B RAIDS , HYPERPLANE ARRANGEMENTS , AND M ILNOR FIBRATIONS Alex Suciu Northeastern University Workshop on Braids, Resolvent Degree and Hilbert’s 13th Problem Institute for Pure and Applied Mathematics, UCLA February 21, 2019 A LEX S UCIU (N ORTHEASTERN ) B RAIDS AND HYPERPLANE ARRANGEMENTS IPAM, F EBRUARY 21, 2019 1 / 25
� � P OLYNOMIAL COVERS AND BRAID MONODROMY P OLYNOMIAL COVERS P OLYNOMIAL COVERS Let X be a path-connected space. A simple Weierstrass polynomial of degree n on X is a map f : X ˆ C Ñ C given by n f p x , z q “ z n ` ÿ a i p x q z n ´ i , i “ 1 with continuous coefficient maps a i : X Ñ C , and with no multiple roots for any x P X . Let E “ E p f q “ tp x , z q P X ˆ C | f p x , z q “ 0 u . The restriction of pr 1 : X ˆ C Ñ X to E defines an n -fold cover π “ π f : E Ñ X , the polynomial covering map associated to f . � X ˆ C E � � π pr 1 X A LEX S UCIU (N ORTHEASTERN ) B RAIDS AND HYPERPLANE ARRANGEMENTS IPAM, F EBRUARY 21, 2019 2 / 25
� P OLYNOMIAL COVERS AND BRAID MONODROMY C ONFIGURATION SPACES C ONFIGURATION SPACES Let Conf n p C q “ t z P C n | z i ‰ z j for i ‰ j u and UConf n p C q “ Conf n p C q{ S n . Since f : X ˆ C Ñ C has no multiple roots, the coefficient map a “ p a 1 , . . . , a n q : X Ñ C n takes values in C n z ∆ n “ UConf n p C q . Over UConf n p C q , there is a canonical n -fold polynomial covering map, π n : E p f n q Ñ UConf n p C q , determined by the W-polynomial ÿ n f n p x , z q “ z n ` i “ 1 x i z n ´ i . We get a pullback diagram of covers, � E p f n q E p f q π f � π n a � B n X A LEX S UCIU (N ORTHEASTERN ) B RAIDS AND HYPERPLANE ARRANGEMENTS IPAM, F EBRUARY 21, 2019 3 / 25
� � � P OLYNOMIAL COVERS AND BRAID MONODROMY C ONFIGURATION SPACES B RAID GROUPS Let B n be the Artin braid group on n strands. Then B n “ π 1 p UConf n p C qq . Ñ Aut p F n q be the Artin representation. We let ψ n : B n ã The coefficient homomorphism , α “ a ˚ : π 1 p X q Ñ B n , is well-defined up to conjugacy. Polynomial covers are those covers π : E Ñ X for which the characteristic homomorphism χ : π 1 p X q Ñ S n factors through the canonical surjection τ n : B n ։ S n , B n α τ n χ π 1 p X q S n A LEX S UCIU (N ORTHEASTERN ) B RAIDS AND HYPERPLANE ARRANGEMENTS IPAM, F EBRUARY 21, 2019 4 / 25
� P OLYNOMIAL COVERS AND BRAID MONODROMY C ONFIGURATION SPACES T HE ROOT MAP Now assume that the W-polynomial f completely factors as ź n f p x , z q “ i “ 1 p z ´ b i p x qq , with continuous roots b i : X Ñ C . Since f is simple, the root map b “ p b 1 , . . . , b n q : X Ñ C n takes values in Conf n p C q . Over Conf n p C q , there is a canonical n -fold cover, π Qn : E p Q n q Ñ Conf n p C q , where Q n p w , z q “ p z ´ w 1 q ¨ ¨ ¨ p z ´ w n q . We get a pullback diagram of covers, � E p Q n q E p f q π f � π Qn b � Conf n p C q X A LEX S UCIU (N ORTHEASTERN ) B RAIDS AND HYPERPLANE ARRANGEMENTS IPAM, F EBRUARY 21, 2019 5 / 25
P OLYNOMIAL COVERS AND BRAID MONODROMY B RAID BUNDLES B RAID BUNDLES Let P n “ ker p τ n : B n ։ S n q be the pure braid group. Then P n “ π 1 p Conf n p C qq . The map β “ b ˚ : π 1 p X q Ñ P n is well-defined up to conjugacy. The polynomial covers which are trivial covers are precisely those for which α “ ι n ˝ β , where ι n : P n ã Ñ B n is the inclusion map. T HEOREM (D. C OHEN , A.S. 1997) Let f : X ˆ C Ñ C be a simple W-polynomial. Let Y “ X ˆ C z E p f q and let p : Y Ñ X be the restriction of pr 1 : X ˆ C Ñ X to Y. The map p : Y Ñ X is a locally trivial bundle, with structure group B n and fiber C n “ C zt n points u . Upon identifying π 1 p C n q with F n , the monodromy of this bundle is ψ n ˝ α : π 1 p X q Ñ Aut p F n q . If f completely factors into linear factors, the structure group reduces to P n , and the monodromy factors as ψ n ˝ ι n ˝ β . A LEX S UCIU (N ORTHEASTERN ) B RAIDS AND HYPERPLANE ARRANGEMENTS IPAM, F EBRUARY 21, 2019 6 / 25
P OLYNOMIAL COVERS AND BRAID MONODROMY B RAID MONODROMY OF PLANE ALGEBRAIC CURVES B RAID MONODROMY OF PLANE ALGEBRAIC CURVES Let C be a reduced algebraic curve in C 2 , defined by a polynomial f “ f p z 1 , z 2 q of degree n . Let π : C 2 Ñ C be a linear projection, and let Y “ t y 1 , . . . , y s u be the set of points in C for which the fibers of π contain singular points of C , or are tangent to C . WLOG, we may assume that π “ pr 1 is generic with respect to C . That is, for each k , the line L k “ π ´ 1 p y k q contains at most one singular point v k of C and does not belong to the tangent cone of C at v k , and, moreover, all tangencies are simple. Let L “ Ť L k . A LEX S UCIU (N ORTHEASTERN ) B RAIDS AND HYPERPLANE ARRANGEMENTS IPAM, F EBRUARY 21, 2019 7 / 25
P OLYNOMIAL COVERS AND BRAID MONODROMY B RAID MONODROMY OF PLANE ALGEBRAIC CURVES C 2 C r r r π C ❄ r r r A LEX S UCIU (N ORTHEASTERN ) B RAIDS AND HYPERPLANE ARRANGEMENTS IPAM, F EBRUARY 21, 2019 8 / 25
P OLYNOMIAL COVERS AND BRAID MONODROMY B RAID MONODROMY OF PLANE ALGEBRAIC CURVES In the chosen coordinates, the defining polynomial f of C may be written as f p x , z q “ z n ` ř n i “ 1 a i p x q z n ´ i . Since C is reduced, for each x R Y , the equation f p x , z q “ 0 has n distinct roots. Thus, f is a simple W-polynomial over C z Y , and π “ π f : C z C X L Ñ C z Y is the associated polynomial n -fold cover. Note that Y p f q “ pp C z Y q ˆ C qzp C z C X L q “ C 2 z C Y L . Thus, the restriction of pr 1 to Y p f q , p : C 2 z C Y L Ñ C z Y , is a bundle map, with structure group B n , fiber C n , and monodromy homomorphism α “ a ˚ : π 1 p C z Y q Ñ B n . A LEX S UCIU (N ORTHEASTERN ) B RAIDS AND HYPERPLANE ARRANGEMENTS IPAM, F EBRUARY 21, 2019 9 / 25
P OLYNOMIAL COVERS AND BRAID MONODROMY B RAID MONODROMY OF PLANE ALGEBRAIC CURVES B RAID MONODROMY PRESENTATION The homotopy exact sequence of fibration p : C 2 z C Y L Ñ C z Y : p ˚ � π 1 p C z Y q � 1 . � π 1 p C n q � π 1 p C 2 z C Y L q 1 This sequence is split exact, with action given by the braid monodromy homomorphism α : π 1 p C z Y q Ñ Aut p π 1 p C n qq . Order the points of Y by decreasing real part, and pick the basepoint y 0 in C z Y with Re p y 0 q ą max t Re p y k qu . Choose loops ξ k : r 0 , 1 s Ñ C z Y based at y 0 , and going around y k . Setting x k “ r ξ k s , identify π 1 p C z Y , y 0 q with F s “ x x 1 , . . . , x s y . Similarly, identify π 1 p C n , ˆ y 0 q with F n “ x t 1 , . . . , t n y . Then π 1 p C 2 z C Y L , ˆ y 0 q “ F n ¸ α F s . A LEX S UCIU (N ORTHEASTERN ) B RAIDS AND HYPERPLANE ARRANGEMENTS IPAM, F EBRUARY 21, 2019 10 / 25
P OLYNOMIAL COVERS AND BRAID MONODROMY B RAID MONODROMY OF PLANE ALGEBRAIC CURVES The corresponding presentation is π 1 p C 2 z C Y L q “ x t 1 , . . . t n , x 1 . . . , x s | x ´ 1 k t i x k “ α p x k qp t i qy . The group π 1 p C 2 z C q is the quotient of π 1 p C 2 z C Y L q by the normal closure of F s “ x x 1 , . . . , x s y . Thus, π 1 p C 2 z C q “ x t 1 , . . . , t n | t i “ α p x k qp t i qy . This presentation can be simplified by Tietze-II moves to eliminate redundant relations. This yields the braid monodromy presentation π 1 p C 2 z C q “ x t 1 , . . . , t n | t i “ α p x k qp t i q , i “ j 1 , . . . , j m k ´ 1 ; k “ 1 , . . . , s y . where m k is the multiplicity of the singular point y k . (Libgober 1986) The 2-complex modeled on this presentation is homotopy equivalent to C 2 z C . A LEX S UCIU (N ORTHEASTERN ) B RAIDS AND HYPERPLANE ARRANGEMENTS IPAM, F EBRUARY 21, 2019 11 / 25
H YPERPLANE ARRANGEMENTS C OMPLEMENT AND INTERSECTION LATTICE H YPERPLANE ARRANGEMENTS An arrangement of hyperplanes is a finite collection A of codimension 1 linear (or affine) subspaces in C ℓ . Intersection lattice L p A q : poset of all intersections of A , ordered by reverse inclusion, and ranked by codimension. P 1 P 2 P 3 P 4 L 4 L 3 P 4 L 2 P 3 L 1 L 1 L 2 L 3 L 4 P 1 P 2 Complement : M p A q “ C ℓ z Ť H P A H . It is a smooth, quasi- projective variety and also a Stein manifold. It has the homotopy type of a finite, connected, ℓ -dimensional CW-complex. A LEX S UCIU (N ORTHEASTERN ) B RAIDS AND HYPERPLANE ARRANGEMENTS IPAM, F EBRUARY 21, 2019 12 / 25
H YPERPLANE ARRANGEMENTS F UNDAMENTAL GROUP F UNDAMENTAL GROUP E XAMPLE (T HE B OOLEAN ARRANGEMENT ) B n : all coordinate hyperplanes z i “ 0 in C n . L p B n q : Boolean lattice of subsets of t 0 , 1 u n . M p B n q : complex algebraic torus p C ˚ q n » K p Z n , 1 q . E XAMPLE (T HE BRAID ARRANGEMENT ) A n : all diagonal hyperplanes z i ´ z j “ 0 in C n . L p A n q : lattice of partitions of r n s : “ t 1 , . . . , n u , ordered by refinement. M p A n q “ Conf n p C q » K p P n , 1 q . For an arbitrary (central) arrangement A , let A 1 “ t H X C 2 u H P A be a generic planar slice. Then the arrangement group, π “ π 1 p M p A qq , is isomorphic to π 1 p M p A 1 qq . A LEX S UCIU (N ORTHEASTERN ) B RAIDS AND HYPERPLANE ARRANGEMENTS IPAM, F EBRUARY 21, 2019 13 / 25
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