H YPERPLANE ARRANGEMENTS : AT THE CROSSROADS OF TOPOLOGY AND COMBINATORICS Alex Suciu Northeastern University Colloquium Goethe University Frankfurt October 25, 2013 A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 1 / 31
H YPERPLANE ARRANGEMENTS H YPERPLANE ARRANGEMENTS An arrangement of hyperplanes is a finite set A of codimension-1 linear subspaces in C ℓ . Intersection lattice L ( A ) : poset of all intersections of A , ordered by reverse inclusion, and ranked by codimension. Complement : M ( A ) = C ℓ z Ť H P A H . The Boolean arrangement B n B n : all coordinate hyperplanes z i = 0 in C n . L ( B n ) : Boolean lattice of subsets of t 0 , 1 u n . M ( B n ) : complex algebraic torus ( C ˚ ) n . The braid arrangement A n (or, reflection arr. of type A n ´ 1 ) A n : all diagonal hyperplanes z i ´ z j = 0 in C n . L ( A n ) : lattice of partitions of [ n ] = t 1 , . . . , n u . M ( A n ) : configuration space of n ordered points in C (a classifying space for the pure braid group on n strings). A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 2 / 31
H YPERPLANE ARRANGEMENTS ‚ x 2 ´ x 4 x 1 ´ x 2 ‚ x 2 ´ x 3 ‚ ‚ x 1 ´ x 3 x 3 ´ x 4 x 1 ´ x 4 F IGURE : A planar slice of the braid arrangement A 4 A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 3 / 31
H YPERPLANE ARRANGEMENTS We may assume that A is essential, i.e., Ş H P A H = t 0 u . Fix an ordering A = t H 1 , . . . , H n u , and choose linear forms f i : C ℓ Ñ C with ker ( f i ) = H i . Define an injective linear map ι A : C ℓ Ñ C n , z ÞÑ ( f 1 ( z ) , . . . , f n ( z )) . This map restricts to an inclusion ι : M ( A ) ã Ñ M ( B n ) . Thus, M ( A ) = ι A ( C ℓ ) X ( C ˚ ) n , a “very affine" subvariety of ( C ˚ ) n . The tropicalization of this sub variety is a fan in R n . Feichtner and Sturmfels: this is the Bergman fan of L ( A ) . A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 4 / 31
H YPERPLANE ARRANGEMENTS M ( A ) has the homotopy type of a connected, finite cell complex of dimension ℓ . In fact, M = M ( A ) admits a minimal cell structure. Consequently, H ˚ ( M , Z ) is torsion-free. The Betti numbers b q ( M ) : = rank H q ( M , Z ) are given by ÿ ℓ ÿ b q ( M ) t q = µ ( X )( ´ t ) rank ( X ) , q = 0 X P L ( A ) where µ : L ( A ) Ñ Z is the Möbius function, defined recursively by µ ( C ℓ ) = 1 and µ ( X ) = ´ ř Y Ľ X µ ( Y ) . The Orlik–Solomon algebra H ˚ ( M , Z ) is the quotient of the exterior algebra on generators t e H | H P A u by an ideal determined by the circuits in the matroid of A . Thus, the ring H ˚ ( M , k ) is determined by L ( A ) , for every field k . A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 5 / 31
M ULTINETS M ULTINETS Let A be an arrangement of planes in C 3 . Its projectivization, ¯ A , is an arrangement of lines in CP 2 . Ñ lines of ¯ Ñ intersection points of ¯ L 1 ( A ) Ð A , L 2 ( A ) Ð A , Ñ incidence structure of ¯ poset structure of L ď 2 ( A ) Ð A . A flat X P L 2 ( A ) has multiplicity q if the point ¯ X has exactly q lines from ¯ A passing through it. D EFINITION (F ALK AND Y UZVINSKY ) A multinet on A is a partition of the set A into k ě 3 subsets A 1 , . . . , A k , together with an assignment of multiplicities, m : A Ñ N , and a subset X Ď L 2 ( A ) , called the base locus, such that: There is an integer d such that ř H P A α m H = d , for all α P [ k ] . 1 If H and H 1 are in different classes, then H X H 1 P X . 2 For each X P X , the sum n X = ř H P A α : H Ą X m H is independent of α . 3 � Ť � Each H P A α H z X is connected. 4 A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 6 / 31
M ULTINETS A multinet as above is also called a ( k , d ) -multinet, or a k -multinet. If m H = 1, for all H P A , the multinet is reduced . If, furthermore, n X = 1, for all X P X , this is a net . In this case, | A α | = | A | / k = d , for all α . Moreover, ¯ X has size d 2 , and is encoded by a ( k ´ 2 ) -tuple of orthogonal Latin squares. ‚ 2 2 ‚ ‚ ‚ 2 A ( 3 , 2 ) -net on the A 3 arrangement A ( 3 , 4 ) -multinet on the B 3 arrangement ¯ ¯ X consists of 4 triple points ( n X = 1) X consists of 4 triple points ( n X = 1) and 3 triple points ( n X = 2) A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 7 / 31
M ULTINETS A ( 3 , 3 ) -net on the Ceva matroid. A ( 4 , 3 ) -net on the Hessian matroid. If A has no flats of multiplicity kr , for some r ą 1, then every reduced k -multinet is a k -net. (Yuzvinsky and Pereira–Yuzvinsky): If A supports a k -multinet with | X | ą 1, then k = 3 or 4; moreover, if the multinet is not reduced, then k = 3. Conjecture (Yuz): The only 4-multinet is the Hessian ( 4 , 3 ) -net. A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 8 / 31
C OHOMOLOGY JUMP LOCI C HARACTERISTIC VARIETIES C OHOMOLOGY JUMP LOCI Let X be a connected, finite cell complex, and let π = π 1 ( X , x 0 ) . Let k be an algebraically closed field, and let Hom ( π , k ˚ ) be the affine algebraic group of k -valued, multiplicative characters on π . The characteristic varieties of X are the jump loci for homology with coefficients in rank-1 local systems on X : V q s ( X , k ) = t ρ P Hom ( π , k ˚ ) | dim k H q ( X , k ρ ) ě s u . Here, k ρ is the local system defined by ρ , i.e, k viewed as a k π -module, via g ¨ x = ρ ( g ) x , and H i ( X , k ρ ) = H i ( C ˚ ( r X , k ) b k π k ρ ) . These loci are Zariski closed subsets of the character group. The sets V 1 s ( X , k ) depend only on π / π 2 . A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 9 / 31
C OHOMOLOGY JUMP LOCI C HARACTERISTIC VARIETIES E XAMPLE (C IRCLE ) We have Ă S 1 = R . Identify π 1 ( S 1 , ˚ ) = Z = x t y and k Z = k [ t ˘ 1 ] . Then: C ˚ ( Ă t ´ 1 � k [ t ˘ 1 ] � k [ t ˘ 1 ] � 0 . S 1 , k ) : 0 For ρ P Hom ( Z , k ˚ ) = k ˚ , we get ρ ´ 1 � k C ˚ ( Ă � k � 0 , S 1 , k ) b k Z k ρ : 0 which is exact, except for ρ = 1, when H 0 ( S 1 , k ) = H 1 ( S 1 , k ) = k . Hence: V 0 1 ( S 1 , k ) = V 1 1 ( S 1 , k ) = t 1 u and V i s ( S 1 , k ) = H , otherwise. E XAMPLE (P UNCTURED COMPLEX LINE ) Identify π 1 ( C zt n points u ) = F n , and x F n = ( k ˚ ) n . Then: $ ( k ˚ ) n & if s ă n , V 1 s ( C zt n points u , k ) = t 1 u if s = n , % H if s ą n . A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 10 / 31
C OHOMOLOGY JUMP LOCI R ESONANCE VARIETIES Let A = H ˚ ( X , k ) . If char k = 2, assume that H 1 ( X , Z ) has no 2-torsion. Then: a P A 1 ñ a 2 = 0. Thus, we get a cochain complex a a � A 1 � A 2 � ¨ ¨ ¨ , ( A , ¨ a ) : A 0 known as the Aomoto complex of A . The resonance varieties of X are the jump loci for the Aomoto-Betti numbers s ( X , k ) = t a P A 1 | dim k H q ( A , ¨ a ) ě s u , R q These loci are homogeneous subvarieties of A 1 = H 1 ( X , k ) . E XAMPLE R 1 1 ( T n , k ) = t 0 u , for all n ą 0. R 1 1 ( C zt n points u , k ) = k n , for all n ą 1. R 1 1 ( Σ g , k ) = k 2 g , for all g ą 1. A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 11 / 31
J UMP LOCI OF ARRANGEMENTS J UMP LOCI OF ARRANGEMENTS Let A = t H 1 , . . . , H n u be an arrangement in C 3 , and identify H 1 ( M ( A ) , k ) = k n , with basis dual to the meridians. s ( M ( A ) , k ) Ă k n lie in the The resonance varieties R 1 s ( A , k ) : = R 1 hyperplane t x P k n | x 1 + ¨ ¨ ¨ + x n = 0 u . R ( A ) = R 1 1 ( A , C ) is a union of linear subspaces in C n . Each subspace has dimension at least 2, and each pair of subspaces meets transversely at 0. R 1 s ( A , C ) is the union of those linear subspaces that have dimension at least s + 1. A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 12 / 31
J UMP LOCI OF ARRANGEMENTS Each flat X P L 2 ( A ) of multiplicity k ě 3 gives rise to a local component of R ( A ) , of dimension k ´ 1. More generally, every k -multinet of a sub-arrangement B Ď A gives rise to a component of dimension k ´ 1, and all components of R ( A ) arise in this way. The resonance varieties R 1 ( A , k ) can be more complicated, e.g., they may have non-linear components. A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 13 / 31
J UMP LOCI OF ARRANGEMENTS E XAMPLE (B RAID ARRANGEMENT A 4 ) ❆ ✁✁ ❆ ✁ ❆ ✁ ❆ ❍ ✟ ✟✟✟✟✟ ❍ ✁ ❍ ❆ ❍ ✁ ❍ ❆ 4 ❍ ✁ ❆ 2 1 5 6 3 R 1 ( A , C ) Ă C 6 has 4 local components (from triple points), and one non-local component, from the ( 3 , 2 ) -net: L 124 = t x 1 + x 2 + x 4 = x 3 = x 5 = x 6 = 0 u , L 135 = t x 1 + x 3 + x 5 = x 2 = x 4 = x 6 = 0 u , L 236 = t x 2 + x 3 + x 6 = x 1 = x 4 = x 5 = 0 u , L 456 = t x 4 + x 5 + x 6 = x 1 = x 2 = x 3 = 0 u , L = t x 1 + x 2 + x 3 = x 1 ´ x 6 = x 2 ´ x 5 = x 3 ´ x 4 = 0 u . A LEX S UCIU (N ORTHEASTERN ) H YPERPLANE ARRANGEMENTS F RANKFURT C OLLOQUIUM 14 / 31
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