Homological properties of determinantal arrangements

Journal of Algebra 471 (2017) 220–239

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Journal of Algebra www.elsevier.com/locate/jalgebra

Homological properties of determinantal arrangements Arnold Yim 1 Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, IN 47907-2067, United States

a r t i c l e

i n f o

Article history: Received 4 June 2015 Available online xxxx Communicated by Luchezar L. Avramov MSC: 13N15 32S22 Keywords: Logarithmic derivations Free divisor Hyperplane Determinantal Arrangement Supersolvable Chordal Homotopy group Poincaré polynomial

a b s t r a c t We explore a natural extension of braid arrangements in the context of determinantal arrangements. We show that these determinantal arrangements are free divisors. Additionally, we prove that free determinantal arrangements defined by the minors of 2 × n matrices satisfy nice combinatorial properties. We also study the topology of the complements of these determinantal arrangements, and prove that their higher homotopy groups are isomorphic to those of S 3 . Furthermore, we find that the complements of arrangements satisfying those same combinatorial properties above have Poincaré polynomials that factor nicely. © 2016 Elsevier Inc. All rights reserved.

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

E-mail address: [email protected]. I would like to express my gratitude to my advisor Uli Walther for his guidance throughout this whole project. I would also like to thank the reviewer for her helpful comments and suggestions. Partial support by the NSF under grant DMS-1401392 is gratefully acknowledged. 1

http://dx.doi.org/10.1016/j.jalgebra.2016.09.019 0021-8693/© 2016 Elsevier Inc. All rights reserved.

A. Yim / Journal of Algebra 471 (2017) 220–239

2. Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Freeness of determinantal arrangements . . . 4. Complements of determinantal arrangements References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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222 224 232 238

1. Introduction In this paper, we investigate a family of hypersurfaces known as determinantal arrangements. Determinantal arrangements are unions of hypersurfaces defined by the minors of a matrix of indeterminates. We focus on determinantal arrangements defined by the 2-minors of a 2 ×n generic matrix. Since these determinantal arrangements can be thought of as natural generalizations of braid arrangements and graphic arrangements, we aim to extend the results known for these hyperplane arrangements to our new setting. In particular, we study the freeness of these arrangements, and the topology of their complements. Let D be a divisor in an n-dimensional complex analytic manifold X. The module of logarithmic derivations DerX (− log D) := {θ ∈ DerX |θ(OX (−D)) ⊆ OX (−D)} are the vector fields on X that are tangent to the smooth points of D. If DerX (− log D) is locally free, then D is called a free divisor. Free divisors were first introduced by Saito [12], motivated by his study of the discriminants of versal deformations of isolated hypersurface singularities. The study of free divisors coming from discriminants of versal deformations has since been a driving force in the theory of singularities (see [9,2,19,18,17]). Aside from versal deformations, free divisors show up naturally in many different settings. For example, many of the classically arising hyperplane arrangements are free (see [10]). This includes braid arrangements and all Coxeter arrangements. In general, it is not clear which divisors are free and which are not. Naturally, one might be interested in freeness for arrangements of more general hypersurfaces. For example, Schenck and Tohˇ aneanu [14] give conditions for when an arrangement of lines 2 and conics on P is free. For determinantal arrangements, Buchweitz and Mond [3] showed that the arrangement defined by the product of the maximal minors of a n × (n + 1) matrix of indeterminates is free. More recently, Damon and Pike [5] showed that certain determinantal arrangements coming from symmetric, skew-symmetric and square generic matrices are free and have complements that are K(π, 1). In both of these cases, the arrangements turn out to be linear free divisors (i.e. the basis for DerX (− log D) is generated by linear vector fields). The vector fields arising in these situations correspond to matrix group actions on the generic matrix which stabilize the divisor D. Many interesting determinantal arrangements, however, are not linear free divisors as our next example shows.

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Example 1.1. Let M be the 2 × 4 matrix of indeterminates 

 M=

x1 y1

x2 y2

x3 y3

x4 y4

,

and for i < j, let Δij be the 2-minor ofM using the i-th and j-th columns, Δij = xi yj − xj yi . Let f be the product f = Δij . Then DerX (− log Var(f )) is free with i
basis consisting of 7 linear derivations (coming from SL(2, C)-action,  column-scaling,  ∂ ∂ and row-scaling on M ), and one derivation of degree 5: θ = Δ24 Δ34 x1 ∂x + y 1 ∂y4 . 4 The fact that there does not exist a basis consisting of only linear derivations follows from Saito’s Criterion which is described in Section 2. This paper focuses on determinantal arrangements like the one in the previous example. Because of their similarities to braid arrangements, we call them determinantal braid arrangements. In Theorem 3.3, we show that determinantal braid arrangements are free divisors. Furthermore, we prove in Theorem 3.5 that free determinantal arrangements satisfy certain combinatorial properties. Interestingly, freeness can also give us topological information. Specifically, Terao proves in [16] that for a free hyperplane arrangement, the Poincaré polynomial for the complement is determined by the degrees of the vector fields in the basis of the module of logarithmic derivations: Theorem 1.2 (Terao). Let A ⊂ Cn be a free central hyperplane arrangement and suppose n  that DerCn (− log A) ∼ C[x1 , . . . , xn ](−bi ), then = i=1

Poin(Cn \ A, t) =

n 

(1 + bi t).

i=1

Inspired by the work done for hyperplane arrangements, we explore the topology for the complements of our determinantal arrangements. In Theorem 4.2, we prove that the complements of our arrangements have higher homotopy groups isomorphic to those of S 3 . Finally, in Theorems 4.5 and 4.6, we show that the Poincaré polynomial of the complement of a free determinantal arrangement factors nicely. 2. Setup We use X = Cn with coordinate ring R  = C[x1 , . . . , xn ]. Let DerX be the free ∂ R-module of vector fields on X generated by ∂x . For f ∈ R reduced and divisor i i=1..n

D = Var(f ), we can define the module of logarithmic derivations as follows:

A. Yim / Journal of Algebra 471 (2017) 220–239

223

Definition 2.1. The module of logarithmic derivations along D = Var(f ) is the R-module DerX (− log D) = {θ ∈ DerX |θ(f ) ∈ (f )}. We want to know when D has a well-behaved singular locus, thus we are interested in when the module of logarithmic derivations along D is free. We say: Definition 2.2. A divisor D in X is free if DerX (− log D) is a free R-module. To determine whether a divisor is free, we use Saito’s criterion [12]: Theorem 2.3 (Saito). A divisor D = Var(f ) is a free divisor if and only if there exist n elements θj =

n

i=1

gij

∂ ∈ DerX (− log D) ∂xi

such that det((gij )) = c · f for some non-zero c ∈ C. We focus on logarithmic derivations for hypersurface arrangements defined by graphs. In the context of hyperplane arrangements, these are called graphic arrangements. Given a graph G with n vertices, we associate a hyperplane arrangement defined by a polynomial f ∈ C[x1 , . . . xn ]. For each edge of G between vertices vi and vj , we include the hyperplane defined by xi − xj = 0 in the arrangement. For example, the graphic argraph on n vertices is the braid arrangement on n rangement associated to a complete  (xi − xj ). variables defined by f = 1≤i
Due to a result by Stanley [15], one knows that a graphic arrangement is free if and only if its corresponding graph is chordal (i.e. a graph for which every cycle of length greater than 3 has an edge between two non-consecutive vertices). More recently, Kung and Schenck [8] improved this result and found that pdimR (DerX (− log D)) ≥ k − 3 where D is a graphic arrangement with longest chord-free induced cycle of length k. We will be using a characterization of chordality given by Fulkerson and Gross [6]: Definition 2.4. A graph G is chordal if and only if there exists an ordering of vertices, such that for each vertex v, the induced subgraph on v and its neighbors that occur before it in the sequence is a complete graph. While freeness is well understood for graphic arrangements, it is still unclear when we consider arrangements of more general hypersurfaces. We investigate certain determinantal arrangements associated to graphs. Specifically, let M be the 2 × n matrix of indeterminates

A. Yim / Journal of Algebra 471 (2017) 220–239

224

 x1 y1

M=

x2 y2

· · · xn · · · yn

 .

For i < j, let Δij denote the 2-minor of M using the i-th and j-th columns, Δij = xi yj − xj yi . Definition 2.5. For each graph G with n vertices, we can associate a determinantal arrangement, AG , consisting of the determinantal varieties Var(Δij ) for each edge between vertices vi and vj of G. 3. Freeness of determinantal arrangements For hyperplane arrangements, the braid arrangement is a well-known example of an arrangement that is free. We define something similar in the context of determinantal arrangements. Consider the determinantal arrangement on  M=

defined by f =



x1 y1

x2 y2

· · · xn · · · yn

 ,

Δij . One can consider points on this arrangement as a selection

1≤i
of n ordered vectors in C2 such that two of these vectors are linearly dependent. Whereas the braid arrangement can be thought of as a selection of n ordered points in C such that two of these points are the same. Because of this analogy, we refer to the arrangement defined by f as the determinantal braid arrangement. In Theorem 3.3, we prove that the determinantal braid arrangement is free: we construct a generating set for the module of logarithmic derivations and show that this set satisfies Saito’s criterion. In Theorem 3.5, we prove that if a graph is not chordal, then the corresponding determinantal arrangement is not free. We show that near a particular point, our arrangement looks like the cyclic graphic arrangement which has projective dimension related to the length of the cycle. Before proving Theorem 3.3, we will need the two following lemmas: Lemma 3.1. For n ∈ Z>0 , let si,j,k denote the degree k symmetric polynomial on the variables zi , . . . , zn that is linear in each variable omitting the variable zj , given by si,j,k =



zα1 zα2 · · · zαk ,

αm =j i≤α1 <···<αk ≤n

and let si,j,0 = 1. Let Ai denote the (n + 1 − i) × (n + 1 − i) matrix (si,j,k ), where the row index j ranges from i to n, and the column index k ranges from 0 to n − i. Then

A. Yim / Journal of Algebra 471 (2017) 220–239

⎛ det(Ai ) = ⎝



225

⎞ (zi − zs )⎠ det(Ai+1 ).

i
Proof. Writing out Ai , we have ⎛

1 ⎜ ⎜1 Ai = ⎜ ⎜ .. ⎝. 1

(zi+1 + zi+2 + · · · + zn ) (zi + zi+2 + · · · + zn ) .. . (zi + zi+1 + · · · + zn−1 )

⎞ · · · (zi+1 zi+2 · · · zn ) ⎟ ··· (zi zi+2 · · · zn ) ⎟ ⎟. .. .. ⎟ . . ⎠ · · · (zi zi+1 · · · zn−1 )

Subtracting the first row from every other row, we have ⎛1 0 ⎜0 ⎜ ⎝ .. . 0

(zi+1 + zi+2 + . . . + zn ) (zi − zi+1 ) (zi − zi+2 ) . . . (zi − zn )

(zi+1 zi+2 + zi+1 zi+3 + · · · + zn−1 zn ) (zi − zi+1 )(zi+2 + zi+3 + · · · + zn ) (zi − zi+2 )(zi+1 + zi+3 + · · · + zn ) . . . (zi − zn )(zi+1 + zi+2 + · · · + zn−1 )

··· ··· ··· .. . ···

⎞ (zi+1 zi+2 · · · zn ) (zi − zi+1 )(zi+2 zi+3 · · · zn ) ⎟ (zi − zi+2 )(zi+1 zi+3 · · · zn ) ⎟ ⎠. . . . (zi − zn )(zi+1 zi+2 · · · zn−1 )

We can factor the lower right (n − i) × (n − i) submatrix as ⎛ ⎜ ⎜ ⎜ ⎜ ⎝

⎞

(zi − zi+1 )

⎟ ⎟ ⎟ ⎟ ⎠

(zi − zi+2 ) ..

. (zi − zn )

⎛

1 (zi+2 + zi+3 + · · · + zn ) ⎜ (zi+1 + zi+3 + · · · + zn ) ⎜1 ×⎜ .. ⎜ .. . ⎝. 1 (zi+1 + zi+2 + · · · + zn−1 ) ⎛ (zi − zi+1 ) ⎜ (zi − zi+2 ) ⎜ =⎜ .. ⎜ . ⎝

⎞ ··· (zi+2 zi+3 · · · zn ) ⎟ ··· (zi+1 zi+3 · · · zn ) ⎟ ⎟ .. .. ⎟ . . ⎠ · · · (zi+1 zi+2 · · · zn−1 ) ⎞ ⎟ ⎟ ⎟ Ai+1 , ⎟ ⎠

(zi − zn ) ⎛ thus det(Ai ) = ⎝



⎞ (zi − zs )⎠ det(Ai+1 ). 2

i
We also require the following lemma on block matrices. While a more general result on the determinants of block matrices exists, the following statement suffices for the proof of the main result of this section.

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 A1 A2 Lemma 3.2. Let A be a block matrix A = with blocks of size n × n with A3 A4 entries in C(z1 , . . . , zn ). If A1 and A3 are diagonal matrices with nonzero entries, then det(A) = det(A1 A4 − A3 A2 ).   A−1 0 1 Proof. Let B be the block matrix B = , then 0 A−1 3 



In In

A−1 1 A2 A−1 3 A4

In 0

A−1 1 A2 −1 A3 A4 − A−1 1 A2

In 0

0 A1 A3

BA =

.

Using row reduction, we find  det(BA) = det  Now, let C be the block matrix C =  det(CBA) = det

In 0

 .



A−1 1 A2 A1 A4 − A3 A2

, then  = det(A1 A4 − A3 A2 ).

Since det(CBA) = det(A), we have det(A) = det(A1 A4 − A3 A2 ). 2 Now, we have our main result of this section: Theorem 3.3. Let G be the complete graph on n vertices for n ≥ 3. The determinantal arrangement AG is free. Proof. If G is the complete graph on n vertices, then the corresponding determinantal arrangement AG is the determinantal braid arrangement defined by f=





Δij =

1≤i
xi yj − xj yi .

1≤i
We provide a set of elements in DerX (− log Var(f )), and show that this set actually forms a basis for DerX (− log Var(f )) according to Saito’s criterion. We first consider several linear derivations: α=

n

k=1

xk

∂ ∂yk

A. Yim / Journal of Algebra 471 (2017) 220–239 n

β=

yk

∂ ∂xk

yk

∂ . ∂yk

k=1

γ=

n

k=1

227

To show that these derivations belong to DerX (− log Var(f )), we show that they stabilize the ideal of each minor, and thus they stabilize the ideal of the product of the minors: α(Δij ) = 0. Since α stabilizes each (Δij ), α ∈ DerX (− log Var(f )). Similarly, β(Δij ) = 0, and γ(Δij ) = Δij , thus β, γ ∈ DerX (− log Var(f )). We also have n linear derivations θ k = xk

∂ ∂ + yk ∂xk ∂yk

for k = 1, 2 . . . , n. We have θk (Δkj ) = Δkj , and θk (Δik ) = Δik . When i, j = k, θk (Δij ) = 0, thus θk stabilizes each (Δij ). This shows that θk ∈ DerX (− log Var(f )). Finally, we have n − 3 elements of degree n + 1. For k = 4, 5, .., n, let τk be a bijection of sets from {1, . . . , n − 4} to {4, . . . , k − 1, k + 1, . . . n}, and let Sn−4 be the symmetric group on the numbers {1, . . . , n − 4}. For m = 0, 1, . . . , n − 4, define am,k =

1 m!(n − 4 − m)!



x(τk ◦σ)(1) · · · x(τk ◦σ)(m) y(τk ◦σ)(m+1) · · · y(τk ◦σ)(n−4) .

σ∈Sn−4

Now, consider the derivations ϕm =

n

k=4

  ∂ ∂ . am,k Δ2k Δ3k x1 + y1 ∂xk ∂yk

If i, j < 4, then ϕm (Δij ) = 0. Now, suppose that i < 4 and j ≥ 4, then ϕm (Δij ) = am,j Δ2j Δ3j (−x1 yi + y1 xi ) . When i = 2, 3, ϕm (Δij ) ∈ (Δij ) ⊆ R, and when i = 1, ϕm (Δij ) = 0.

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228

If i, j ≥ 4, ϕm (Δij ) = am,i Δ2i Δ3i Δ1j − am,j Δ2j Δ3j Δ1i . Note that each term in am,i either has a factor of xj or yj , and also note that the terms in am,j are exactly the terms in am,i , with xi and yi instead of xj and yj respectively, thus it is enough to show that xj Δ2i Δ3i Δ1j − xi Δ2j Δ3j Δ1i and yj Δ2i Δ3i Δ1j − yi Δ2j Δ3j Δ1i are divisible by Δij . Using Plücker relations, we can write: xj Δ2i Δ3i Δ1j − xi Δ2j Δ3j Δ1i

=

xj Δ3i (Δ1i Δ2j − Δ12 Δij ) − xi Δ2j Δ3j Δ1i

=

Δ1i Δ2j (xj Δ3i − xi Δ3j ) − xj Δ3i Δ12 Δij

=

Δ1i Δ2j (−x3 Δij ) − xj Δ3i Δ12 Δij ∈ (Δij ),

and similarly, yj Δ2i Δ3i Δ1j − yi Δ2j Δ3j Δ1i = Δ1i Δ2j (−y3 Δij ) − yj Δ3i Δ12 Δij ∈ (Δij ). Since ϕm stabilizes each (Δij ), ϕm ∈ DerX (− log Var(f )). It remains to show that {α, β, γ, θ1 , . . . , θn , ϕ0 , . . . , ϕn−4 } in DerX (− log Var(f )) form a basis. According to Saito’s criterion, these derivations form a basis if and only if the determinant of the coefficient matrix is a nonzero constant multiple of f . With our elements, we have the coefficient matrix: ⎛

y1 ⎜ y2 ⎜ ⎜y ⎜ 3 ⎜ ⎜ y4 ⎜ . ⎜ . ⎜ . ⎜ ⎜ yn ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞

x1 x2 x3

xn

a0,4 Δ24 Δ34 x1 .. . a0,n Δ2n Δ3n x1

··· .. . ···

yn

a0,4 Δ24 Δ34 y1 .. . a0,n Δ2n Δ3n y1

··· .. . ···

x4 .. x1 x2 x3 x4 .. . xn

y1 y2 y3 y4 .. . yn

.

y1 y2 y3 y4 ..

.

⎟ ⎟ ⎟ ⎟ ⎟ an−4,4 Δ24 Δ34 x1 ⎟ ⎟ .. ⎟ ⎟ . ⎟ an−4,n Δ2n Δ3n x1 ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ an−4,4 Δ24 Δ34 y1 ⎟ ⎟ ⎟ .. ⎠ . an−4,n Δ2n Δ3n y1

We swap some rows to organize our matrix into blocks (this could potentially change the determinant by a sign, but that is unimportant in checking Saito’s criterion):

A. Yim / Journal of Algebra 471 (2017) 220–239

⎛y

⎞

x1

1

⎜ y2 ⎜ y3 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ y4 ⎜ . ⎜ . ⎜ . ⎜ yn ⎜ ⎜ ⎜ ⎝

229

x2 x3 x1 x2 x3

y1 y2 y3

y1 y2 y3 x4 ..

. xn

x4 .. . xn

y4 .. . yn

y4 ..

. yn

a0,4 Δ24 Δ34 x1 .. . a0,n Δ2n Δ3n x1 a0,4 Δ24 Δ34 y1 .. . a0,n Δ2n Δ3n y1

··· .. . ··· ··· .. . ···

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ an−4,4 Δ24 Δ34 x1 ⎟ . ⎟ .. ⎟ . ⎟ an−4,n Δ2n Δ3n x1 ⎟ ⎟ an−4,4 Δ24 Δ34 y1 ⎟ ⎟ .. ⎠ . an−4,n Δ2n Δ3n y1



 A 0 Denote the matrix above by N , with blocks N = . Since N is a triangular C D block matrix, det(N ) = det(A) det(D). By explicit computation, we find that det(A) = Δ12 Δ13 Δ23 .

(3.1)

To calculate the determinant of D, we split the matrix into more blocks: ⎛  D=

D1 D3

D2 D4



x4

⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ y4 ⎜ ⎜ ⎝

..

. xn

..

. yn

a0,4 Δ24 Δ34 x1 .. . a0,n Δ2n Δ3n x1 a0,4 Δ24 Δ34 y1 .. . a0,n Δ2n Δ3n y1

⎞ · · · an−4,4 Δ24 Δ34 x1 ⎟ .. .. ⎟ . . ⎟ · · · an−4,n Δ2n Δ3n x1 ⎟ ⎟ ⎟. · · · an−4,4 Δ24 Δ34 y1 ⎟ ⎟ .. ⎟ .. ⎠ . . · · · an−4,n Δ2n Δ3n y1

Using Lemma 3.2, we know that det(D) and det(D1 D4 − D3 D2 ) agree over the field of fractions C(x1 , . . . , xn , y1 , . . . , yn ). But since these determinants are polynomials, we must have det(D) = det(D1 D4 − D3 D2 ). Now, ⎞ a0,4 Δ24 Δ34 (y1 x4 − x1 y4 ) · · · an−4,4 Δ24 Δ34 (y1 x4 − x1 y4 ) ⎟ ⎜ .. .. .. D1 D4 − D3 D2 = ⎝ ⎠ . . . a0,n Δ2n Δ3n (y1 xn − x1 yn ) · · · an−4,n Δ2n Δ3n (y1 xn − x1 yn ) ⎞ ⎛ a0,4 Δ24 Δ34 Δ14 · · · an−4,4 Δ24 Δ34 Δ14 ⎟ ⎜ .. .. .. = −⎝ ⎠ . . . a0,n Δ2n Δ3n Δ1n · · · an−4,n Δ2n Δ3n Δ1n ⎛ ⎞⎛ ⎞ Δ24 Δ34 Δ14 a0,4 · · · an−4,4 ⎜ ⎟ ⎜ .. .. ⎟ .. .. = −⎝ ⎠⎝ . . . . ⎠ Δ2n Δ3n Δ1n a0,n · · · an−4,n ⎛

=: −D5 D6 .

230

A. Yim / Journal of Algebra 471 (2017) 220–239

Observe that

det(D5 ) =

3  n 

Δij ,

(3.2)

i=1 j=4

therefore it remains to show that det(D6 ) is a nonzero constant multiple of the product of all minors using the last n − 3 columns of M . We show that each Δij for i, j ≥ 4 divides det(D6 ) by showing that det(D6 ) vanishes on Var(Δij ). Indeed, Δij vanishes when columns i and j of M are scalar multiples of each other. Write xj = cxi and yj = cyi . Looking to rows i and j of D6 , we have am,j = cam,i , and since these rows are scalar multiples of each other, det(D6 ) vanishes here which implies that each Δij divides det(D6 ). The degree of the product of the n−3 minors, 2 = (n − 3)(n − 4), is the same as the degree of det(D6 ), hence 2 det(D6 ) is a constant multiple of the product of the minors. To check that det(D6 ) is not identically zero, we substitute yk = 1 into D6 to get the matrix in Lemma 3.1 on the variables x4 , . . . , xn , thus if x4 = x5 = · · · = xn , then det(D6 ) = 0. With equations (3.1) and (3.2), we find det(N ) = (−1)n−3 det(A) det(D5 ) det(D6 ) is a constant multiple of the product of all of the minors of M . By Saito’s criterion, {α, β, γ, θ1 , . . . , θn , ϕ0 , . . . , ϕn−4 } form a basis for DerX (− log Var(f )), hence our determinantal arrangement is free. 2 We believe that our work with determinantal arrangements on 2 × n generic matrices only scratches the surface of a broader class of free divisors. For example, we can change the size of our generic matrix. In the case where m = 3 and n = 4, one knows that the arrangement is a linear free divisor (see [3,7]). However, in the next case, m = 3 and n = 5, we already do not know whether or not the arrangement is free. More generally, one can ask: Question 3.4. Let M be the m × n matrix of indeterminates, and let f be the product of all maximal minors of M . While it is known that Var(f ) is free when M is n × (n + 1) (see [3,4]), is the arrangement defined by f free for any n > m > 2? One can also consider determinantal arrangements defined by subgraphs of the complete graph. Much like hyperplane arrangements, we find that the freeness of the determinantal arrangement is related to whether or not the graph is chordal. Theorem 3.5. If a determinantal arrangement AG is free, then G is chordal. Moreover, if G has a chord-free induced cycle of length k, then pdimR (DerX (− log AG )) ≥ k − 3.

A. Yim / Journal of Algebra 471 (2017) 220–239

231

Proof. Suppose that G is not chordal, then G has an chord-free induced cycle of length k where 4 ≤ k ≤ n. We can reorganize the columns of M so that this chord-free induced cycle occurs on the first k vertices of AG . To show that AG = Var(f ) is not free, we will localize to a neighborhood of the point  p=

1 0

··· 1 1 1 ··· 1 ··· 0 1 2 ··· n − k

 .

We consider f in the local ring C[x1 , . . . , xn , y1 , . . . , yn ]mp where mp is the maximal ideal associated to the point p. In this local ring, Δij is a unit if i or j is greater than k. Thus, around p, AG looks like Var(Δ12 Δ23 · · · Δ(k−1)k Δ1k ) ⊂ C2n whose associated graph is the cyclic graph on k vertices. We show that p is in the non-free locus of Var(Δ12 Δ23 · · · Δ(k−1)k Δ1k ). In our local ring, xi is a unit for all i, thus  Var(Δ12 Δ23 · · · Δ(k−1)k Δ1k ) = Var

xk−2 xk−2 1 k Δ12 Δ23 · · · Δ(k−1)k Δ1k x22 x23 · · · x2k−1

 .

But, xk−2 xk−2 1 k Δ12 Δ23 · · · Δ(k−1)k Δ1k x22 x23 · · · x2k−1 xk−2 xk−2 1 k (x1 y2 − x2 y1 )(x2 y3 − x3 y2 ) · · · (xk−1 yk − xk yk−1 )(x1 yk − xk y1 ) x22 x23 · · · x2k−1      x1 x k x 1 xk x1 xk x1 x k y2 − xk y1 y3 − y2 · · · x1 yk − yk−1 (x1 yk − xk y1 ). = x2 x3 x2 xk−1

=

Now, making a change of coordinates z1 z2 .. . zk−1 zk

↔ ↔ .. . ↔ ↔

xk y1 x1 xk x2 y2

.. . x1 xk xk−1 yk−1 x1 yk ,

we have that Var(Δ12 Δ23 · · · Δ(k−1)k Δ1k ) = Var((z2 −z1 )(z3 −z2 ) · · · (zk −zk−1 )(zk −z1 )). Since our point p corresponds to zi = 0 for i = 0, . . . , k on the cyclic graphic arrangement Var((z2 − z1 )(z3 − z2 ) · · · (zk − zk−1 )(zk − z1 )), we know that p is in the non-free locus of Var(Δ12 Δ23 · · · Δ(k−1)k Δ1k ), and thus AG is not free. Moreover, this is a generic hyperplane arrangement so by Rose and Terao [11],

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pdimR (DerX (− log(Δ12 Δ23 · · · Δ(k−1)k Δ1k ))) = k − 3. Since localization is an exact functor, pdimR (DerX (− log (A)G )) ≥ k − 3. 2 Remark 3.6. The converse of Theorem 3.5 is not true. For example, for any chordal graph with a vertex v of degree 2, if the induced subgraph v with its neighbors is not a cycle then the corresponding determinantal arrangement is not free. In this case, the arrangement locally behaves like f = Δ12 Δ13 , and one can check that this arrangement is not free. However, evidence suggests that many of the arrangements with chordal graphs are indeed free. For example, direct computations of small cases suggest that arrangements corresponding to doubly-connected (graphs that remain connected after removing any single vertex) chordal graphs are free. 4. Complements of determinantal arrangements In this section, we investigate the topology of the complements of free determinantal arrangements. We exploit the combinatorial structure of free determinantal arrangements to construct a fibration for the complement. In Theorem 4.2, we use this fibration to show that the higher homotopy groups of the complement behave like the homotopy groups of S 3 . In Theorems 4.5 and 4.6, we show that the Poincaré polynomial factors over Q and give the explicit Poincaré polynomial for the complement of the determinantal braid arrangement. ⎛ ⎞  Δij ⎠ denote the determinantal braid arrangement on a 2 × n Let An = Var ⎝ 1≤i
generic matrix. Now, consider the arrangement in the ambient space of 2 × n matrices with coefficients in C. Let Un = C2n \ An be the complement of the arrangement. To study the topology of Un , consider the fibration p : Un → Un−1 , where p is the projection onto the first (n − 1) columns. This map is well defined because the columns of Un are pairwise linearly independent, and so the first (n − 1) columns are also pairwise linearly independent. The fiber of this map is a selection of a last column that is linearly independent from the first (n − 1) columns. Thus the fibers are homotopy equivalent to C2 minus (n − 1) lines through the origin. This fibration can also be generalized to any free determinantal arrangement on a 2 × n generic matrix M . From Theorem 3.5, we know that the graph associated to the arrangement is chordal. From Definition 2.4, we can order the vertices such that for each vertex v, the induced subgraph on v and its neighbors that occur before it in the sequence is a complete graph. Without loss of generality, assume that our free determinantal arrangement is associated to a graph G with vertices {v1 , . . . , vn } labeled according to the reverse perfect elimination ordering. Let Gk denote the induced subgraph on {v1 , . . . , vk }, then for k = 2, . . . , n, let Uk = C2k \ AGk . Now, for each k = 3, . . . , n, we have a fibration pk : Uk → Uk−1 where pk is the projection onto the first (k − 1) columns. This map is well defined because for each Var(Δij ) ⊂ AGk−1 , we must have

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Var(Δij ) ⊂ AGk from the way the arrangements are defined. Suppose that induced subgraph on vk and its neighbors in Gk is the complete graph on m vertices, then the fibers, Fk , of pk are homotopy equivalent to C2 minus (m − 1) lines through the origin. We also have a fibration p2 : U2 → C2 \ {0} with fibers homotopy equivalent to C2 \ C. Note that pk is only a fibration when the graph associated to the arrangement is chordal. If the graph is not chordal, the fibers are not homotopy equivalent. Example 4.1. Consider the cyclic arrangement on 4 vertices: f = Δ12 Δ23 Δ34 Δ14 , we can follow our procedure of projecting the complement onto the first three columns, however, some fibers look like C2 minus 2 lines (when the first and third column are linearly independent) and other fibers look like C2 minus 1 line (when the first and third column are linearly dependent). When the graph is chordal, this is no longer an issue since all of the relevant columns are guaranteed to be linearly independent. We now use this fibration to prove statements about the topology of Un . Theorem 4.2. Let G be a chordal graph on n vertices labeled according to the reverse perfect elimination ordering. Let Gk denote the induced subgraph of G on the first k vertices, let Uk = C2k \ AGk , and let pk be the fibration described above with fibers Fk . Then for k = 2, . . . , n the following sequence is exact: 0 → π1 (Fk ) → π1 (Uk ) → π1 (Uk−1 ) → 0. Furthermore, πi (Uk ) ∼ = πi (S 3 ) for i ≥ 2. Proof. Denote U1 = C2 \ 0, then for each fibration pk : Uk → Uk−1 for k = 2, . . . n, consider the homotopy long exact sequence (note that our spaces are path-connected, so the reduced homotopy π0 is zero): · · · → π2 (Fk ) → π2 (Uk ) → π2 (Uk−1 ) → π1 (Fk ) → π1 (Uk ) → π1 (Uk−1 ) → 0.

(4.1)

By Proposition 5.6 in [10], every central line arrangement is K(π, 1), so since each Fk is a central line arrangement πi (Fk ) = 0 for i ≥ 2 for each k. From (4.1), πi (Uk ) ∼ = πi (Uk−1 ) for i ≥ 3. Since U1 = C2 \ {0} ∼ = S 3 , for each k, πi (Uk ) ∼ = πi (S 3 ) for i ≥ 3. Furthermore, consider the segment 0∼ = π2 (Fk ) → π2 (Uk ) → π2 (Uk−1 ).

(4.2)

When k = 2, the group on the right in (4.2) is π2 (S 3 ) = 0, which implies that π2 (U2 ) = 0. By induction on k, π2 (Uk ) = 0 for all k. Plugging in π2 (Uk−1 ) = 0 into (4.1) we get the short exact sequence 0 → π1 (Fk ) → π1 (Uk ) → π1 (Uk−1 ) → 0.

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Remark 4.3. Using the short exact sequence in Theorem 4.2, we can compute π1 (U2 ) ∼ = Z. Beyond U2 , we do not know the relations between the generators of π1 (Un ). Inspired by Theorem 1.2, we attempted to find a connection between the generators of the module of logarithmic derivations and the Poincaré polynomial of the complement of a free determinantal arrangement. Unfortunately, there is not a nice relation like the one given by Terao, but we are able to calculate the Poincaré polynomial nonetheless. To calculate the Poincaré polynomial, we will be using the cohomology Serre spectral sequence for the fibration described in this chapter. Since the terms on the E2 page are calculated using local coefficients, we show that the fundamental group on the base space induces the trivial monodromy action on the cohomology of the fiber. This allows us to use constant coefficients to describe the terms on the E2 page of the spectral sequence. As a first step, we will try to understand the fundamental group of our determinantal arrangement complements. Lemma 4.4. Let G be the complete graph on n vertices and let Un = C2n \ AG , then π1 (Un ) is generated by loops γ : [0, 2π] → Un given by  γ(t) =

eit 0

0 1

1 2

1 + e−it

1 3

1 + e−it

··· ···

1 n−1

1 + e−it

 ,

and δ : [0, 2π] → Un given by  δ(t) =

eit eit

1 2

1 3

··· ···

 1 n

,

and loops constructed by permuting the columns of γ and δ. Proof. We will proceed by induction on n. For the base case n = 2, we know  that U2 eit 0 is GL(2, C), and we know that π1 (GL(2, C)) = Z and is generated by γ(t) = 0 1 (which can also be continuously deformed into δ(t)). Assume that the lemma is true for π1 (Un ). To find the generators for π1 (Un+1 ), we use Theorem 4.2. The short exact sequence 0 → π1 (Fn+1 ) → π1 (Un+1 ) → π1 (Un ) → 0 tells us that π1 (Un+1 ) is generated by lifts of generators in π1 (Un ) and the images of generators from π1 (Fn+1 ). To lift the generators from π1 (Un ), we simply add a last column to γ, δ, and  their 1 permutations. For γ and its permutations, we . For δ 1  add on  the column + e−it n 1 and its permutations, we add on the column . n+1

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It remains to look at the images of generators from π1 (Fn+1 ). Recall that Fn+1 is the complement of a central line arrangement. Pick a coordinate system so that one of the lines is Var(x1 ). Consider the Hopf bundle h : C2 → CP1 with fiber C∗ . Note that the h restricted to the C2 \ Var(x1 ) has image isomorphic to C, therefore h : C2 \ Var(x1 ) → C is a trivial bundle. Now, if we restrict h further to Fn+1 , we see that its image is isomorphic to C with n − 1 points removed. So we have  that Fn+1 is homotopy  equivalent to (C \(n −1)points) ×C∗ which has homotopy type S 1 ×S 1 . Therefore n−1

π1 (Fn+1 ) is generated by the meridians around n − 1 lines, and a loop around the origin. The image of the meridian around the line generated by the first column is homotopic to the loop:  0 1

1 2

1 + e−it

1 3

1 + e−it

··· ···

1 n

1 + e−it

eit 0



for t ∈ [0, 2π]. The (1, n + 1)-minor of this loop is eit , thus it is a meridian to the subvariety x1 yn+1 − xn+1 y1 = 0. For 2 ≤ j ≤ n, the (j, n + 1)-minor is 1j eit − 1, and all other minors are constant, thus this loop contracts to a point in the complements of the subvarieties xj yk − xk yj = 0 for j, k = 1, (n + 1). The images of the meridians around other lines are simply the loop above with its columns permuted, so these loops are permutations of γ in C2(n+1) . The image of a loop around the origin is homotopic to the loop:  1 1 2 3

··· 1 eit · · · n + 1 eit



for t ∈ [0, 2π], so this loop is a permutation of δ in C2(n+1) .

2

Theorem 4.5. Let G be the complete graph on n vertices for n ≥ 2. Let Un = C2n \ AG , then Poin(Un , t) = (1 + t3 )(1 + t)n−1

n−2 

(1 + kt).

k=1

Proof. We proceed by induction on n. For the base case n = 2, the complement U2 is GL(2, C). The fibration p2 : U2 → C2 \ {0}, where p2 is the projection onto the first column of a matrix in GL(2, C), with fibers homotopy equivalent to C2 minus a line. The base space C2 \ {0} is homotopy equivalent to S 3 , and the fiber is homotopy equivalent to S 1 . Considering the cohomology Serre spectral sequence, E2p,q ∼ = H p (S 3 , H q (S 1 )),

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we do not have to worry about local coefficients, because S 3 is simply connected. Since the target for dr : Erp,q → Erp+r,q−r+1 is always zero for r ≥ 2, the spectral sequence collapses at the E2 -page. Thus, Poin(U2 , t) = Poin(S 3 , t) · Poin(S 1 , t) = (1 + t3 )(1 + t). Similarly, the fibration pn+1 : Un+1 → Un , where pn is the projection onto the first n columns, with fiber Fn+1 homotopy equivalent to C2 minus n lines. The cohomology Serre spectral sequence gives us E2p,q ∼ = H p (Un , Hq (Fn+1 )) ⇒ H p+q (Un+1 ),

(4.3)

where Hq (Fn+1 ) is the cohomology with respect to the local coefficient system on Un . To show that we can use constant coefficients H q (Fn+1 ), we show that the action of the fundamental group of the base on the homology of the fiber is the identity. Consider the loops γ and δ as defined in Lemma 4.4. We can permute the columns of γ and δ to get all of the generators of π1 (Un ), thus it is enough to understand how these two loops act on the cohomology of the fiber. Since our fiber is the complement of a central arrangement of lines, by Lemma 5 in [1], the elements of H 1 (Fn+1 ) generate H 2 (Fn+1 ) via the cup product. Hence it is enough to understand how γ and δ act on H 1 (Fn+1 ). n  span(vj ). We Now, denote the columns of γ by vj for j = 1, . . . , n. Our fiber is C2 \     j=1 0 eiθ can consider the loops in the fiber given by α1 = v1 + ε and αj = vj + ε eiθ 0 for j ≥ 2 and 0 ≤ θ ≤ 2π. For ε sufficiently small, the loops αj are meridians to the lines Cvj . These meridians can be contracted in the complements C2 \ Cvk for k = j, therefore they generate H 1 . Since γ and δ are globally defined on Un and since αj at the start and end points of the loops are the same, the action of γ on H 1 (Fn+1 ) is the identity. Thus, in equation (4.3), E2p,q ∼ = H p (Un , H q (Fn+1 )).    n+1 n + 1 Since Var(f ) has . Now, = n(n+1) components, dim(H 1 (Un+1 )) = 2 2 2 1,0 0,1 dim(E∞ ) + dim(E∞ ) = dim H 1 (Un+1 ) =

n(n + 1) . 2

1,0 1,0 Note that, Er1,0 is not the target of dr for any r, therefore E21,0 ∼ . = E3 ∼ = ··· ∼ = E∞ 1,0 Using the induction hypothesis, we can calculate dim(E∞ ) to be the coefficient of t in Poin(Un , t), thus

1,0 dim(E∞ ) = (n − 1) +

n−2

k=1

k=

(n − 1)n . 2

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To compute the Poincaré polynomial for Fn+1 (which is the complement of a central line arrangement), we use Theorem 1.2. Note that the module of logarithmic derivations for a central line arrangement is free with a basis consisting of the Euler vector field (which has degree 1), and another of vector field of degree n − 1 (by Saito’s criterion). Thus Poin(Fn+1 , t) = (1 + t)(1 + (n − 1)t), which implies that dim(E20,1 ) = n. Now, n(n + 1) 1,0 0,1 1,0 = dim(E∞ ) + dim(E∞ ) ≤ dim(E∞ ) + dim(E20,1 ) 2 n(n + 1) (n − 1)n +n= , = 2 2 0,1 thus we must have dim(E∞ ) = dim(E20,1 ), and hence dr (Er0,1 ) = 0, for all r ≥ 2. Since elements of H 1 (Fn+1 ) generate H 2 (Fn+1 ) and differentials on cup products are derivations, dr (Er0,2 ) = 0 for all r ≥ 2. Any element of E2p,q can be written as a linear combination of products of α ∈ E2p,0 and β ∈ E20,q , hence d2 (αβ) = βd2 (α) + αd2 (β) = 0. p,q Inductively, dr = 0 for r ≥ 2, thus E2p,q ∼ . Furthermore, = E∞

Poin(Un+1 , t) = Poin(Un , t) · Poin(Fn+1 , t)   n−2  (1 + kt) ((1 + t)(1 + (n − 1)t)) = (1 + t3 )(1 + t)n−1 k=1

= (1 + t3 )(1 + t)n

n−1 

(1 + kt).

2

k=1

Following the same proof as in Theorem 4.5 and using the fibration described earlier in this chapter we have the following result: Theorem 4.6. Let G be a chordal graph, then Poincaré polynomial of U = C2n \ AG factors over Q into a product of a cubic with 2|AG | − 3 linear terms. In a survey of hyperplane arrangements, Schenck [13] posed the problem to define supersolvability for hypersurface arrangements. For hyperplane arrangements, supersolvability is a combinatorial property on the lattice of intersections, and arrangements that are supersolvable are free. In particular, fiber-type arrangements (arrangements whose complement can be written as a fibration by projecting onto the first n − 1 coordinates) are supersolvable. For arrangements of more general hypersurfaces, it is not clear whether or not the intersection lattice gives us any useful information, but we can still have fiber-type arrangements. In the context of determinantal arrangements on a 2 × n generic matrix, we have fiber-type arrangements when the corresponding graph is chordal. It is tempting to extend this notion to determinantal arrangements on an m × n generic matrix, however, this cannot be done with our fibration.

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Example 4.7. Let the determinantal arrangement A be defined by the product of all maximal minors of a 3 × 7 generic matrix, and let U = C3×7 \ A be the complement. If we consider the projection p of U onto the first 6 columns, the fibers are not homotopy equivalent in general. For a generic choice of a basepoint x, p−1 (x) is the ⎛ complement of a central generic arrangement of 15 hyperplanes in C3 . The fiber ⎞ 1 −1 0 0 1 −1 ⎜ ⎟ p−1 ⎝ 0 0 1 −1 1 −1 ⎠, however, is not the complement of a generic arrange1 1 1 1 1 1 ment, thus our projection does not give us a fibration of the complement U . Although our approach does not extend to generic matrices of larger sizes, it does not mean that a fibration does not exist under certain conditions. We believe that finding such conditions for constructing fibrations would be a start to defining a notion for supersolvability for determinantal arrangements and for hypersurface arrangements in general. References [1] Egbert Brieskorn, Sur les groupes de tresses [d’après V.I. Arnol d], in: Séminaire Bourbaki, 24ème Année, 1971/1972, in: Lecture Notes in Math., vol. 317, Springer, Berlin, 1973, pp. 21–44, Exp. No. 401, MR 0422674 (54 #10660). [2] J.W. Bruce, Functions on discriminants, J. Lond. Math. Soc. (2) 30 (3) (1984) 551–567, MR 810963 (87e:58028). [3] Ragnar-Olaf Buchweitz, David Mond, Linear free divisors and quiver representations, in: Singularities and Computer Algebra, in: London Math. Soc. Lecture Note Ser., vol. 324, Cambridge Univ. Press, Cambridge, 2006, pp. 41–77, MR 2228227 (2007d:16028). [4] Ragnar-Olaf Buchweitz, Brian Pike, Lifting free divisors, Proc. Lond. Math. Soc. 112 (5) (2016) 799–826. [5] James Damon, Brian Pike, Solvable group representations and free divisors whose complements are K(π, 1)’s, Topology Appl. 159 (2) (2012) 437–449, MR 2868903. [6] D.R. Fulkerson, O.A. Gross, Incidence matrices and interval graphs, Pacific J. Math. 15 (1965) 835–855, MR 0186421 (32 #3881). [7] Michel Granger, David Mond, Alicia Nieto-Reyes, Mathias Schulze, Linear free divisors and the global logarithmic comparison theorem, Ann. Inst. Fourier (Grenoble) 59 (2) (2009) 811–850, MR 2521436 (2010g:32047). [8] Joseph P.S. Kung, Hal Schenck, Derivation modules of orthogonal duals of hyperplane arrangements, J. Algebraic Combin. 24 (3) (2006) 253–262, MR 2260017 (2007k:05045). [9] E.J.N. Looijenga, Isolated Singular Points on Complete Intersections, London Mathematical Society Lecture Note Series, vol. 77, Cambridge University Press, Cambridge, 1984, MR 747303 (86a:32021). [10] Peter Orlik, Hiroaki Terao, Arrangements of Hyperplanes, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 300, Springer-Verlag, Berlin, 1992, MR 1217488 (94e:52014). [11] Lauren L. Rose, Hiroaki Terao, A free resolution of the module of logarithmic forms of a generic arrangement, J. Algebra 136 (2) (1991) 376–400, MR 1089305 (93h:32048). [12] Kyoji Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (2) (1980) 265–291, MR 586450 (83h:32023). [13] Hal Schenck, Hyperplane arrangements: computations and conjectures, in: Arrangements of Hyperplanes, Sapporo, 2009, in: Adv. Stud. Pure Math., vol. 62, Math. Soc. Japan, Tokyo, 2012, pp. 323–358, MR 2933802. aneanu, Freeness of conic-line arrangements in P2 , Comment. Math. [14] Hal Schenck, Ştefan O. Tohˇ Helv. 84 (2) (2009) 235–258, MR 2495794 (2010d:52030). [15] R.P. Stanley, Supersolvable lattices, Algebra Universalis 2 (1972) 197–217, MR 0309815 (46 #8920).

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