Minimal free resolutions for subschemes of star configurations

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Minimal free resolutions for subschemes of star configurations Alfio Ragusa 1 , Giuseppe Zappalà ∗,1 Dip. di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy

a r t i c l e

i n f o

Article history: Received 10 February 2015 Received in revised form 21 May 2015 Available online xxxx Communicated by S. Iyengar MSC: 13H10; 14N20; 13D40

a b s t r a c t We study some subschemes of a star configuration whose support is defined in a combinatorial way. On the other hand we study some schemes arising from special square matrices. We see that in many important cases such subschemes coincide with the previous subschemes of a star configuration. This permits to give a graded minimal free resolution for them, getting in a special case a Gorenstein scheme of codimension 3. © 2015 Published by Elsevier B.V.

0. Introduction Star configurations of projective spaces were studied very intensively in the last few years (for basic facts on star configurations see for instance [4]). These configurations of linear varieties arise upon studying secant varieties of a variety defined by suitable reduced forms. On the other side they appear also as extremal sets of points with maximal Hilbert function or as support of certain fat point schemes. These studies were concentrated essentially on properties as Cohen–Macaulayness, minimal free resolutions, homological dimensions (see for instance [5,8]), meanwhile other researchers used such configurations for studying the powers and the symbolic powers of their defining ideal (see for instance [1,6]). Now, since many of the properties of the star configurations, geometrical, algebraic and homological, are related to the combinatorial nature of their support it seems natural to try to study those subschemes of the star configurations which can be defined by suitable supports with good combinatorial properties. The idea to relate properties of schemes to the combinatorial nature of their support goes back for instance to the paper [10] and in [3] there were characterized the supports producing Cohen–Macaulay ideals. Thus if R = k[x1 , . . . , xn ] and if we set Cc,n the set of subsets of {1, . . . , n} with cardinality c, to every subset S of Cc,n we can associate

* Corresponding author. 1

E-mail addresses: [email protected] (A. Ragusa), [email protected] (G. Zappalà). Fax: +39095330094.

http://dx.doi.org/10.1016/j.jpaa.2015.06.010 0022-4049/© 2015 Published by Elsevier B.V.

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an equidimensional squarefree monomial ideal IS :=



(xa1 , . . . , xac ).

{a1 ,...,ac }∈S

Vice versa the primary decomposition of a squarefree monomial ideal of R provides a subset of Cc,n . In the particular case in which S = Cc,n , IS defines the star configuration for which it is easy to check its Cohen Macaulyness and to compute its minimal free resolution. When c = 2, IS is minimally generated by the n squarefree monomials in R of degree n − 1. In this paper we are interested to investigate the subschemes of a star configuration whose support Sn,h is a union of n − 2h consecutive diagonals Dk , where Dk = {{i, j} ∈ C2,n | |i − j| = k}. The case h = 0 recovers trivially the star configuration. The case h = 1 produces the complement of an n-gon in C2,n . Except the h = 0 case, all these schemes are not aCM, nevertheless in many cases we will be able to produce a minimal free resolution of them (see Corollary 2.11 and Theorem 3.6). Studying the h = 1 case (the complement in C2,n of an n-gon), we get a special square matrix of size n and rank n − 1 which in some sense defines such a scheme. This matrix can be generalized producing a squarefree monomial ideal In,r which is strictly linked to the scheme defined by Sn,h , for h = n − 1 − r. For these ideals we are able to produce a minimal free resolution, so when they coincide with Sn,h (Theorem 3.6) we obtain a minimal free resolution for these last schemes. As a special case, when n is odd and h = n−1 2 , we prove that In,h is a Gorenstein ideal of height 3 (Theorem 3.3). 1. Preliminaries and basic facts Throughout the paper k will be a field and R := k[x1 , . . . , xn ] = ⊕d Rd , n ≥ 3, will be the standard graded polynomial k-algebra. For reasons of convenience we will associate to every integer i ∈ Z a variable xi with the condition that xi = xj iff i − j is a multiple of n. We will denote Z+ := {r ∈ Z | r > 0}. If r ∈ Z+ we will set [r] := {1, . . . , r}. If c, r ∈ Z+ we will denote by Cc,r the set of the subsets of [r] of cardinality c. Let S ⊆ Cc,n ; we can associate to S an equidimensional squarefree monomial ideal IS :=



(xa1 , . . . , xac ).

{a1 ,...,ac }∈S

When S = Cc,n , IS defines a c-codimensional scheme called star configuration. Such scheme is aCM and its minimal free resolution is well known (see for instance [4]). In particular, when c = 2, IS is minimally generated by the n squarefree monomials in R of degree n − 1. Note that a 2-codimensional star configuration can be described as tower scheme (see [3] for details about tower schemes) just considering the tower set T = {(i, j) | 1 ≤ i < j ≤ n}. So, a graded minimal free resolution for the ideal IS defining a star configuration of codimension 2 is 0 → R(−n)n−1 → R(−n + 1)n → IS . In the paper [3] the authors characterized the sets S ⊂ C2,n , such that IS is aCM i.e. pd R/IS = 2. Here we would like to investigate the projective dimension for special subschemes of the star configurations. To do that we will use the following result which can be found in [2, Theorem 4] and can be deduced by results contained in [7].

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Theorem 1.1. Let I be a monomial ideal in the polynomial ring R. Suppose that the maximal height of an associated prime of I is d. Then pd R/I ≥ d. 2. Ideals associated to special matrices We start by defining the n-gon in C2,n     Pn = {i, j} ∈ C2,n | |i − j| = 1 ∪ {1, n} . The set Sn,1 consisting of the diagonals of Pn is Sn,1 = C2,n \ Pn . In this section we will compute the minimal graded free resolution of ISn,1 , from which we will get an n × n-matrix of a special shape. Thereafter, generalizing such matrix we will define a class of ideals which will be related to certain subschemes of a star configuration. Remark 2.1. We recall that a subset S ⊆ C2,n is said to be connected if for every α, β ∈ S there is γ ∈ S such that α ∩ γ = ∅ and β ∩ γ = ∅. In the paper [3] it was shown that if the ideal IS is aCM, then S is connected. As we will see below ISn,1 is not aCM, nevertheless for n ≥ 5 the set Sn,1 is connected (it is enough to observe that C2,n \ Sn,1 = Pn does not contain any 4-gon). For n = 4, S4,1 = {{1, 3}, {2, 4}} which is not connected. For n = 3, S3,1 = ∅. The study of the schemes supported on Sn,1 was originally motivated by seeking non-aCM schemes whose support is connected. Now if we denote Dh = {{i, j} ∈ C2,n | |i − j| = h}, then Sn,1 = C2,n \ (D1 ∪ Dn−1 ). n We set πn = i=1 xi . The minimal set of monomial generators for ISn,1 is given by {f1 , . . . , fn } where fi = πn /xi xi+1 for i = 1, . . . , n. Indeed, we first observe that J = ISn,1 ∩ (x1 , xn ) is aCM, since it is easy to see to be the ideal of a tower scheme. A minimal graded free resolution of J is well known, in particular the minimal set of monomial generators of J is {f1 , . . . , fn−1 }. Obviously fn ∈ ISn,1 and it is very simple matter to verify that ISn,1 = (f1 , . . . , fn ). Now using the short exact sequence 0 → J → ISn,1 ⊕ (x1 , xn ) → (x1 , xn , fn ) → 0, since the graded minimal free resolution of J is 0 → R(−(n − 1))n−2 → R(−(n − 2))n−1 → J → 0 and since (x1 , xn , fn ) is a complete intersection ideal one gets that a graded minimal free resolution of ISn,1 has the following form ϕ

0 → R(−n) → R(−(n − 1))n −→ R(−(n − 2))n → ISn,1 → 0.

(1)

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If we order the generators in this way: fn , f1 , . . . , fn−1 , a matrix associated to ϕ is ⎛ An,n−2

⎜ ⎜ ⎜ =⎜ ⎜ ⎝

x1 0 .. .

−xn x2 .. .

0 −x1 .. .

... ... .. .

0 −xn−1

0 0

... 0

xn−1 ...

0 0 .. .

⎞

⎟ ⎟ ⎟ ⎟. ⎟ −xn−2 ⎠ xn

This subscheme of the star configuration leads to two possible generalizations. The first one consists of studying the subschemes defined by Sn,h where Sn,h is recursively defined by n−2 . 2

Sn,h = Sn,h−1 \ (Dh ∪ Dn−h ), 2 ≤ h ≤ Note that

  Sn,h = {i, j} ∈ C2,n | h + 1 ≤ |i − j| ≤ n − 1 − h . The second one comes from generalizing the matrix An,n−2 to the following matrix ⎛ An,r

⎜ ⎜ ⎜ =⎜ ⎜ ⎝

x1 0 .. .

−xr+2 x2 .. .

0 −xr+3 .. .

... ... .. .

0 −xr+1

0 0

... 0

xn−1 ...

⎞

0 0 .. .

⎟ ⎟ ⎟ ⎟ , 0 ≤ r ≤ n − 1. ⎟ −xn+r ⎠ xn

Note that det An,r = 0 for every n and r. Indeed it is enough to observe that the matrix ⎛

a1 ⎜ 0 ⎜ ⎜ . M = ⎜ .. ⎜ ⎝ 0 b1

b2 a2 .. .

0 b3 .. .

... ... .. .

0 0

. . . an−1 0 ...

0 0 .. .

⎞

⎟ ⎟ ⎟ ⎟ ⎟ bn ⎠ an

has det M =

n

ai + (−1)n+1

i=1

n

bi .

i=1

So rank An,r = n − 1, for every n and r. Now we denote by Mij the matrix obtained by M by deleting the i-th row and the j-th column. An easy computation shows that

i det Mij =

h=1 bh j−1 h=1 ah

j−1

h=i+1 ah i h=j+1 bh

n

h=j+1 bh

n

h=i+1

ah

if i < j if i ≥ j.

Analogously we will denote by (An,r )ij the matrix obtained by An,r by deleting the i-th row and the j-th column. Therefore we have (by replacing ah with xh and bh with −xh+r )

det(An,r )ij =

i j−1 n (−1)i−j+n h=1 xh+r h=i+1 xh h=j+1 xh+r j−1 i n (−1)i−j h=1 xh h=j+1 xh+r h=i+1 xh

if i < j if i ≥ j

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hence

det(An,r )ij =

r+i j−1 r+n (−1)i−j+n h=r+1 xh h=i+1 xh h=r+j+1 xh j−1 r+i n (−1)i−j h=1 xh h=r+j+1 xh h=i+1 xh

if i < j if i ≥ j.

r+i Now we set fri = h=i+1 xh , for r > 0 and f0i = 1. In the sequel the product of r consecutive variables (recalling that xn and x1 are consecutive since x1 = xn+1 ) will be called an r-cycle monomial. Now we would like to prove that, if we denote by (An,r )−,j the matrix obtained by An,r by deleting the j-th column, the maximal minors of (An,r )−,j have as greatest common divisor the monomial fn−r−1,r+j . To do this we need the following technical lemma whose proof will be delayed to the end of the paper. Lemma 2.2. With the above notation det(An,r )ij = (−1)σ(n,i,j) fri fn−r−1,r+j , where  σ(n, i, j) =

i−j+n i−j

if i < j if i ≥ j.

Proposition 2.3. Given j, GCD({det(An,r )ij | 1 ≤ i ≤ n}) = fn−r−1,r+j . Proof. We get the assertion just using Lemma 2.2 and observing that GCD(fr1 , . . . , frn ) = 1.

2

Because of Proposition 2.3, we will associate to the matrix An,r , in a natural way, the ideal In,r = ({det(An,r )ij /fn−r−1,r+j | 1 ≤ i ≤ n}), which does not depend on j by Proposition 2.3. Proposition 2.4. In,r = (fr1 , . . . , frn ). Proof. It follows directly by the definition of In,r and by Proposition 2.3. 2 Remark 2.5. In this remark we deal with the trivial cases r = 0, r = 1 and r = n − 1. If r = 0, In,0 = R. If r = 1, In,1 = (x1 , . . . , xn ), i.e. In,1 is the irrelevant maximal ideal in R. If r = n − 1, In,n−1 is the ideal generated by the n squarefree monomials of degree n − 1, i.e. is the ideal of the star configuration associated to C2,n . From now on, by Remark 2.5, we will assume that 2 ≤ r ≤ n − 2. Let F , G be free R-modules of rank n, and let ϕ : F → G a map of R-modules associated to the matrix An,r with respect to suitable bases. In the case r = n − 2, by the exact sequence (1), we deduce that Ker ϕ is a rank 1 free module. Moreover, since ISn,1 = In,n−2 , we get that Coker ϕ is isomorphic to In,n−2 . In order to generalize these results we state the following lemma (which is probably well known even if not explicitly stated).

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Lemma 2.6. Let S be a UFD and let F , G be free S-modules of rank n. Let ϕ : F → G be a map of n S-modules of rank n − 1. Let (e1 , . . . , en ) be a basis of F and let u = h=1 ah eh ∈ Ker ϕ, u = 0, such that GCD(a1 , . . . , an ) = 1. Then Ker ϕ = (u). In particular Ker ϕ is a free S-module of rank 1. n Proof. Let v ∈ Ker ϕ, v = h=1 bh eh . Using the field of fractions of S, we get that there exist λ, μ ∈ S, GCD(λ, μ) = 1, such that λu = μv, i.e. λah = μbh , for 1 ≤ h ≤ n; consequently μ is a divisor of ah , for 1 ≤ h ≤ n, i.e. μ is a unit and therefore v ∈ (u). 2 In the sequel we will denote by ϕn,r : R(−(r + 1))n → R(−r)n the map of graded R-modules associated to the matrix An,r , with respect to the canonical bases. Proposition 2.7. The graded R-module Coker ϕn,r admits the following graded minimal free resolution ϕn,r

τ

πn,r

0 → R(−n) −→ R(−(r + 1))n −−−−→ R(−r)n −−−→ Coker ϕn,r → 0, where a matrix associated to τ is ⎛

Bn,r

⎞ fn−r−1,r+1 ⎠. =⎝ ... fn−r−1,r+n

Proof. It is a straightforward consequence of Lemma 2.6. 2 Now we would like to give information on Coker ϕn,r , in particular we would like to see when Coker ϕn,r is isomorphic to In,r . To do that we consider the map ρn,r : R(−r)n → In,r defined by ρn,r (ei ) = fri , 1 ≤ i ≤ n (here e1 , . . . , en is the canonical basis of R(−r)n ). Since Ker πn,r = Im φn,r ⊆ Ker ρn,r , these maps induce a surjective map λn,r : Coker ϕn,r → In,r . We will see when λn,r is an isomorphism. Let us consider the minimal set of generators of In,r , G = {fr1 , . . . , frn }. Let M ⊆ Rn be the module of the first syzygies acting on the elements of G. We set sij = (c1 , . . . , cn ) ∈ Rn , for 1 ≤ i < j ≤ n with ci = − lcm(fri , frj )/fri , cj = lcm(fri , frj )/frj , ch = 0, for h = i, j. Since G is the set of the monomial generators of In,r , the set Σ = {sij | 1 ≤ i < j ≤ n} generates M . Now we set Σ = {si,i+1 | 1 ≤ i ≤ n}, by setting sn,n+1 = −s1,n . In the sequel we will denote by (Σ ) the submodule of Rn generated by Σ .

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Lemma 2.8. With the above notation, the syzygy sij = (c1 , . . . , cn ) ∈ (Σ ) iff ci is either multiple of xi+r+1 or multiple of xi . n Proof. If sij ∈ (Σ ), then sij = h=1 αh sh,h+1 . Since ci = 0, either αi−1 or αi is different from zero. Moreover the i-th component of sh,h+1 is zero for every h = i − 1, i, so we get that ci is either multiple of the i-th component of si−1,i or multiple of the i-th component of si,i+1 , i.e. either xi or xi+r+1 . Now let us suppose that ci is multiple of xi+r+1 and let d = deg ci . We note that ci is a d-cycle monomial starting from xi+r+1 . Indeed by recalling the definition of ci , since fri and frj are r-cycle monomials, we deduce that ci is a d-cycle monomial. Moreover xi+r divides fri , therefore it does not divide ci , consequently ci starts from xi+r+1 . We will show that h ci k=i+1 xk sij = αh sh,h+1 , where αh = h+r+1 is a monomial. k=i+r+1 xk h=i j−1 

(2)

Since xi+r+1 divides ci and it does not divide fri , we get that xi+r+1 divides frj , so j + 1 ≤ i + r. Consequently if j + r ≤ n, d = deg ci = 2r − (i + r − j) − r = j − i. If j + r > n and j + r − n < i + 1 we get analogously d = deg ci = j − i. On the other hand if j + r − n ≥ i + 1 then d = deg ci = 2r − (i + r − j) − r − (j + r − n − i) = n − r. h+r+1 When d = j − i then ci is multiple of k=i+r+1 xk , so αh is a monomial. When d = n − r then h

h h k=i+1 xk = h+r+1−n = xk . xk k=i+n+1 xk k=i+1 k=h+r+2−n

k=i+1 αh = h+r+1

xk

Up to now we showed that in any case αh is a monomial. In order to verify the equality (2), we need to show that the u-th component of the right side vanishes for u = i, j and it coincides with cu , for u = i, j. Note that the u-th component of the right side, for u = i, j, is −αu−1 xu + αu xu+r+1 = 0. ri The i-th component of the right side is αi xi+r+1 = ci and observed that cj = −ci ffrj we see the j-th component of the right side is −αj−1 xj = cj . Similarly if ci is multiple of xi , then cj is multiple of xj+r+1 and analogously to the previous case we get that cj is a d-cycle monomial starting from xj+r+1 . By repeating the same arguments as in the previous case, one can prove that

sij =

n+i−1  h=j

h cj k=j+1 xk βh sh,h+1 , where βh = h+r+1 . k=j+r+1 xk

2

Theorem 2.9. With the above notation, M is minimally generated by the set Σ iff 2r ≥ n − 1. Proof. Let us suppose that M is minimally generated by the set Σ . This implies that sij ∈ (Σ ), for every 1 ≤ i < j ≤ n. In particular s1,r+2 ∈ (Σ ) and, since xr+2 does not divide c1 , by Lemma 2.8, x1 divides c1 , therefore x1 divides fr,r+2. So n + 1 ≤ 2r + 2 i.e. 2r ≥ n − 1. Let 2r ≥ n − 1. It is enough to prove that for every sij = (c1 , . . . , cn ) ∈ Σ, sij ∈ (Σ ). If ci is multiple of xi+r+1 , by Lemma 2.8, we are done. So let us suppose that ci is not multiple of xi+r+1 . Then frj is not multiple of xi+r+1 too, consequently i + r + 1 < j + 1, i.e. r ≤ j − i − 1. So we have n ≤ 2r + 1 ≤ j − i − 1 + r + 1 = j − i + r ⇒ j + 1 ≤ i + n ≤ j + r

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hence xi = xi+n divides frj and since xi does not divide fri , we deduce that xi divides ci . Using Lemma 2.8 we get the conclusion. 2 Now we are ready to state the condition in order that λn,r is an isomorphism. Theorem 2.10. The map λn,r : Coker ϕn,r → In,r is an isomorphism iff 2r ≥ n − 1. Proof. Note that, by definition, Coker ϕn,r = Rn /(Σ ). Now let us suppose that 2r ≥ n − 1. Let u + (Σ ) ∈ Ker λn,r . Then u ∈ M = (Σ). Applying Theorem 2.9 we see that u ∈ (Σ ), hence λn,r is injective, so it is an isomorphism. Conversely if λn,r is injective we deduce that M = (Σ ). Again by Theorem 2.9, we get 2r ≥ n − 1. 2 Corollary 2.11. The graded minimal free resolution of In,r is τ

ϕn,r

πn,r

0 → R(−n) −→ R(−(r + 1))n −−−−→ R(−r)n −−−→ In,r → 0 iff

n−1 2

≤ r ≤ n − 2.

Proof. It is an immediate consequence of Theorem 2.10. 2 Using the previous corollary we can get easily the Hilbert function of R/In,r . Corollary 2.12. If

n−1 2

≤ r ≤ n − 2, for the Hilbert function H of R/In,r we have ⎧ ⎨i + 1 Δn−2 H(i) = i + 1 − n ⎩ 0

Proof. It is an immediate consequence of Corollary 2.11.

for 0 ≤ i ≤ r − 1 for r ≤ i ≤ n − 2 for i ≥ n − 1. 2

3. Subschemes of star configurations In this section we will put into relations the ideals In,r which arise from the matrices An,r with the ideals ISn,h , defining particular subschemes of the star configurations. In the following proposition we state the numerical conditions such that a monomial prime ideal is a minimal prime for In,r . Proposition 3.1. Let 1 ≤ a1 < . . . < at ≤ n. We set at+1 = a1 + n and at+2 = a2 + n. Then 1) In,r ⊆ (xa1 , . . . , xat ) iff ai+1 − ai ≤ r for 1 ≤ i ≤ t. 2) (xa1 , . . . , xat ) is a minimal prime for In,r iff ai+1 − ai ≤ r and ai+2 − ai ≥ r + 1 for 1 ≤ i ≤ t. Proof. 1) If aj+1 − aj ≥ r + 1 for some j then fraj ∈ / In,r , so In,r  (xa1 , . . . , xat ). Conversely let us suppose that ai+1 − ai ≤ r for 1 ≤ i ≤ t. Since In,r = (fr1 , . . . , frn ) we need to show that frh ∈ (xa1 , . . . , xat ) for

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every h. Say j is the integer such that aj ≤ h < aj+1 . Then aj+1 ≤ aj + r ≤ h + r, therefore xaj+1 divides frh . 2) If aj+2 − aj ≤ r for some j then In,r ⊆ (xa1 , . . . , xaj , xaj+2 , . . . , xat ) (since the conditions of item 1 are verified), so (xa1 , . . . , xat ) is not a minimal prime for In,r . Conversely it is enough to show that every ideal of the type (xa1 , . . . , xaj , xaj+2 , . . . , xat ) does not contain In,r . It is true since aj+2 − aj ≥ r + 1 by the assumptions. 2 Corollary 3.2. We have In,r ⊆ (xi , xj ) iff n − r ≤ |i − j| ≤ r. In particular ht In,r = 2 iff n ≤ 2r. Proof. It is a straightforward consequence of Proposition 3.1. 2 From Corollary 2.11 and Corollary 3.2, we deduce the following result. Theorem 3.3. If n is odd then In, n−1 is a Gorenstein ideal of height 3. 2

Proof. By Corollary 2.11 a graded minimal free resolution of In, n−1 is 2

n ϕ n−1 n π n−1    n, n, n−1 n+1 2 0 → R(−n) −→ R − −−−−−−→ R − −−−−2−→ In, n−1 → 0. 2 2 2 τ

n−1 Furthermore by Remark 3.2, In, n−1 ⊆ (xi , xj ) iff n+1 2 ≤ |i − j| ≤ 2 , i.e. the primary decomposition of 2 In, n−1 does not contain any prime ideal of height 2. Thus since by Theorem 1.1 the maximal height of a 2 prime ideal in the primary decomposition of In, n−1 is less than or equal to its projective dimension i.e. 3, 2 we get that In, n−1 is equidimensional of height 3, i.e. it is a Gorenstein ideal. 2 2

Proposition 3.4. For every r such that

n 2

≤ r ≤ n − 2, we have In,r ⊆ ISn,n−1−r .

Proof. Since, by definition, 

ISn,n−1−r =

(xi , xj ),

n−r≤|i−j|≤r

the result follows by Corollary 3.2.

2

Now we set In,r =



(xi , xj )

Λ

  where Λ = {i, j} ∈ C2,n | In,r ⊆ (xi , xj ) . Corollary 3.5. For every r such that

n 2

≤ r ≤ n − 2, we have In,r = ISn,n−1−r .

Proof. It is a direct consequence of Proposition 3.4 and Corollary 3.2.

2

In Theorem 3.6 we will see when In,r defines a 2-codimensional equidimensional subscheme of a star configuration.

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Theorem 3.6. Let r be an integer,

n 2

≤ r ≤ n − 2. We have

In,r = ISn,n−1−r

iff

2n − 2 ≤ r ≤ n − 2. 3

Proof. Let p = (x1 , xn−r , xr+2 ), with n2 ≤ r ≤ n − 2. Since (n − r) − 1 < r, (r + 2) − (n − r) = 2r − n + 2 ≤ r and (n + 1) − (r + 2) = n − r − 1 < r, we see that In,r ⊆ p, by Proposition 3.1. Now we would like to see when p is redundant in the primary decomposition of In,r . Note that In,r  (x1 , xn−r ) and In,r  (x1 , xr+2 ). Therefore p is redundant iff In,r ⊆ (xn−r , xr+2 ) iff n − r ≤ (r + 2) − (n − r) ≤ r iff n − r ≤ 2r − n + 2 iff 2n − 2 ≤ 3r. Therefore if 2n − 2 > 3r, In,r is not equidimensional, hence In,r = ISn,n−1−r . Now let us suppose that 2n−2 ≤ r ≤ n − 2. By Proposition 3.5 and by Theorem 1.1, it is enough to 3 prove that every prime ideal (xa , xb , xc ), 1 ≤ a < b < c ≤ n, containing In,r , is redundant in its primary decomposition. Let In,r ⊆ (xa , xb , xc ), with 1 ≤ a < b < c ≤ n. By Proposition 3.1, the following system of inequalities holds ⎧ ⎨b − a ≤ r c−b≤r ⎩ n + a − c ≤ r. If n + a − b ≤ r, then In,r ⊆ (xa , xb ). If n + b − c ≤ r, then In,r ⊆ (xb , xc ). So we can assume that n + a − b ≥ r + 1 and n + b − c ≥ r + 1. Then we get 2r + 2 ≤ a − c + 2n ≤ a − c + 3r + 2 ⇒ c − a ≤ r, hence In,r ⊆ (xa , xc ), i.e. the prime ideal (xa , xb , xc ) is redundant in the primary decomposition of In,r .

2

Remark 3.7. From Corollary 2.12 and from Theorem 3.6 we can deduce the Hilbert function of ISn,h , for 1 ≤ h ≤ n−1 3 . Now we set Un,h =

h 

Dn−k , 1 ≤ h ≤ n − 1.

k=1

Un,h is a towerizable set (see Definition 3.2 and Theorem 2.6 in [3]), so IUn,h is a Cohen–Macaulay ideal for every h. Moreover one can verify easily that Un,h ∪ Sn,h = Un,n−1−h , 1 ≤ h ≤

n−2 . 2

As an application of the previous result we get the following result. Proposition 3.8. If 1 ≤ h ≤

n−1 3

then IUn,h + ISn,h is a Gorenstein ideal of codimension 3.

Proof. Let us consider the following short exact sequence 0 → IUn,n−1−h → IUn,h ⊕ ISn,h → IUn,h + ISn,h → 0. 2n−2 If 1 ≤ h ≤ n−1 ≤ n − 1 − h ≤ n − 2, so, by Theorem 3.6, we get that ISn,h = In,n−1−h . Therefore, 3 , then 3 by Corollary 2.11, the graded minimal free resolution of ISn,h is

0 → R(−n) → R(−(n − h))n → R(−(n − 1 − h))n → ISn,h → 0.

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On the other hand we know the graded minimal free resolution of IUn,h , for every h. More precisely 0 → R(−(h + 1))h → R(−h)h+1 → IUn,h → 0 and 0 → R(−(n − h))n−h−1 → R(−(n − 1 − h))n−h → IUn,n−1−h → 0. Using the mapping cone construction one can see that pd(IUn,h + ISn,h ) ≤ 3, so ht(IUn,h + ISn,h ) ≤ 3. On the other hand no primes of height 2 can appear in the primary decomposition of IUn,h + ISn,h since it should appear in both the primary decompositions of IUn,h and ISn,h . Moreover all the minimal generators of IUn,n−1−h are a part of a minimal set of generators of ISn,h and the same happens for the syzygies, so by deleting redundant terms, we get the following graded minimal free resolution for IUn,h + ISn,h 0 → R(−n) → R(−(h + 1))h ⊕ R(−(n − h))h+1 → R(−h)h+1 ⊕ R(−(n − 1 − h))h → IUn,h + ISn,h → 0. Therefore IUn,h + ISn,h is a Gorenstein ideal of height 3. 2 The previous result can be set within the context of a property of liaison theory proved in Remark 1.4 in [9] and studied also in [11]. Proof of Lemma 2.2. If i < j we have to show that

fri fn−r−1,r+j =

r+i

j−1

xh

h=r+1

r+n

xh

h=i+1

xh .

h=r+j+1

We need to distinguish some cases. 1) r + j + 1 ≤ n and j − 1 ≤ r + i. Then j−1

fn−r−1,r+j =

n

xh

h=1

xh ,

h=j+r+1

so

fri fn−r−1,r+j =

i h=1

j−1

xh

x2h

h=i+1

i+r

n

xh

h=j

xh .

h=j+r+1

Meanwhile r+i h=r+1

xh

j−1 h=i+1

xh

r+n

xh =

h=r+j+1

r+i

xh

h=1

=

i h=1

n h=j+r+1

xh

j−1 h=i+1

j−1

xh

x2h

xh

h=i+1 i+r h=j

xh

n h=j+r+1

xh .

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2) r + j + 1 ≤ n and j − 1 > r + i. Then

fn−r−1,r+j =

j−1

n

xh

h=1

xh ,

h=j+r+1

so

fri fn−r−1,r+j =

i

i+r

xh

h=1

j−1

x2h

h=i+1

n

xh

h=i+r+1

xh .

h=j+r+1

Meanwhile r+i

j−1

xh

h=r+1

r+n

xh

h=i+1

xh =

h=r+j+1

r+i

xh

h=1

=

i

n h=j+r+1

xh

h=1

i+r

j−1

xh

j−1

x2h

h=i+1

xh

h=i+1 n

xh

h=i+r+1

h=j+r+1

3) r + j + 1 > n, j − 1 ≤ r + i and i + 1 ≤ r + j + 1 − n. Then j−1

fn−r−1,r+j =

xh ,

h=r+j+1−n

so

fri fn−r−1,r+j =

r+j−n

j−1

xh

h=i+1

x2h

h=r+j−n+1

r+i

xh .

h=j

Meanwhile r+i h=r+1

xh

j−1 h=i+1

xh

r+n

r+i

xh =

h=r+j+1

h=r+j+1−n

=

r+j−n

j−1

xh

j−1

xh

h=i+1

xh

h=i+1

x2h

h=r+j−n+1

4) r + j + 1 > n, j − 1 ≤ r + i and i + 1 > r + j + 1 − n. Then j−1

fn−r−1,r+j =

xh ,

h=r+j+1−n

so

fri fn−r−1,r+j =

i h=r+j+1−n

xh

j−1 h=i+1

x2h

r+i h=j

xh .

r+i h=j

xh .

xh .

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Meanwhile r+i

j−1

xh

h=r+1

xh

h=i+1

r+n

r+i

xh =

h=r+j+1

xh

h=r+j+1−n i

=

j−1

xh

h=i+1 j−1

xh

h=r+j+1−n

x2h

h=i+1

r+i

xh .

h=j

5) r + j + 1 > n, j − 1 > r + i and i + 1 ≤ r + j + 1 − n. As before we get fri fn−r−1,r+j =

r+j−n

i+r

xh

h=i+1

j−1

x2h

h=r+j−n+1

xh .

h=i+r+1

Meanwhile r+i

xh

h=r+1

j−1

xh

h=i+1

r+n

r+i

xh =

h=r+j+1

h=r+j+1−n

=

r+j−n

j−1

xh

i+r

xh

h=i+1

xh

h=i+1

x2h

h=r+j−n+1

j−1

xh .

h=i+r+1

6) r + j + 1 > n, j − 1 > r + i and i + 1 > r + j + 1 − n. As before we get i

fri fn−r−1,r+j =

i+r

xh

h=r+j+1−n

j−1

x2h

h=i+1

xh .

h=i+r+1

Meanwhile r+i h=r+1

xh

j−1 h=i+1

xh

r+n

xh =

h=r+j+1

r+i

xh

h=r+j+1−n

=

i

j−1

xh

h=i+1

xh

h=r+j+1−n

i+r

x2h

h=i+1

j−1

xh .

h=i+r+1

Analogously one can verify that

fri fn−r−1,r+j =

j−1 h=1

when i ≥ j.

xh

r+i h=r+j+1

xh

n

xh

h=i+1

2

References [1] C. Bocci, B. Harbourne, Comparing powers and symbolic powers of ideals, J. Algebr. Geom. 19 (2010) 399–417. [2] S. Faridi, The projective dimension of sequentially Cohen–Macaulay monomial ideals, arXiv:1310.5598v2. [3] G. Favacchio, A. Ragusa, G. Zappalà, Tower sets and other configurations with the Cohen–Macaulay property, J. Pure Appl. Algebra 219 (2015) 2260–2278. [4] A.V. Geramita, B. Harbourne, J. Migliore, Star configurations in Pn , J. Algebra 376 (2013) 279–299. [5] A.V. Geramita, J. Migliore, L. Sabourin, On the first infinitesimal neighborhood of a linear configuration of points in P 2 , J. Algebra 298 (2006) 563–611.

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[6] B. Harbourne, C. Huneke, Are symbolic powers highly evolved?, J. Ramanujan Math. Soc. 28A (2013) 247–266. [7] S. Morey, R.H. Villarreal, Edge ideals: algebraic and combinatorial properties, in: Progress in Commutative Algebra 1, de Gruyter, Berlin, 2012, pp. 85–126. [8] J.P. Park, Y. Shin, The minimal free graded resolution of a star-configuration in P n , J. Pure Appl. Algebra 219 (2015) 2124–2133. [9] C. Peskine, L. Szpiro, Liaison des variétés algébriques. I, Invent. Math. 26 (1974) 271–302. [10] A. Ragusa, G. Zappalà, Partial intersection and graded Betti numbers, Beitr. Algebra Geom. 44 (1) (2003) 285–302. [11] A. Ragusa, G. Zappalà, Properties of 3-codimensional Gorenstein schemes, Commun. Algebra 29 (1) (2001) 303–318.