aa r X i v : . [ m a t h . AG ] J u l HOMOLOGICAL PROPERTIES OF DETERMINANTALARRANGEMENTS
ARNOLD YIM
Abstract.
We explore a natural extension of braid arrangements in the con-text of determinantal arrangements. We show that these determinantal ar-rangements are free divisors. Additionally, we prove that free determinantalarrangements defined by the minors of 2 × n matrices satisfy nice combinatorialproperties.We also study the topology of the complements of these determinantal ar-rangements, and prove that their higher homotopy groups are isomorphic tothose of S . Furthermore, we find that the complements of arrangements sat-isfying those same combinatorial properties above have Poincar´e polynomialsthat factor nicely. Contents
1. Introduction 12. Setup 33. Freeness of Determinantal Arrangements 44. Complements of Determinantal Arrangements 10References 141.
Introduction
Let D be a divisor on an an n -dimensional complex analytic manifold X . The module of logarithmic derivations Der X ( − log D ) := { θ ∈ Der X | θ ( O X ( − Y )) ⊆ O X ( − Y ) } are the vector fields on X that are tangent along D . If Der X ( − log D ) islocally free, then D is called a free divisor . The simplest example of free divisorsare normal crossing divisors.Free divisors were first introduced by Saito [11], motivated by his study of thediscriminants of versal deformations of isolated hypersurface singularities. Thestudy of free divisors coming from discriminants of versal deformations has sincebeen a driving force in the theory of singularities (see [8, 2, 18, 17, 16]).Aside from versal deformations, free divisors show up naturally in many differentsettings. For example, many of the classically arising hyperplane arrangements arefree (see [9]). This includes braid arrangements and all Coxeter arrangements. Mathematics Subject Classification.
Key words and phrases. logarithmic derivations, free divisor, hyperplane, determinantal, ar-rangement, supersolvable, chordal, homotopy group, Poincar´e polynomial.I would like to express my gratitude to my advisor Uli Walther for his guidance throughout thiswhole project. Partial support by the NSF under grant DMS-1401392 is gratefully acknowledged.
Surprisingly, freeness can also give us topological information. Specifically, Teraoproves in [15] that for a free hyperplane arrangement, the Poincar´e polynomial forthe complement is determined by the degrees of the vector fields in the basis of themodule of logarithmic derivations:
Theorem 1.1 (Terao) . Let A ⊂ C n be a free central hyperplane arrangement andsuppose that Der C n ( − log A ) ∼ = n M i =1 C [ x , . . . , x n ]( − b i ) , then Poin( C n \ A , t ) = n Y i =1 (1 + b i t ) . Observe that Poincar´e polynomials are topological invariants that are not spe-cific to hyperplane arrangements, and neither are the degrees of logarithmic vectorfields for graded free divisors. Naturally, one might be interested in freeness forarrangements of more general hypersurfaces and how freeness might be connectedto topology. For example, Schenck and Tohˇaneanu [13] give conditions for when anarrangement of lines and conics on P is free.We are particularly interested in determinantal arrangements , which are con-figurations of determinantal varieties. Buchweitz and Mond [3] showed that thearrangement defined by the product of the maximal minors of a n × ( n + 1) matrixof indeterminates is free. Recently, Damon and Pike [4] show that certain determi-nantal arrangements coming from symmetric, skew-symmetric and square generalmatrices are free and have complements that are K ( π, X ( − log D )is generated by linear vector fields). The vector fields arising in these situationscorrespond to matrix group actions on the generic matrix which stabilize the divi-sor D . Many interesting determinantal arrangements, however, are not linear freedivisors as our next example shows. Example 1.2.
Let M be the 2 × M = (cid:18) x x x x y y y y (cid:19) , and for i < j , let ∆ ij be the 2-minor of M using the i -th and j -th columns, ∆ ij = x i y j − x j y i . Let f be the product f = Y i We look at divisors on X = C n with coordinate ring R = C [ x , . . . , x n , y , . . . , y n ].Let Der X be the free R -module of vector fields on X generated by n ∂∂x i , ∂∂y i o i =1 ..n .For any divisor f on X , we are interested in the following object: Definition 2.1. The module of logarithmic derivations along f is the R -moduleDer X ( − log f ) = { θ ∈ Der X | θ ( f ) ∈ ( f ) } . We want to know when f has a well-behaved singular locus, thus we are interestedin when the module of logarithmic derivations along f is free. We say: Definition 2.2. A divisor f on X is free if Der X ( − log f ) is a free R -module.To determine whether a divisor is free, we use Saito’s criterion [11]: Theorem 2.3 (Saito) . A divisor f ∈ C [ x , . . . , x n ] is a free divisor if and only ifthere exists n elements θ j = n X i =1 g ij ∂∂x i ∈ Der X ( − log f ) such that det(( g ij )) = c · f for some non-zero c ∈ C . We focus on logarithmic derivations for hypersurface arrangements defined bygraphs. In the context of of hyperplane arrangements, these are called graphicarrangements. Given a graph G with n vertices, we associate a hyperplane ar-rangement defined by a polynomial f ∈ C [ x , . . . x n ]. For each edge of G betweenvertices v i and v j , we include the hyperplane defined by x i − x j = 0 in the arrange-ment. For example, the graphic arrangement associated to a complete graph on n vertices is the braid arrangement on n variables defined by f = Y ≤ i A graph G is chordal if and only if there exists an ordering ofvertices, such that for each vertex v , the induced subgraph on v and its neighborsthat occur before it in the sequence is a complete graph.While freeness is well understood for graphic arrangements, it is still unclearwhen we consider arrangements of more general hypersurfaces. We investigatecertain determinantal arrangements associated to graphs. Specifically, let M bethe 2 × n matrix of indeterminates M = (cid:18) x x · · · x n y y · · · y n (cid:19) . For i < j , let ∆ ij denote the 2-minor of M using the i -th and j -th columns,∆ ij = x i y j − x j y i . Definition 2.5. For each graph G with n vertices, we can associate a determinantalarrangement, A G , consisting of the determinantal varieties Var(∆ ij ) for each edgebetween vertices v i and v j of G . ARNOLD YIM Freeness of Determinantal Arrangements For hyperplane arrangements, the braid arrangement is a well-known example ofan arrangement that is free. The braid arrangement is made up of hyperplanes thatare defined by any two coordinates being equal. We define something similar in oursetting. Consider C n as a collection of n two-dimensional vectors, and thus thehypersurface defined by the vanishing of a minor of M is the hypersurface definedby two vectors being linearly dependent. Our analog of the braid arrrangement isthe determinantal arrangement defined by any two columns being linearly depen-dent. If we think about these arrangements as coming from graphs, both the braidarrangement and our analog come from the complete graph on n vertices.In Theorem 3.3, we prove that our analog of the braid arrangement is free: weconstruct a generating set for the module of logarithmic derivations and show thatthis set satisfies Saito’s criterion. In Theorem 3.5, we prove that if a graph isnot chordal, then the corresponding determinantal arrangement is not free. Weshow that near a particular point, our arrangement looks like the cyclic graphicarrangement 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 > , let s i,j,k denote the degree k symmetric polynomial onthe variables z i , . . . , z n that is linear in each variable omitting the variable z j , givenby s i,j,k = X α m = ji ≤ α < · · · < α k ≤ n z α z α · · · z α k , and let s i,j, = 1 .Let A i denote the ( n + 1 − i ) × ( n + 1 − i ) matrix ( s i,j,k ) , where the row index j ranges from i to n , and the column index k ranges from to n − i . Then det( A i ) = Y i Writing out A i , we have A i = z i +1 + z i +2 + · · · + z n ) · · · ( z i +1 z i +2 · · · z n )1 ( z i + z i +2 + · · · + z n ) · · · ( z i z i +2 · · · z n )... ... . . . ...1 ( z i + z i +1 + · · · + z n − ) · · · ( z i z i +1 · · · z n − ) . Subtracting the first row from every other row, we have z i +1 + z i +2 + . . . + z n ) ( z i +1 z i +2 + z i +1 z i +3 + · · · + z n − z n ) · · · ( z i +1 z i +2 · · · z n )0 ( z i − z i +1 ) ( z i − z i +1 )( z i +2 + z i +3 + · · · + z n ) · · · ( z i − z i +1 )( z i +2 z i +3 · · · z n )0 ( z i − z i +2 ) ( z i − z i +2 )( z i +1 + z i +3 + · · · + z n ) · · · ( z i − z i +2 )( z i +1 z i +3 · · · z n )... ... ... . . . ...0 ( z i − z n ) ( z i − z n )( z i +1 + z i +2 + · · · + z n − ) · · · ( z i − z n )( z i +1 z i +2 · · · z n − ) . OMOLOGICAL PROPERTIES OF DETERMINANTAL ARRANGEMENTS 5 We can factor the lower right ( n − i ) × ( n − i ) submatrix as ( z i − z i +1 ) ( z i − z i +2 ) . . . ( z i − z n ) z i +2 + z i +3 + · · · + z n ) · · · ( z i +2 z i +3 · · · z n )1 ( z i +1 + z i +3 + · · · + z n ) · · · ( z i +1 z i +3 · · · z n )... ... . . . ...1 ( z i +1 + z i +2 + · · · + z n − ) · · · ( z i +1 z i +2 · · · z n − ) = ( z i − z i +1 ) ( z i − z i +2 ) . . . ( z i − z n ) A i +1 , thus det( A i ) = Y i Let A be a block matrix A = (cid:18) A A A A (cid:19) with blocks of size n × n with entries in C ( z , . . . , z n ) . If A and A are diagonal matrices with nonzeroentries, then det( A ) = det( A A − A A ) .Proof. Let B be the block matrix B = (cid:18) A − A − (cid:19) , then BA = (cid:18) I n A − A I n A − A (cid:19) .Using row reduction, we finddet( BA ) = det (cid:18) I n A − A A − A − A − A (cid:19) . Now, let C be the block matrix C = (cid:18) I n A A (cid:19) , thendet( CBA ) = det (cid:18) I n A − A A A − A A (cid:19) = det( A A − A A ) . Since det( CBA ) = det( A ), we have det( A ) = det( A A − A A ). (cid:3) Now, we have our main result of this section: Theorem 3.3. Let G be the complete graph on n vertices for n ≥ . The determi-nantal arrangement A G is free.Proof. If G is the complete graph on n vertices, then the corresponding determi-nantal arrangement A G is defined by f = Y ≤ i 4, define a m,k = 1 m !( n − − m )! X σ ∈ S n − x ( τ k ◦ σ )(1) · · · x ( τ k ◦ σ )( m ) y ( τ k ◦ σ )( m +1) · · · y ( τ k ◦ σ )( n − . Now, consider the derivations ϕ m = n X k =4 a m,k ∆ k ∆ k (cid:18) x ∂∂x k + y ∂∂y k (cid:19) . OMOLOGICAL PROPERTIES OF DETERMINANTAL ARRANGEMENTS 7 If i, j < 4, then ϕ m (∆ ij ) = 0. Now, suppose that i < j ≥ 4, then ϕ m (∆ ij ) = n X k =4 a m,k ∆ k ∆ k (cid:18) x ∂∂x k + y ∂∂y k (cid:19)! ( x i y j − x j y i )= a m,j ∆ j ∆ j (cid:16) x ∂∂x j + y ∂∂y j (cid:17) ( x i y j − x j y i )= a m,j ∆ j ∆ j ( − x y i + y x i ) . When i = 2 , ϕ m (∆ ij ) ∈ (∆ ij ) ⊆ R , and when i = 1, ϕ m (∆ ij ) = 0.If i, j ≥ ϕ m (∆ ij ) = n X k =4 a m,k ∆ k ∆ k (cid:18) x ∂∂x k + y ∂∂y k (cid:19)! ( x i y j − x j y i )= (cid:16) a m,i ∆ i ∆ i (cid:16) x ∂∂x i + y ∂∂y i (cid:17) + a m,j ∆ j ∆ j (cid:16) x ∂∂x j + y ∂∂y j (cid:17)(cid:17) ( x i y j − x j y i )= a m,i ∆ i ∆ i ( x y j − y x j ) + a m,j ∆ j ∆ j ( − x y i + y x i )= a m,i ∆ i ∆ i ∆ j − a m,j ∆ j ∆ j ∆ i . Note that each term in a m,i either has a factor of x j or y j , and also note thatthe terms in a m,j are exactly the terms in a m,i , with x i and y i instead of x j and y j respectively, thus it is enough to show that x j ∆ i ∆ i ∆ j − x i ∆ j ∆ j ∆ i and y j ∆ i ∆ i ∆ j − y i ∆ j ∆ j ∆ i are divisible by ∆ ij . Using Pl¨ucker relations, we canwrite: x j ∆ i ∆ i ∆ j − x i ∆ j ∆ j ∆ i = x j ∆ i (∆ j ∆ i ) − x i ∆ j ∆ j ∆ i = x j ∆ i (∆ i ∆ j − ∆ ∆ ij ) − x i ∆ j ∆ j ∆ i = ∆ i ∆ j ( x j ∆ i − x i ∆ j ) − x j ∆ i ∆ ∆ ij = ∆ i ∆ j ( x j x y i − x j x i y − x i x y j + x i x j y ) − x j ∆ i ∆ ∆ ij = ∆ i ∆ j ( x j x y i − x i x y j ) − x j ∆ i ∆ ∆ ij = ∆ i ∆ j ( − x ∆ ij ) − x j ∆ i ∆ ∆ ij ∈ (∆ ij ) , and similarly, y j ∆ i ∆ i ∆ j − y i ∆ j ∆ j ∆ i = y j ∆ i (∆ j ∆ i ) − y i ∆ j ∆ j ∆ i = y j ∆ i (∆ i ∆ j − ∆ ∆ ij ) − y i ∆ j ∆ j ∆ i = ∆ i ∆ j ( y j ∆ i − y i ∆ j ) − y j ∆ i ∆ ∆ ij = ∆ i ∆ j ( y j x y i − y j x i y − y i x y j + y i x j y ) − y j ∆ i ∆ ∆ ij = ∆ i ∆ j ( − y j x i y + y i x j y ) − x j ∆ i ∆ ∆ ij = ∆ i ∆ j ( − y ∆ ij ) − x j ∆ i ∆ ∆ ij ∈ (∆ ij ) . Since ϕ m stabilizes each (∆ ij ), ϕ m ∈ Der X ( − log f ).It remains to show that this set of elements in Der X ( − log f ) form a basis.According to Saito’s criterion, these derivations form a basis if and only if thedeterminant of the coefficient matrix is a nonzero constant multiple of f . With our ARNOLD YIM elements, we have the coefficient matrix: y x y x y x y x a , ∆ ∆ x · · · a n − , ∆ ∆ x ... . . . ... . . . ... y n x n a ,n ∆ n ∆ n x · · · a n − ,n ∆ n ∆ n x x y y x y y x y y x y y a , ∆ ∆ y · · · a n − , ∆ ∆ y ... ... . . . ... . . . ... x n y n y n a ,n ∆ n ∆ n y · · · a n − ,n ∆ n ∆ n y . We swap some rows to organize our matrix into blocks (this could potentiallychange the determinant by a sign, but that is unimportant in checking Saito’scriterion): y x y x y x x y y x y y x y y y x a , ∆ ∆ x · · · a n − , ∆ ∆ x ... . . . ... . . . ... y n x n a ,n ∆ n ∆ n x · · · a n − ,n ∆ n ∆ n x x y y a , ∆ ∆ y · · · a n − , ∆ ∆ y ... ... . . . ... . . . ... x n y n y n a ,n ∆ n ∆ n y · · · a n − ,n ∆ n ∆ n y . Denote the matrix above by N , with blocks N = (cid:18) A C D (cid:19) . Since N is a tri-angular block matrix, det( N ) = det( A ) det( D ). By explicit computation, we findthat(3.1) det( A ) = ∆ ∆ ∆ . To calculate the determinant of D , we split the matrix into more blocks: D = (cid:18) D D D D (cid:19) = x a , ∆ ∆ x · · · a n − , ∆ ∆ x . . . ... . . . ... x n a ,n ∆ n ∆ n x · · · a n − ,n ∆ n ∆ n x y a , ∆ ∆ y · · · a n − , ∆ ∆ y . . . ... . . . ... y n a ,n ∆ n ∆ n y · · · a n − ,n ∆ n ∆ n y . OMOLOGICAL PROPERTIES OF DETERMINANTAL ARRANGEMENTS 9 By Lemma 3.2, we have det( D ) = det( D D − D D ). Now, D D − D D = a , ∆ ∆ ( y x − x y ) · · · a n − , ∆ ∆ ( y x − x y )... . . . ... a ,n ∆ n ∆ n ( y x n − x y n ) · · · a n − ,n ∆ n ∆ n ( y x n − x y n ) = − a , ∆ ∆ ∆ · · · a n − , ∆ ∆ ∆ ... . . . ... a ,n ∆ n ∆ n ∆ n · · · a n − ,n ∆ n ∆ n ∆ n = − ∆ ∆ ∆ . . . ∆ n ∆ n ∆ n a , · · · a n − , ... . . . ... a ,n · · · a n − ,n =: − D D . Observe that(3.2) det( D ) = Y i =1 n Y j =4 ∆ ij , therefore it remains to show that det( D ) is a nonzero constant multiple of theproduct of all minors using the last n − M .We show that each ∆ ij for i, j ≥ D ) by showing that det( D )vanishes on Var(∆ ij ). Indeed, ∆ ij vanishes when columns i and j of M are scalarmultiples of each other. Write x j = cx i and y j = cy i . Looking to rows i and j of D , we have a m,j = ca m,i , and since these rows are scalar multiples of each other,det( D ) vanishes here which implies that each ∆ ij divides det( D ). The degree ofthe product of the minors, 2 (cid:18) n − (cid:19) = ( n − n − , is the same as the degreeof det( D ), hence det( D ) is a constant multiple of the product of the minors. Tocheck that det( D ) is not identically zero, we substitute y k = 1 into D to get thematrix in Lemma 3.1 on the variables x , . . . , x n , thus if x = x = · · · 6 = x n , thendet( D ) = 0.With equations (3.1) and (3.2), we find det( N ) = ( − n − det( A ) det( D ) det( D )is a constant multiple of the product of all of the minors of M . By Saito’s crite-rion, { α, β, γ, θ , . . . , θ n , ϕ , . . . , ϕ n − } form a basis for Der X ( − log f ), hence ourdeterminantal arrangement is free. (cid:3) We believe that our work with determinantal arrangements on 2 × n genericmatrices 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], [6]). However,in the next case, m = 3 and n = 5, we already don’t know whether or not thearrangement is free. More generally, one can ask: Question 3.4. Let M be the m × n matrix of indeterminates with n > m > f be the product of all maximal minors of M . Is the arrangement definedby f free?One can also consider determinantal arrangements defined by subgraphs of thecomplete graph. Much like hyperplane arrangements, we find that the freeness ofthe determinantal arrangement is related to whether or not the graph is chordal. Theorem 3.5. If a determinantal arrangement A G is free, then G is chordal.Moreover, if G has a chord-free induced cycle of length k , then pdim(Der X ( − log A G )) ≥ k − . Proof. Suppose that G is not chordal, then G has an chord-free induced cycle oflength 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 A G . To show that A is not free, wewill localize to a neighborhood of the point p = (cid:18) · · · · · · · · · · · · n − k (cid:19) .We will consider our divisor in the local ring C [ x , . . . , x n , y , . . . , y n ] m p where m p is the maximal ideal associated to the point p . In this local ring, ∆ ij is a unit if i or j is greater than k . Thus, around p , A G looks like Var(∆ ∆ · · · ∆ ( k − k ∆ k )whose associated graph is the cyclic graph on k vertices.We show that p is in the non-free locus of Var(∆ ∆ · · · ∆ ( k − k ∆ k ). In our lo-cal ring, x i is a unit for all i , thus Var(∆ ∆ · · · ∆ ( k − k ∆ k ) = Var (cid:16) x k − x k − k x x ··· x k − ∆ ∆ · · · ∆ ( k − k ∆ k (cid:17) .But, x k − x k − k x x ··· x k − ∆ ∆ · · · ∆ ( k − k ∆ k = x k − x k − k x x ··· x k − ( x y − x y )( x y − x y ) · · · ( x k − y k − x k y k − )( x y k − x k y )= (cid:16) x x k x y − x k y (cid:17) (cid:16) x x k x y − x x k x y (cid:17) · · · (cid:16) x y k − x x k x k − y k − (cid:17) ( x y k − x k y ) . Now, making a change of coordinates z ↔ x k y z ↔ x x k x y ... ... ... z k − ↔ x x k x k − y k − z k ↔ x y k , we have that Var(∆ ∆ · · · ∆ ( k − k ∆ k ) = Var(( z − z )( z − z ) · · · ( z k − z k − )( z k − z )) . Since our point p , corresponds to z i = 0 for the cyclic graphic arrangementVar(( z − z )( z − z ) · · · ( z k − z k − )( z k − z )), we know that p is in the non-free locusof Var(∆ ∆ · · · ∆ ( k − k ∆ k ), and thus A G is not free. Moreover, this is a generichyperplane arrangement so by Rose and Terao [10], pdim(Der X ( − log(∆ ∆ · · · ∆ ( k − k ∆ k ))) = k − 3. Since localization is an exact functor, pdim(Der X ( − log ( A ) G )) ≥ k − (cid:3) Remark 3.6. The converse of Theorem 3.5 is not exactly true. For example, forany chordal graph with a vertex v of degree 2, if the induced subgraph v with itsneighbors is not a cycle then the corresponding determinantal arrangement is notfree. In this case, the arrangement locally behaves like f = ∆ ∆ , and one cancheck that this arrangement is not free. However, evidence suggests that many ofthe arrangements with chordal graphs are indeed free. For example, arrangementscorresponding to doubly-connected (graphs that remain connected after removingany single vertex) chordal graphs seem to be free.4. Complements of Determinantal Arrangements Terao’s theorem ([15]) relating the Poincar´e polynomial for the complement of afree hyperplane arrangement to the degrees of the basis for the module of logarith-mic derivation is very interesting to us. Since neither the Poincar´e polynomial nor OMOLOGICAL PROPERTIES OF DETERMINANTAL ARRANGEMENTS 11 the degrees of vector fields are specific to hyperplane arrangements, we investigatehere how these things are related in general. For free determinantal arrangements,although the degrees of the basis does not give the factorization of the Poincar´epolynomial directly, we do find that the Poincar´e polynomial factors.In Theorems 4.1 and 4.2, we show that the Poincar´e polynomial complement ofa free determinantal arrangement factors nicely. We construct a fibration of thecomplement, then show that the corresponding Serre spectral sequences collapsesat the E page (which implies that the Poincar´e polynomial for our complement isthe product of the Poincar´e polynomials of the base and the fiber). In Theorem4.4, we use the homotopy long exact sequence for our fibration to prove that thehigher homotopy groups for the complement are isomorphic to those of S . Theorem 4.1. Let G be the complete graph on n vertices. Let U n = C n \ A G ,then Poin( U n , t ) = (1 + t )(1 + t ) n − n − Y k =1 (1 + kt ) . Proof. We proceed by induction on n . For the base case n = 2, the complement U is GL(2 , C ). Consider the fibration p : U → C \ { } , where p is the projection ontothe first column of a matrix in GL(2 , C ), with fibers homotopic to C minus a line.The base space C \ { } is homeomorphic to S , and the fiber is homeomorphic to S . Considering the cohomology Serre spectral sequence, E p,q ∼ = H p ( S , H q ( S )) , we do not have to worry about local coefficients, because S is simply connected.Since the target for d r : E p,qr → E p + r,q − r +1 r is always zero for r ≥ 2, the spectralsequence collapses at the E -page. Thus,Poin( U , t ) = Poin( S , t ) · Poin( S , t ) = (1 + t )(1 + t ) . Similarly, we have a fibration p : U n +1 → U n , where p is the projection onto thefirst n columns, with fiber F homotopic to C minus n lines. The cohomology Serrespectral sequence gives us(4.1) E p,q ∼ = H p ( U n , H q ( F )) ⇒ H p + q ( U n +1 ) . To show that we have constant coefficients again, H q ( F ), we show that the actionof the fundamental group of the base on the homology of the fiber is the identity.Consider the loop γ : [0 , π ] → U n , given by γ ( t ) = (cid:18) e it · · · 10 1 + e − it + e − it · · · n − + e − it (cid:19) . The (1 , γ is e it , thus γ is a meridian to the subvariety x y − x y = 0.For j ≥ 3, the (1 , j )-minor is j − e it + 1, and all other minors are constant, thus γ contracts to a point in the complements of the subvarieties x j y k − x k y j = 0 for j, k = 1 , 2. We can permute the columns of γ , to get loops around any particularsubvariety x j y k − x k y j = 0; thus it is enough to understand the action of γ on thehomology of the fiber. Since our fiber is the complement of a central arrangementof lines (which is a braid space), elements of H ( F ) generate H ( F ) via the cupproduct [1], hence it is enough to understand how γ acts on H ( F ). Now, denote the columns of γ by v j for j = 1 , . . . , n . Our fiber is C \ n [ j =1 span( v j ).We can consider the loops in the fiber given by α = v + ε (cid:18) e iθ (cid:19) and α j = v j + ε (cid:18) e iθ (cid:19) for j ≥ ≤ θ ≤ π . For ε sufficiently small, the loops α j aremeridians to the lines C v j , and can be contracted in the complements C \ C v k for k = j , therefore they generate H .Since γ is globally defined on U n and since α j at γ (0) is defined exactly the sameas α j at γ (1), the action of γ on H ( F ) is the identity. Thus, in equation (4.1), E p,q ∼ = H p ( U n , H q ( F )) . Since Var( f ) has (cid:18) n + 12 (cid:19) components, dim( H ( U n +1 )) = (cid:18) n + 12 (cid:19) = n ( n +1)2 . Now, dim( E , ∞ ) + dim( E , ∞ ) = dim H ( U n +1 ) = n ( n + 1)2 . Note that, E , r is not the target of d r for any r , therefore E , ∼ = E , ∼ = · · · ∼ = E , ∞ .Using the induction hypothesis, we can calculate dim( E , ∞ ) to be the coefficient of t in Poin( U n , t ), thusdim( E , ∞ ) = ( n − 1) + n − X k =1 k = ( n − n . To compute the Poincar´e polynomial for F , we use Theorem 1.1. Note that themodule of logarithmic derivations for a central line arrangement is free with a basisconsisting of the Euler vector field (which has degree 1), and another of vector fieldof degree n − F, t ) = (1 + t )(1 + ( n − t ), whichimplies that dim( E , ) = n .Now, n ( n + 1)2 = dim( E , ∞ )+dim( E , ∞ ) ≤ dim( E , ∞ )+dim( E , ) = ( n − n n = n ( n + 1)2 , thus we must have dim( E , ∞ ) = dim( E , ), and hence d r ( E , r ) = 0, for all r ≥ H ( F ) generate H ( F ), and since differentials on cup productsare derivations, d r ( E , r ) = 0 for all r ≥ 2. Any element of E p,q can be writtenas a linear combination of products of α ∈ E p, and β ∈ E ,q , hence d ( αβ ) = βd ( α )+ αd ( β ) = 0. Inductively, d r = 0 for r ≥ 2, thus E p,q ∼ = E p,q ∞ . Furthermore,Poin( U n +1 , t ) = Poin( U n , t ) · Poin( F, t )= (1 + t )(1 + t ) n − n − Y k =1 (1 + kt ) ! ((1 + t )(1 + ( n − t ))= (1 + t )(1 + t ) n n − Y k =1 (1 + kt ) . (cid:3) Following the same proof: Theorem 4.2. Let G be a chordal graph, then Poincar´e polynomial of U = C n \ A G factors over Q into a product of a cubic with | A G | − linear terms. OMOLOGICAL PROPERTIES OF DETERMINANTAL ARRANGEMENTS 13 Proof. Since G is chordal, there exists an ordering of vertices, such that for eachvertex v , the induced subgraph on v and its neighbors that occur before it is acomplete graph. Reorganize the columns of M according to this sequence, then wecan write X as a fibration similar to the one used in the proof of Theorem 4.1. (cid:3) Note that our fibration only works when we have a chordal graph. If the graphis not chordal, the fibers are not homotopy equivalent. Example 4.3. Consider the cyclic arrangement on 4 vertices: f = ∆ ∆ ∆ ∆ ,we can follow our procedure of projecting the complement onto the first threecolumns, however some fibers look like C minus 2 lines (when the first and thirdcolumn are linearly independent) and other fibers look like C minus 1 line (whenthe first and third column are linearly dependent).When the graph is a chordal, this is no longer an issue since all of the rele-vant columns are guaranteed to be linearly independent and thus the fibers alwayslook the same. This notion of having homotopic fibers for the complement of thedeterminantal arrangements is analogous to fiber-type hyperplane arrangements.This fibration of the complement also gives us information on the homotopygroups. Theorem 4.4. Let G be a chordal graph on n vertices, and U = C n \ A G , then π i ( U ) ∼ = π i ( S ) for i ≥ .Proof. Without loss of generality, assume that the columns of M are ordered ac-cording to the chordal ordering. Let U k denote the complement of the arrangementrestricted to the first k columns. Consider the Serre fibrations p k : U k → U k − for 2 ≤ k ≤ n , where p k is the projection of 2 × k matrices onto the first k − F k homotopic to C minus k − k , considerthe homotopy long exact sequence(4.2)0 ← π ( U k ) ← π ( F k ) ← π ( U k − ) ← π ( U k ) ← π ( F k ) ← π ( U k − ) ← · · · . By Proposition 5.6 in [9], every central 2-(hyperplane)arrangement is K ( π, k , π i ( F k ) = 0 for i ≥ i = 0. From (4.2), π i ( U k ) ∼ = π i ( U k − ) for i ≥ 3. Since U = C \ { } ∼ = S , for each k , π i ( U k ) ∼ = π i ( S ) for i ≥ π ( U k − ) ← π ( U k ) ← π ( F k ) . When k = 2, the group on the left in (4.3) is π ( S ) = 0, by induction on k , π ( U k ) = 0 for all k . (cid:3) Remark 4.5. Although we have the short exact sequence0 → π ( F k ) → π ( U k ) → π ( U k − ) → , it is not clear what π ( U k ) is in general.In a survey of hyperplane arrangements, Schenck [12] posed the problem to de-fine supersolvability for hypersurface arrangements. For hyperplane arrangements,supersolvability is a combinatorial property on the lattice of intersections, and ar-rangements that are supersolvable are free. In particular, fiber-type arrangementsare supersolvable. For arrangements of more general hypersurfaces, it is not clear whether or notthe 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 genericmatrix, 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. Example 4.6. Let the determinantal arrangement A defined by the product of allmaximal minors of a 3 × U = C × \ A be the complement.If we consider the projection p of U onto the first 6 columns, the fibers are nothomotopy equivalent in general. For a generic choice of a basepoint x , p − ( x ) is thecomplement of a central generic arrangement of 15 hyperplanes in C . The fiber p − − − 10 0 1 − − 11 1 1 1 1 1 , however, is not the complement of a genericarrangement, 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, itdoes not mean that a fibration does not exist under certain conditions. We believethat finding such conditions for constructing fibrations would be a start to defininga notion for supersolvability for determinantal arrangements and for hypersurfacearrangements in general. References 1. Egbert Brieskorn, Sur les groupes de tresses [d’apr`es V. I. 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Zakalyukin, Reconstructions of fronts and caustics depending on a parameter, and ver-sality of mappings , Current problems in mathematics, Vol. 22, Itogi Nauki i Tekhniki, Akad.Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983, pp. 56–93. MR 735440(85h:58029) Department of Mathematics, Purdue University, 150 North University Street, WestLafayette, IN 47907-2067 E-mail address ::