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EFFICIENT COMPUTATION OF RESONANCE VARIETIES VIA GRASSMANNIANS 9 PAULOLIMA-FILHOANDHALSCHENCK1 0 0 2 Abstract. AssociatedtothecohomologyringAofthecomplementX(A)of n a hyperplane arrangement A in Cℓ are the resonance varieties Rk(A). The a moststudiedoftheseisR1(A),whichistheunionofthetangentconesat1to J the characteristicvarietiesofπ1(X(A)). R1(A)maybedescribedintermsof Fittingideals,orasthelocuswhereacertainExtmoduleissupported. Both 2 these descriptions give obvious algorithms for computation. In this note, we showthatinterpretingR1(A)asthelocusofdecomposabletwo-tensorsinthe O] Orlik-Solomon ideal of A leads to a description of R1(A) as the intersection of a Grassmannian with a linear space, determined by the quadratic gener- C ators of the Orlik-Solomon ideal. This method is much faster than previous . alternatives. h t a m [ 1. Motivation: cohomology rings of arrangement complements 2 Let A={H ,...,H } be a centralcomplex hyperplane arrangementin Cℓ, and 1 n v let X(A) = Cℓ \ A. In [16], Orlik and Solomon determined a presentation for 7 A=H∗(X(A),Z): 2 0 Definition 1.1. A = H∗(X(A),Z) is the quotient of the exterior algebra E = 2 (Zn)ongeneratorse ,...,e indegree1bytheidealI generatedbyallelementsof 8. tVheform∂e := 1(−1)q−n1e ···e ···e ,forwhichcodimH ∩···∩H <r. 0 i1...ir Pq i1 iq ir i1 ir c 8 Since Aisaquotientofanexterioralgebra,multiplicationbyanelementa∈A1 0 gives a degree one differential on A, yielding a cochain complex (A,a): : v i (1.1) (A,a): 0 //A0 a //A1 a //A2 a // ··· a //Aℓ // 0 . X Aomoto [1] studied this complex in connection with his work on hypergeometric r a functions, and the complex was subsequently studied in relation to local system cohomology by Esnault, Schechtman and Viehweg in [8]. The complex (A,a) is exact as long as n a 6= 0, and in [22], Yuzvinsky showed that in fact (A,a) is Pi=1 i generically exact except at the last position ℓ. Fix afield k, we willwrite A=H∗(X(A),k) for the Orlik-Solomonalgebraover k. The resonance varieties of A consist of points a= n a e ↔(a :···:a ) in P(A1)∼=Pn−1 for which (A,a) fails to be exact. So foPr eia=c1hiki≥1, 1 n Rk(A)={a∈Pn−1 |Hk(A,a)6=0}. Falk initiated the study of R1(A) in [9], obtaining necessary and sufficient combi- natorialconditionsfora∈R1(A). Falk alsoconjecturedthatR1(A)isthe unionof 2000 Mathematics Subject Classification. Primary52C35,Secondary 13D07,20F14. Key words and phrases. Grassmannian,Hyperplanearrangement, Resonancevariety. 1PartiallysupportedbyNSF0707667, NSAH98230-07-1-0052. 1 2 PAULOLIMA-FILHOANDHALSCHENCK 5 3 1 @ (cid:0) t (cid:0) (cid:0)@ @ 4 (cid:0) (cid:0) @ (cid:0)(cid:0)@(cid:0)@@@ 2 (cid:16)t(cid:0)(cid:0)(cid:16)(cid:0)4(cid:16)(cid:0)tP(cid:16)P(cid:16)P(cid:16)3tP(cid:16)P(cid:16)@Pt@2P@P@t 1 5 0 0 Figure 1. The braid arrangement and its matroid asubspacearrangement. Thiswasprovedin[4]. Byusingthe Cartanclassification of affine Kac-Moody Lie algebras, Libgober and Yuzvinsky [14] also obtained this result, and showed that R1(A) is in fact a union of disjoint, positive dimensional subspaces. For the higher resonancevarieties, Cohen and Orlik show in [2] that all components of the Rk(A) are linear subvarieties. For all this, characteristic zero is necessary, see [10]. A major impetus for studying R1(A) is a conjecture of Suciu in[21], relating R1(A) to the LCSranksof the fundamental group. Results onthis conjecture appear in [13], [15], [18]. In [7], Eisenbud, Popescu, and Yuzvinsky prove that the complex (A,d ), re- a garded as a complex of S = Sym(kn) modules, is a free resolution of the cokernel F(A) of the final nonzero map. Combining with results of [3],[4], the paper [20] shows that: R1(A)=V(ann(Extℓ−1(F(A),S))). In [5], this result is generalized to: Rk(A)= V(annExtℓ−k′(F(A),S)) [ k′≤k In particular, the resonance varieties of hyperplane arrangements may be realized as support loci of appropriate Ext modules. Example 1.2. Let A be the braid arrangement in P2, with defining polynomial Q=xyz(x−y)(x−z)(y−z). Fromthematroid(seeFigure1),itiseasytoseethat theOrlik-SolomonalgebraAisthequotientoftheexterioralgebraE ongenerators e ,...,e by the ideal I = h∂e ,∂e ,∂e ,∂e ,∂e i, where ijkl runs over 0 5 145 235 034 012 ijkl all four-tuples; it turns out that the elements ∂e are redundant. ijkl The minimal free resolution of A as a module over E begins: 0oo A oo E oo ∂1 E4(−2)oo ∂2 E10(−3)oo ∂3 E15(−4)⊕E6(−5)oo ···, where ∂ = ∂e ∂e ∂e ∂e , and ∂ is equal to 1 145 235 034 012 2 (cid:0) (cid:1) e1−e4 e1−e5 0 0 0 0 0 0 e3−e0 e2−e0   0 0 e2−e3 e2−e5 0 0 0 0 e0−e1 e0−e4 .  0 0 0 0 e0−e3 e0−e4 0 0 e1−e5 e2−e5    0 0 0 0 0 0 e0−e1 e0−e2 e3−e5 e4−e5 The resonance variety R1(A) ⊂ P5 has 4 local components, corresponding to the triple points, and1 essentialcomponent (i.e., one that does not come fromany EFFICIENT COMPUTATION OF RESONANCE VARIETIES VIA GRASSMANNIANS 3 propersub-arrangement),correspondingtotheneighborlypartitionΠ=(05|13|24): {x +x +x =x =x =x =0}, {x +x +x =x =x =x =0}, 1 4 5 0 2 3 2 3 5 0 1 4 {x +x +x =x =x =x =0}, {x +x +x =x =x =x =0}, 0 3 4 1 2 4 0 1 2 3 4 5 {x +x +x =x −x =x −x =x −x =0}. 0 1 2 0 5 1 3 2 4 The last two columns of the matrix representing ∂ correspond to a pair of linear 2 syzygies on I , which arise from the essential component of R1(A): 2 ∂e +∂e +∂e −∂e =(e −e −e +e )∧(e −e +e −e ). 012 034 145 235 0 1 3 5 1 2 3 4 If we write the two-form above as λ∧µ= a f ∈I , then these syzygies are: i i 2 P 0=λ∧λ∧µ= a λf and 0=λ∧µ∧µ= a µf . X i i X i i Thisexamplemotivatedinvestigationsin[19]ontheconnectionbetweenR1(A)and the linear syzygies of A, where A is viewed as a module over the exterior algebra E. In this example, the syzygies arising from R1(A) are independent, but this is not the case in general. Thisconcludesourbriefintroductiontohyperplanearrangementsandresonance varieties. For additional details on arrangements, see Orlik-Terao [17]. 2. Grassmannians We write G(k,V) for the Grassmannianof k-planes in a vector space V. This is anaffinecone,andcanbe thoughtofasthe projectivevarietyG(k−1,P(V))). Let W ⊂P(E )×P(ΛkE ) denote the open subset W := {([a],[ρ])|a∧ρ 6=0}. The k 1 1 k various maps we need are displayed in the following diagram: (2.1) W µk //P(Λk+1E ) ||yyπy1yyyyyy k HHHHHπHH2HH$$ 1 P(E ) P(ΛkE ), 1 1 where µ denote the multiplication map ([a],[ρ])7→[a∧ρ] and the π ’s denote the k i projections. Let Θ ⊂ P(E )×P(ΛkE ) ×P(Λk+1E ) denote the graph of µ . If π := k 1 1 1 k 23 π ×π : Θ → P(ΛkE )×P(Λk+1E ) is the projection onto the two last factors, 2 3 k 1 1 denote (2.2) Γ :=π (Θ )⊂P(ΛkE )×P(Λk+1E ). k 23 k 1 1 Given 0 6= a ∈ E , let Lk ⊂ ΛkE denote the image of the multiplication map 1 a 1 a: Λk−1E → ΛkE , and let [Lk] ⊂ P(ΛkE ) denote the corresponding projective 1 1 a 1 linearsubspace. IfwewriteΛkE asaninternaldirectsumLk⊕V thecomplement 1 a Ua =P(ΛkE1)\[Lka]iseasilyseentobeisomorphictothetotalspaceofOP(V)⊗Lka, in other words, it is isomorphic to sum of O(1)’s over the projective space P(V). It is easy to see that π : W → P(E ) is a fiber bundle whose fiber π−1([a]) is 1 k 1 1 isomorphicto U . In particular,we conclude that dimΘ =dimW = n +n−2. a k k (cid:0)k(cid:1) Now we consider the case k =1. Here we can identify Γ with the image of the 1 flag variety F(1,2;E ) of lines in a plane in E under the sequence of embeddings 1 1 F(1,2;E ) ⊂ P(E )×G(2,E )−I−d×−→℘ P(E )×P(Λ2E ), 1 1 1 1 1 4 PAULOLIMA-FILHOANDHALSCHENCK where℘isthePlu¨ckerembedding. Furthermore,theprojectionπ : Θ →F(1,2;E ) 23 1 1 is the evident C∗-bundle. The following observation is the key to computing the first resonance variety in terms of the Grassmann geometry described above. Let I ⊂ ΛkE denote the k 1 homogeneous component of degree k of the ideal I, and let [I ]⊂P(ΛkE ) denote k 1 the corresponding linear subspace. Proposition 2.1. Using the notation in (2.1) R1(A)=π (µ−1( [I ] ). 1 1 2 In other words, if [I ]dec := G(2,E )∩[I ] ⊂ P(Λ2E ) denotes the decompos- 2 1 2 1 able elements in [I ], then R1(A) = p (p−1([I ]dec), where p and p denote the 2 1 2 2 1 2 projections from F(1,2;E ) onto P(E ) and G(2,E ), respectively. 1 1 1 Remark 2.2. In practice, a point of Υ := G(2,E )∩[I ] corresponds to a line 1 2 in P(E ). The resonance variety R1(A) is simply the collection ∪ G(1,L) of all 1 L∈Υ lines in P(E ) that correspond to points of Υ. 1 The situationwith higher resonancevarieties Rk(A) is more complicated, but it still can be described as follows. Again, we refer to diagram (2.1) for notation. Proposition 2.3. The resonance variety Rk(A) can be described as Rk(A)=π µ−1[I ]\π−1[I ] , 1(cid:0) k k+1 2 k (cid:1) where {···} denotes Zariski closure. 3. Examples and Code Inthissection,wecomputeseveralexamples,comparingthetimeofthecomputa- tionusingtheGrassmannianagainstthetimeofthecomputationusingannihilator of Ext modules. Example 3.1. We compute the first resonance variety of the A arrangement of 3 Example 1.2 using the approach of Proposition 2.1. First, we find Idec observing 2 that u∈Λ2E is decomposable iff u∧u=0, by the Grassmann-Plu¨cker relations. 1 Denote the basis of I by ρ := ∂e ; ρ := ∂e ; ρ := ∂e and ρ := ∂e 2 1 145 2 012 3 034 4 235 and, given i∈{0,...,5} denote e =e ∧···∧e ∧e ∧···∧e ∈Λ5C6. i 0 i−1 i+1 5 A direct calculation gives b ρ ∧ρ =∂e , ρ ∧ρ =−∂e , ρ ∧ρ = ∂e , 1 2 3 1 3 2 1 4 0 ρ ∧ρ =∂be , ∂ ∧ρ = ∂eb , ρ ∧ρ =−∂be . 2 3 5 2 4 4 3 4 1 Hence, given 06=u= 4 bt ρ ∈I one can wbrite 0=u∧u=∂bω, where Pi=1 i i 2 ω =t t e −t t e +t t e +t t e +t t e −t t e . 1 2 3 1 3 2 1 4 0 2 3 5 2 4 4 3 4 1 Now, ∂ω = 0 iff ω =bλ∂(e b ), for sobme λ, sinbce Λ6C6bis one-dibmensional and 012345 ∂: E →E is acyclic. This gives λ=t t =t t =−t t =−t t =t t =−t t . 1 4 3 4 1 3 1 2 2 4 2 3 If λ 6= 0 then t 6= 0 for all i. In particular, since t 6= 0 implies t = t = −t i 1 2 3 4 and t 6= 0 implies t = t = −t , one concludes that u = t(ρ +ρ +ρ −ρ ) for 2 3 4 1 1 2 3 4 some t6=0. It is easy to see that ρ +ρ +ρ −ρ =(e −e −e +e )∧(e −e +e −e ) 1 2 3 4 0 1 3 5 1 2 3 4 EFFICIENT COMPUTATION OF RESONANCE VARIETIES VIA GRASSMANNIANS 5 andthat this decomposable vectorcorrespondsprecisely to the only essentialcomponent of R1(A). If λ=0 and t 6=0, the equations above give t =0 for all k 6=i. This gives the i k four additional elements ρ , i=1,2,3,4,in Idec which correspondto the four local i 2 components. i1 : load "Rscript" i2 : time R1A A3 5*P 0 -- used .036 seconds --The EPY script produces the module F(A) described in Section 1. i3 : time ann(Ext^2(EPY(A3),S)) -- used 0.125 seconds The ann(Ext2(F(A),S)) computation takes place in P(E ), while the Grass- 1 mannian computation takes place in P(Λ2(E )). So the output 5∗P indicating 1 0 that R1(A) consists of five points means five points in G(2,E ), so five lines in 1 P(E ). 1 The four-fold speedup seems small, but next we tackle a larger example. Example 3.2. The Hessian arrangement consists of the twelve lines passing thru the nine inflection points of a smooth plane cubic curve. There are 4 lines incident ateachofthenineinflectionpoints,sothatR1(A)willcontain9localcomponents, each of dimension two. t t t @ (cid:0)@ (cid:0) (cid:0) (cid:0) @ @ (cid:0)t @t(cid:0) @t @ (cid:0)@ (cid:0) (cid:0) (cid:0) @ @ t(cid:0) @t(cid:0) @t Figure 2. The Hessian arrangement i4 : time R1A Hessian 54*P + 10*P 0 2 -- used 14.004 seconds --Again, we test against the time to find the annilihator of Ext^2. o5 : time ann(Ext^2(EPY(hessian),S)) -- used 9038.345 seconds Thiscomputationindicatesthatforthe Hessianconfiguration,R1(A)hastencom- ponents; in the Grassmannian, the tenth component is two-dimensional. Since it corresponds to a linear subvariety of P(E ), and the space of lines in a fixed P2 is 1 6 PAULOLIMA-FILHOANDHALSCHENCK two-dimensional, this means that (in contrast to the previous example), the non- local component of R1(A) is a P2. The Hessian configuration is the only example knownwithanonlocalcomponentofR1(A)ofdimensiongreaterthanone,see[11]. The previous computations were performed on a ubuntu 7.10 system with a 2.2 GHz AMD processorand64GB of RAM. We close with a shortsectionillustrating how to implement this using the Macaulay2 package of Grayson and Stillman. gring = (k,n)->(S = sort subsets(n,k); vlist = apply(S, i->w_i); ZZ/31991[vlist]); --produce a ring where the variables are indexed by subsets, i.e. plucker ring. --variables lex ordered in the indices. ----------------------------------------------------------------------------------- g2n=(n)->(G=gring(2,n); T=sort subsets(n,4); pluckers = ideal matrix {apply(T, i-> w_{i#0,i#1}*w_{i#2,i#3}-w_{i#0,i#2}*w_{i#1,i#3}+w_{i#0,i#3}*w_{i#1,i#2})}) --script takes input n, and builds a ring with variables w_ij, return ideal --of pluckers for affine G(2,n). ----------------------------------------------------------------------------------- OSrelns = (L)->(L1=apply(L, i->w_{i#0,i#1}-w_{i#0,i#2}+w_{i#1,i#2}); L2 = jacobian matrix {L1}) --this takes a list of the rank 2 dependencies. For example, for A_3 --we have {{0,1,2},{0,3,4},{2,3,5},{1,4,5}}. Dependencies are decomposable --two tensors, so give a point on the Grassmannian. The resulting matrix is --the set of such points in P(\Wedge^2(K^n)). ----------------------------------------------------------------------------------- pointideal1 = (m)->(v=transpose vars G; minors(2,(v|m))) --compute the ideal of a point. pointsideal1 = (m)->( t=rank source m; J=pointideal1(submatrix(m, ,{0})); scan(t-1, i->( J=intersect(J, pointideal1(submatrix(m, ,{i+1}))))); J) --pointsideal1 takes a matrix with columns representing points, and returns --the ideal of the points. So, to get the linear subspace spanned by the --points, we’ll need to take the degree one part of the ideal J. ----------------------------------------------------------------------------------- R1A = (M)->(t1=max mingle M; --determine n g2n(t1+1); --build pluckers and ring P = pointsideal1(OSrelns M); --ideal of points on G(2,n) LL = select(P_*, f -> first degree f <= 1); --get the linear forms R1 = pluckers + ideal LL; --intersect G(2,n) with LL hilbertPolynomial coker gens R1) --script to take the dependent sets of a matroid, then build G(2,n), find the --linear span of the points of M on G(2,n), and intersect that linear span --with G(2,n), yielding ideal of R^1(A) in G(2,n). Print Hilbert poly. ----------------------------------------------------------------------------------- Acknowledgements: ComputationsinMacaulay2[12]wereessentialtoourwork. We also thank Mike Falk for useful suggestions. 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MR2002k:14029 [22] S.Yuzvinsky, Cohomology of Brieskorn-Orlik-Solomon algebras, Comm.Algebra23(1995), 5339–5354. MR97a:5202 Departmentof Mathematics,Texas A&MUniversity, CollegeStation,TX77843 E-mail address: [email protected] URL:http://www.math.tamu.edu/~paulo.lima-filho Departmentof Mathematics,University ofIllinois, Urbana,IL 61801 E-mail address: [email protected] URL:http://www.math.uiuc.edu/~schenck