ∈ Zc ,
+ +
α ≤ β iff π (α) ≤ π (β) for every i∈ [c].
i i
We will denote by (Zc )∗ := {(a ,...,a ) ∈ Zc | a 6= a for every i 6= j}.
+ 1 c + i j
Let T ⊂ Zc be a finite set. Let 1 ≤ t ≤ c−1 be an integer and let α ∈ Zt .
+ +
We set
T := {γ ∈Zc−t | (γ,α) ∈ T}
α +
and
Tα := {γ ∈ Zc−t |(α,γ) ∈ T}
+
Definition 1.1. The function ϕ : (Zc )∗ → C such that ϕ(a ,...,a ) =
+ c,n 1 c
{a ,...,a }, will be called forgetful function. A function ω : C → (Zc )∗
1 c c,n +
will be called ordinante iff ϕ◦ω = id .
Cc,n
Let L ⊂ Zc be a finite set. L is said left segment if for every α ∈ L and
+
β ∈ Zc with β ≤ α it follows that β ∈ L.
+
Let L ⊂ Zc be a left segment. The set {α ,...,α } ⊆ L is called set
+ 1 r
of generators for L if for every α ∈ L, α ≤ α for some i. The element
i
(maxπ (L),...,maxπ (L)) ∈Zc is said the size of L.
1 c +
Let L ⊂ Zc be a left segment of size (m ,...,m ), with c < n. For
+ 1 c
1 ≤ i ≤ c, let F = {f ,...,f } be c families of generic linear forms
i i1 imi
belonging to R. For every α = (a ,...,a )∈ L we set I := (f ,...,f ).
1 c α 1a1 cac
We recall that the scheme defined by the ideal I (F ,...,F ) := I
L 1 c α∈L α
is called partial intersection, with support on L and with respect to the
T
families F ,...,F .
1 c
Towers sets 3
If α ∈ Zc we set v(α) := c π (α). If L ⊂ Zc is a left segment, the
+ i=1 i +
Hilbert function of L is
P
H (i) := |{α ∈ L | v(α) = i+c}|.
L
We remind that H coincides with the Hilbert function of a partial intersec-
L
tionsupportedonL(fordetails aboutleftsegmentsandpartialintersections
see [RZ]).
In the sequel, if M is a matrix of rank r, with entries in R, we will denote
by I(M) the ideal generated by the minors of size r in M.
2. Towers sets
Many recent papers dealt with special configurations of linear subvari-
eties of projective spaces which raised up to Cohen-Macaulay varieties, for
instance partial intersections studied in [RZ], k-configurations studied in
[GHS], star configurations studied in [GHM]. In this section we would like
to generalize all these configurations in such a way to preserve the Cohen-
Macaulayness.
Definition 2.1. Let T ⊂ Zc be a finite set. We say that T is a tower set if
+
for every t ∈ [c−1] and for every α,β ∈ Zt , with α < β, T 6= ∅, we have
+ α
T ⊇ T .
α β
Note that when c= 1 every finite subset of Z is a tower set.
+
Remark 2.2. If T ⊂ Zc is a tower set and α ∈ Zt then T ⊂ Zc−t is also a
+ + α +
tower set.
Definition 2.3. Let T ⊂ Zc be a tower set. Let R := k[x ,...,x ], with
+ 1 n
2 ≤ c ≤ n−1. Let F = {f | j ∈ π (T)}, 1 ≤ i ≤ c, where f ∈ R , such
i ij i ij dij
that f and f are coprime when j 6= h and for every α = (a ,...,a ) ∈ T
ij ih 1 c
the sequence (f ,...,f ) is regular. We will denote by I the complete
1a1 cac α
intersection ideal generated by f ,...,f . We set
1a1 cac
I (F ,...,F ) := I .
T 1 c α
α∈T
\
It defines a c-codimensional subscheme of Pn called tower scheme, with
support on T, with respect to the families F ,...,F .
1 c
Note that if T is a c-left segment then T is a tower set, so every partial
intersection is a tower scheme.
Recently many people investigated special subschemes called star con-
figurations. We recall that a star configuration is defined as follows. Let
R := k[x ,...,x ], s,c ∈ Z such that c ≤ min{s,n − 1}. Take a set F
1 n +
consisting of s forms f ,...,f ∈ R such that any c of them are a regular
1 s
sequence. If s ≥ a > ... > a ≥ 1 are integers and α = {a ,...,a } we set
1 c 1 c
I := (f ,...,f ). A star configuration is the subscheme V (F,Pn) ⊂ Pn
α a1 ac c
definedbytheideal I whereαrunsoverallthesubsetsof[s]ofcardinality
α
c. For more details on star configurations see, for instance, [GHM].
T
Remark 2.4. A star configuration is a particular tower scheme. Namely, let
T = {(a ,...,a ) ∈ Zc |s ≥ a > ... > a ≥ 1},
1 c + 1 c
4 Towers sets
and let us consider the families of forms
F = (f ,...,f ), for 1 ≤ i≤ c.
i s−i+1 c−i+1
T is trivially a tower set and V(I (F ,...,F )) = V (F,Pn).
T 1 c c
Inthequotedpapersitwasshownthatpartialintersections,k-configurations
andstarconfigurationsareallaCMschemes. Nowweprovethateverytower
scheme is an aCM scheme and this generalizes those results. We need the
following lemma, which is a slight generalization of Lemma 1.6 in [RZ].
Lemma 2.5. Let c,r ≥ 2 be integers. Let V ⊇ ... ⊇ V be (c − 1)-
1 r
codimensional aCM subschemes of Pn and A = V(f ) hypersurfaces, 1 ≤
j j
j ≤ r, with degf = d . We set Y := V ∩A and let us suppose that Y is
j j i i i i
c-codimensional for each i and that Y and Y have no common components
i j
r
for i6= j. We set d:= d , Y := Y ∪...∪Y and X := Y ∪Y . Then
i=1 i 1 r−1 r
the following sequence of graded R-modules
P
f ϕ
0 → I (−(d−d )) → I → I /(f) → 0
Yr r X Y
r−1
is exact, where f = f and ϕ is the natural map. Moreover
i
i=1
Q
I = I +f I +f f I +···+f ...f I +(f ...f ).
X V1 1 V2 1 2 V3 1 r−1 Vr 1 r
Proof. The proof is analogous to that of Lemma 1.6 in [RZ]. We report it
for convenience of the reader.
Observe that the exactness of the above sequence in the middle depends
on the fact that f is regular modulo I , since Y and Y have no common
Yr i j
components for i 6= j. So, the only not trivial fact to prove is that the map
ϕ is surjective. For this we use induction on r. For r = 2, since V is aCM,
1
I = I +(f ), therefore every element in I /(f ) looks like z+(f ) with
Y V1 1 Y 1 1
z ∈ I ⊆ I . Hence, z ∈ I ∩ I = I . So, ϕ is surjective and the
V1 V2 Y1 Y2 X
sequence is exact. Now, from the exactness of this sequence it follows that
I is generated by I and f I , i.e. I = I +f I +(f f ).
X V1 1 Y2 X V1 1 V2 1 2
Let us suppose the lemma true for r−1. This means, in particular, that
I = I + f I + ··· + f ...f I + (f ...f ). Therefore, every
Y V1 1 V2 1 r−2 Vr−1 1 r−1
element z ∈ I /(f ...f ) has the form x+(f ...f ) with x ∈ I +
Y 1 r−1 1 r−1 V1
f I +···+f ...f I .Hence, x ∈ I whichimplies x ∈ I ∩I = I .
1 V2 1 r−2 Vr−1 Vr Y Yr X
Again, by the exactness of our sequence we get that I is generated by
X
f ...f I and by I + f I + ··· + f ...f I , i.e. I = I +
1 r−1 Yr V1 1 V2 1 r−2 Vr−1 X V1
f I +···+f ...f I +(f ...f ). (cid:3)
1 V2 1 r−1 Vr 1 r
We are ready to prove our result.
Theorem 2.6. Every tower scheme is aCM.
Proof. Let X be a tower scheme of codimension c. To show that X is aCM
we use induction on c. The property is trivially true for c = 1, so we can
assume that every tower scheme of codimension c−1 is aCM. Let T be the
support of X and let F = {f | j ∈ π (T)}, for 1 ≤ i ≤ c, be the families
i ij i
defining X. Let π (T) = {a ,...,a }, with a < ... < a . For every i ∈ [s]
c 1 s 1 s
we denote by V the tower scheme of codimension c−1 supported on T . If
i ai
i < j then V ⊇ V . By inductive hypotheses each V is aCM. Moreover we
i j i
Towers sets 5
denote by A the hypersurface defined by f , with i∈ [s] and Y = V ∩A .
i cai i i i
Note that by the hypotheses on F ’s Y is aCM of codimension c.
h i
Therefore we have X = Y . Now we use induction on s. For s = 1
i
1≤i≤s
X = Y is aCM. Suppose thaSt Y = Y is aCM and show that X =
1 i
1≤i≤s−1
Y ∪Y is aCM. Applying the previous lSemma we get the exact sequence
s
0 → I (−degf)→ I → I /(f) → 0
Ys X Y
s−1
wheref = f fromwhichweseethataresolution ofI canbeobtained
cai X
i=1
as direct suQm of the resolutions of IYs(−degf) and IY/(f); since both have
resolutions of length c the same is true for I and we are done. (cid:3)
X
Our next aim is to compute the Hilbert function of a tower scheme in the
case when the defining families consist of linear forms. To do this, if T is a
tower set, we define a map σ : T → Zc as follows. Let α = (a ,...,a ) ∈ T,
+ 1 c
we set
h (α) :=|{i | i≤ a , (i,a ,...,a )∈ T} |,
1 1 2 c
h (α) :=|{i | i≤ a , T 6= ∅} |, for 2 ≤ j ≤ c
j j (i,aj+1,...,ac)
and finally
σ(a ,...,a ) := (h (α),...,h (α)).
1 c 1 c
The map σ is trivially injective. We set T# := σ(T).
Proposition 2.7. For every tower set T, T# is a left segment.
Proof. Let α′ = (a′,...,a′) ∈ T# and β′ = (b′,...,b′) ∈ Zc , such that
1 c 1 c +
β′ ≤ α′. We have to prove that β′ ∈ T#, i.e. we have to find β ∈ T such
that σ(β) = β′. Let α = (a ,...,a ) ∈ T be such that σ(α) = α′, hence
1 c
a′ = h (α). Since b′ ≤ h (α), there is a unique element b such that T 6= ∅
i i c c c bc
and | {i | i ≤ b , T 6= ∅} |= b′. Now, since b′ ≤ h (α), we have that
c i c c−1 c−1
T(b′c−1,ac) 6= ∅ and, since T is a tower set, T(b′c−1,bc) 6= ∅, therefore there is
a unique element b such that T 6= ∅ and | {i | i ≤ b , T 6=
c−1 (bc−1,bc) c−1 (i,bc)
∅} |= b′ .Byiteratingthesameargumentwewillsetb theuniqueelement
c−1 j
such that T 6= ∅ and | {i | i ≤ b , T 6= ∅} |= b′,
(bj,bj+1,...,bc) j (i,bj+1,...,bc) j
for 1 ≤ j ≤ c. Now we set β = (b ,...,b ). By definition β ∈ T and
1 c
σ(β) = β′. (cid:3)
Remark 2.8. Note that if T,U ⊂ Zc are tower sets such that T ⊆ U then
+
T# ⊆ U#.
Proposition 2.9. Let T ⊂ Zc be a tower set. Let X = V(I (F ,...,F )).
+ T 1 c
Let Y be a tower scheme supported on T#, with respect to the same families
F ,...,F . Then H = H .
1 c X Y
Proof. If c = 1 then T is a finite subset of Z, say r = |T|, so T# = [r].
ThereforeI andI areprincipalidealsgeneratedbyaformofsamedegree,
X Y
hence H = H .
X Y
So we may assume that c ≥ 2 and we proceed by induction on c. We
consider the set π (T) = {m ,...,m }, m < ... < m . Since T is a tower
c 1 s 1 s
set, T ⊇ ... ⊇ T are (c−1)-tower sets. Let X bethe scheme definedby
m1 ms i
6 Towers sets
I (F ,...,F ). Then each X is an aCM scheme of codimension c−1
Tmi 1 c−1 i
by Theorem 2.6 and X ⊇ ... ⊇ X . Moreover, by Remark 2.8, (T )# ⊇
1 s m1
... ⊇ (T )#. Let Y be the scheme defined by I (F ,...,F ). By
ms i (Tmi)# 1 c−1
the inductive hypothesis H = H . Now, let F = {f ,...,f }, we set
Xi Yi c 1 s
X := X ∩ V(f ) and Y = Y ∩ V(f ). Since X and Y are aCM then
i i i i i i i i
H = H . Finally, using induction on s and the exact sequences (see
Xi Yi
Lemma 2.5)
0→ I (−δ) → I → I /(f ...f )→ 0
Xs X Xi∪...∪Xs−1 1 s−1
0 → I (−δ) → I → I /(f ...f ) → 0
Ys Y Yi∪...∪Ys−1 1 s−1
where δ = deg(f ...f ), we get the conclusion. (cid:3)
1 s−1
Proposition 2.9 allows us to find a formula for the Hilbert function of a
tower scheme.
Remark 2.10. Note that, according to the exact sequence of Lemma 2.5,
the Hilbert function of a tower scheme depends on the tower set and on the
degrees of the forms in the families.
Now we associate to a tower scheme X, supported on a left segment L, a
partial intersection Y with support on a suitable left segment L such that
D
H = H .
X Y
If L is a left segment of size (a ,...,a ) and D = {d }, 1 ≤ i ≤ c and
1 c ij
1 ≤ j ≤ a are positive integers, we define a new left segment, which we will
i
be denoted by L , in the following way. If L is (minimally) generated by
D
K ,...,K , then L is the left segment generated by K′,...,K′ where, if
1 r D 1 r
K = (k ,...,k ) then K′ = ( k1 d ,..., kc d ).
i 1 c i j=1 1j j=1 cj
Thus, let X be a tower scheme supported on the left segment L and let
P P
F = {f | j ∈ π (L)}, for 1 ≤ i ≤ c, be the families defining X. Set now
i ij i
π (L) = {a ,...,a }, with a < ... < a and D = {d } where d = degf ,
c 1 c 1 c ij ij ij
1 ≤ i ≤ c and 1 ≤ j ≤ a . Since, by Remark 2.10, H depends just on D
i X
we may assume that f = dij lh, where each lh is a linear form. Now
ij h=1 ij ij
we denote by Y the partial intersection supported on L with respect the c
D
Q
ordered families of linear forms
L =(l ,...,l ,l ,...,l ,...,l ,...,l ).
i i11 i1di1 i21 i2di2 iai1 iaidiai
Proposition 2.11. Given a tower scheme X supported on the left segment
L with respect to the families of forms F = {f | j ∈ π (L)}, for 1 ≤ i ≤ c
i ij i
and D ={d } where d = degf . Then H = H .
ij ij ij X LD
Proof. By definition I = ∩ (f ,...,f ). Now we denote by Y
X (j1,...,jc)∈L 1j1 cjc
the partial intersection supported on L with respect the c ordered families
D
of linear forms L = (l ,...,l ,l ,...,l ,...,l ,...,l ), 1 ≤
i≤ c. Now if α isian init1e1ger suci1hdit1hait211 ≤ αi≤2di2 ai diai1we setiaidiai
s=1 is
tα := max{j | di1+...+dij <Pα}+1
and
tα−1
h := α− d
α is
s=1
X
Generalized tower sets: a characterization of aCM property 7
and p := l . Then, with this notation
iα itαhα
I = (p ,...,p ).
Y 1α1 cαc
(α1,...\,αc)∈LD
It is a matter of computation to show that I = I and this completes the
X Y
proof. (cid:3)
InthenextcorollaryweleadbackthecomputationoftheHilbertfunction
of a tower scheme to that of a partial intersection. The Hilbert function of
a partial intersection was explicitly computed in [RZ].
Corollary 2.12. If X is a tower scheme supported on a tower set T with
respect to families of forms of degrees D, then H = H .
X (T#)D
Proof. It follows just using Propositions 2.9 and 2.11. (cid:3)
3. Generalized tower sets: a characterization of aCM
property
In this section we will generalize tower sets in such a way to characterize
aCM squarefree monomial ideal of codimension 2.
Let I ⊂ k[x ,...,x ] be an equidimensional squarefree monomial ideal of
1 n
codimension c and let I = p ∩...∩p be its primary decomposition. Each
1 t
p is a prime ideal of the type (x ,...,x ). So we can consider the subset
i ai1 aic
S(I) := {{a ,...,a } | 1≤ i ≤ t} of C .
i1 ic c,n
Vice versa to S ⊆ C we can associate an equidimensional squarefree
c,n
monomial ideal
I := (x ,...,x ).
S a1 ac
{a1,..\.,ac}∈S
Definition 3.1. Let S ⊆ C . We will say S aCM if I is an aCM ideal.
c,n S
Definition 3.2. Let S ⊆ C . We will say that S is towerizable if there
c,n
exists a permutation τ on [n] and an ordinante function ω : C → (Zc )∗
c,n +
such that τ(ω(S)) is a tower set.
Remark 3.3. Let S ⊆ C . Note that S is towerizable if there exists a tower
c,n
set T and families F ⊆ {x ,...,x }, such that S(I (F ,...,F )) = S.
i 1 n T 1 c
By Theorem 2.6 if S is towerizable then S is aCM, however there are
aCM equidimensional squarefree monomial ideals I such that S(I) is not
towerizable. Here it is an example in codimension 2.
Example 3.4. Let S = {1,2},{3,4},{5,6},{4,6},{1,4},{1,6} . Then it is
easy to check that I is the determinantal ideal generated by the order 3
S
(cid:8) (cid:9)
minors of the matrix
x 0 0
1
x x x
 2 3 5,
0 x 0
4
0 0 x6
 
soS is aCM.LetussupposethatS istowerizable. Thenthereexists atower
scheme X with support on a tower set T such that S(I ) = S. Of course
X
|T| = 6 and there is not a variable x such that the ideal (x ) contains
k k
4 of the 6 minimal primes of I . Consequently, |π (T)| ≤ 3 and for every
S 2
a ∈ π (T) |T | ≤ 3 so we have only three possibilities
2 a
8 Generalized tower sets: a characterization of aCM property
1) π (T) = {a,b} with |T | = 3 and |T | = 3;
2 a b
2) π (T) = {a,b,c} with |T | = 2, |T | = 2 and |T | = 2;
2 a b c
3) π (T) = {a,b,c} with |T | = 3, |T | = 2 and |T | = 1.
2 a b c
The first two cases cannot occur since I = I does not contain monomials
X S
of degree two.
Therefore T ⊃ T ⊃ T and T = {h ,h ,h }, T = {h ,h }, T = {h }
a b c a 1 2 3 b 1 2 c 1
for some h ’s and thus T = (h ,a),(h ,a),(h ,a),(h ,b),(h ,b),(h ,c) .
i 1 2 3 1 2 1
But the numbers 2, 3 and 5 belong each to one only element of S whereas
(cid:8) (cid:9)
in T there are only two such numbers, precisely h and c.
3
Because of the previous example it is natural to ask which sets S ⊆ C
c,n
are aCM. We will give a characterization in codimension 2 (see Theorems
3.19 and 3.35).
Definition 3.5. Let h ∈ Z . Let S ⊆ C . We set
+ 2,n
S :h := {A ∈ S | h6∈ A}.
If S ⊆ Z2 we set
+
S :h := {α ∈ S |π (α) 6= h and π (α) 6= h}.
1 2
Remark 3.6. Note that if S is aCM then S : h is aCM. Indeed, if M is an
Hilbert-Burch matrix for I then I is generated by the maximal minors
S S:h
of the matrix obtained from M by replacing x with 1.
h
In the sequel we will use the following result which shows that if S ⊆ C
2,n
isaCMthenalsotheschemeobtainedbyreplacing(x ,x ) ⊇ I with(h ,h ),
i j S i j
generic complete intersections, is aCM.
Proposition 3.7. Let S ⊆ C be an aCM set, with I ⊂ k[x ,...,x ]. Let
2,n S 1 n
h ,...,h ∈ k[y ,...,y ] be forms, such that depth(h ,h ) = 2 for every
1 n 1 m i j
{i,j} ∈ S and
depth(h ,h ,h ,h ) ≥ 3
i j u v
forevery{i,j},{u,v} ∈S,{i,j} =6 {u,v}.ThentheidealJ = (h ,h )
{i,j}∈S i j
is aCM.
T
Proof. We consider the following vectors
x =(x ,...,x ) and h= (h ,...,h ).
1 n 1 n
Since I is aCM we can consider M = M(x), an Hilbert-Burch matrix
S
for I . We claim that N = M(h) is an Hilbert-Burch matrix for J. We have
S
to prove that J = I(N). Let g ∈ I(N) be a maximal minor of N. Then
g = f(h), with f(x) ∈ I . Therefore f(x) = λ (x)x + µ (x)x for every
S i i j j
{i,j} ∈ S, consequently f(h) ∈ (h ,h ) for every {i,j} ∈ S. So I(N) ⊆ J.
i j
To conclude the proof it is enough to show that degI(N) =degJ. By the
generality of the forms h ,...,h , we have that
1 n
degJ = (degh )(degh ).
i j
{i,j}∈S
X
Now we proceed by induction on n. If n = 2 then S = {1,2} and I(N) =
(h ,h ). So we can suppose that degI(N) = (degh )(degh ), when
1 2 {i,j}∈S i j
S ⊆ C . We can write S = (S : n)∪S where S :n = {α ∈ S | n 6∈ α}
2,n−1 (n)P
Generalized tower sets: a characterization of aCM property 9
and S = S \ (S : n). By Remark 3.6, S : n is aCM. Let M(x) be an
(n)
Hilbert-Burch matrix for I . We set N = M(h). Therefore
S:n
c
I = I(M) = (I(M) :x )∩(x , x ) = I(M)∩(x , x ).
S n n b uc n u
u∈YS(n) u∈YS(n)
c
Hence, using the inductive hypothesis, we get
degI(N) = deg(I(N))+deg(h , h )) =
n u
u∈YS(n)
b
(degh )(degh )+(degh ) degh = (degh )(degh ).
i j n u i j
{i,j}X∈(S:n) {u,nX}∈S(n) {iX,j}∈S
(cid:3)
We recall that if T ⊆ Z2 and i∈ Z then
+ +
Ti = {j ∈ π (T) | (i,j) ∈ T}
2
and
T = {j ∈ π (T) |(j,i) ∈ T}.
i 1
Remark 3.8. Let T ⊂ (Z2)∗ be a tower set. Then
+
1) a < b and T 6= ∅ ⇒ (a,b) 6∈ T. Indeed, the assumption implies
a
T ⊇ T . Since (a,a) 6∈ T, we have a 6∈ T , therefore a 6∈ T i.e.
a b a b
(a,b) 6∈ T.
2) a < b and (b,a) ∈ T ⇒ (a,b) 6∈ T. Indeed, the assumption implies
T 6= ∅ so, by the previous item, (a,b) 6∈ T.
a
3) {(a,b),(b,a)} 6⊆ T for every a and b. It follows by item 2.
Note that by item 3, |T|= |ϕ(T)| where ϕ is the forgetful function.
Proposition 3.9. Let T ⊂ Z2 be a tower set. Then Ti and Th are compa-
+
rable under inclusion for every i and h.
Proof. Let j ∈ Ti be such that j 6∈ Th, we have to show that Th ⊂ Ti. Let
k ∈ Th, i.e. h ∈ T ; but h 6∈ T therefore since T is a tower set we have that
k j
T ⊂ T , so i∈ T i.e. (i,k) ∈ T that implies that k ∈Ti. (cid:3)
j k k
Proposition 3.10. Let T ⊂ (Z2)∗ be a tower set.
+
1) Let h ∈ π (T) be such that T ⊇ T for every j ∈ π (T). Then
2 h j 2
h6∈ π (T).
1
2) Let h ∈ π (T) be such that Th ⊇ Ti for every i ∈ π (T). Then
1 1
h6∈ π (T).
2
Proof. 1) If (h,j) ∈ T then h∈ T ⊆ T , i.e. (h,h) ∈ T.
j h
2) Using Proposition 3.9 the proof is analogous to item 1.
(cid:3)
Let T ⊂ (Z2)∗ be a tower set. Let h∈ π (T)∩π (T). We set
+ 1 2
F (h) := {j ∈ π (T) |T ⊂ T and (h,j) 6∈ T}.
T 2 h j
Note that if j ∈ F (h) then j <h.
T
10 Generalized tower sets: a characterization of aCM property
Definition 3.11. Let U ⊆ C . We say that U is connected if for every
2,n
A,B ∈ U there is C ∈ U such that A ∩ C 6= ∅ and B ∩ C 6= ∅. Let
S ⊂ (Z2)∗. We say that S is connected if ϕ(S) is connected.
+
Definition 3.12. Let S ⊂ (Z2)∗ be a finite set. We say that S is a gener-
+
alized tower set if
1) S is connected;
2) S = T ∪S where T is a tower set
0
and S has the following further properties
0
3) for every (i,j) ∈ S , i6∈ π (T)∪π (T) and j ∈ π (T)∩π (T);
0 1 2 1 2
4) for every (i,j) ∈ S and h ∈ F (j), (i,h) ∈ S .
0 T 0
Definition 3.13. Let S ⊂ (Z2)∗ be a generalized tower set. Let R =
+
k[x ,...,x ],n ≥ 3.LetF = {f | j ∈ π (S)},1 ≤ i≤ 2,whereeachf isa
1 n i ij i ij
formsatisfyingsuchconditionsofgenericity: forevery(a ,a ) ∈S,f ,f
1 2 1a1 2a2
are coprime and for every (a ,a ), (b ,b ) ∈ S, with {a ,a } 6= {b ,b },
1 2 1 2 1 2 1 2
depth(f ,f ,f ,f ) ≥ 3. If α= (a ,a ) ∈ S, we will denote by I the
1a1 2a2 1b1 2b2 1 2 α
complete intersection ideal generated by f ,f . We set
1a1 2a2
I (F ,F ) := I .
S 1 2 α
α∈S
\
Itdefinesa2-codimensionalsubschemeofPncalledgeneralizedtowerscheme,
with support on S, with respect to the families F ,F .
1 2
In the sequel if S ⊂ (Z2)∗ we will set for short I := I , consequently
+ S ϕ(S)
S will be said aCM if I is aCM.
S
In order to prove our results on the Cohen-Macaulayness of such schemes
we need several lemmas.
Lemma 3.14. Let S = T ∪S be a generalized tower set. Let i ∈ π (S )
0 1 0
and let m = min{j | (i,j) ∈ S }. Then F (m)= ∅.
0 T
Proof. Lets ∈F (m);thens < mandbyDefinition3.12,item4,(i,s) ∈ S ,
T 0
which is a contradiction. (cid:3)
Lemma 3.15. With the above notation, if h∈ π (T)∩π (T) then for every
1 2
j ∈ π (T)∩π (T)\{h} we have F (j) ⊆ F (j).
1 2 T:h T
Proof. If b ∈ F (j) then (T : h) ⊂ (T : h) and (j,b) 6∈ T : h, with j 6= h
T:h j b
and b 6= h, so (j,b) 6∈ T. Moreover there is a such that (a,b) ∈ T : h and
(a,j) 6∈ T : h. Since a 6= h this implies that T 6⊆ T . Since T is a tower set
b j
we get that T ⊂ T .
j b
(cid:3)
Lemma 3.16. Let S = T ∪S be a generalized tower set. Let h ∈ π (T)∩
0 1
π (T). Then S :h isa generalized tower setwith respect to the decomposition
2
S : h= (T :h)∪(S :h).
0
Proof. Of course S : h =(T :h)∪(S :h).
0
1) Since S is connected then S : h is connected too.
2) Let a,b ∈ π (T : h), a < b. Let i ∈ (T : h) ; then (i,b) ∈ T : h i.e.
2 b
i∈ T ⊆ T ; since i6= h and b 6= h then i∈ (T :h) .
b a a