Discriminantal arrangement

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1 Discriminantal arrangement YAMAGATA So C k n arrangement C n discriminantal arrangement 1989 Manin-Schectman Braid arrangement Discriminantal arrangement Gr(3, n) S.Sawada S.Settepanella 1 A arrangement 1989 Manin Schctman Braid arrangement discriminantal arrangement [12] A = {H 0 1, H0 2,..., H0 n} C k n (generic arrangement) C n C n discriminantal arrangement B(n, k) B(n, k) B(n, 1) ( [14]) arrangement [12], [1], [3] Zamolodchikov equation [7] [15] [12] Z combinatorics ( combinatorics n k ) B(n, k) Z 1994 Falk [5] [13] sect. 8, [14] [9] A Z discriminantal arrangement combinatorics A B(n, k) combinatorics A = {H 0 1, H0 2,..., H0 n} 1997 Bayer Brandt [3] discriminantal arrangement very generic non very generic so.yamagata.math@gmail.com. The author is supported by The Ministry of Education, Culture, Sports, Science and Technology through Program for Leading Graduate School (Hokkaido University Ambitious Leader s Program )

2 very generic discriminantal arrangement 1999 Athanasiadeis [1] non very generic combinatorics 2016 Libgober Settepanella [11] [11] [10] combinatorics F V F k A = {H 1, H 2,..., H n } (arrangement of hyperplanes) F C V (k 1)- H α = (α 1, α 2,..., α k ) F k 0 H H = {v V α v = 0}, α v = (α 1, α 2,..., α k ) (v 1, v 2,..., v k ) = α i v i. J = {v V α v = a, a C} V C k+1 k- P(V) = (V \ {0})/C V C k+1 H k- H = {P = [U] P(V) U H} 1 i k A A B A A A = {H 1, H 2,..., H n } V P(V) A V P(V) A generic arrangement 1 p n {H i1, H i2,..., H ip } A, p k dim(h i1 H i2,..., H ip ) = k p, {H i1, H i2,..., H ip } A, p > k H i1 H i2,..., H ip =.

3 p 1234 p 1256 p l l 56 l H 1 H 2 H H 4 H 5 H 6 8 C 3 1: Intersection semilattice of L(A) (intersection poset) L(A) = { H B H B A}, x < y x y x < y x y Combinatorics semilattice lattice C 3 generic arrangementa = {H 1, H 2,..., H 6 } 1 l i j = H i H j p i jkl = l i j l kl 2.2 Discriminantal arrangement A = {H 0 1, H0 2,..., H0 n} C k generic arrangement H 0 1, H0 2,..., H0 n S(H 0 1,..., H0 n) (Hi 0 S ) n- H 1,..., H n i = 1,..., n H i Hi 0 = H i = Hi 0 S = {(H 1, H 2,..., H n ) H i H 0 i = or H i = Hi 0 (i = 1, 2,..., n)}.

4 S C n A S ([11]). Generic arrangement A generic arrangement S L = {i 1,..., i k+1 } [n] := {1,..., n} D L = {(H 1, H 2,..., H n ) S i j L [n], L =k+1 H i j } Discriminantal arrangement L {D L } L [n], L =k+1 B(n, k, A) Mannin Schectman [12]. Discriminantal arrangement normal vector Bayer Brandt [3]. Definition 2.1 (Bayer and Brandt [3]). A = {H 0 1, H0 2,..., H0 n} C k generic arrangement α 1, α 2..., α n H 0 1, H0 2..., H0 n normal vector normal vector section 2.1 (α 1, α 2..., α n ) (v 1, v 2..., v n ) = n α i v i. Discriminantal arrangement B(n, k, A) C n normal vector i=1 k+1 α L = ( 1) i det(α s1,..., αˆ si,..., α sk+1 )e si, (1) i=1 {s 1 < s 2 < < s k+1 } [n] {e j } 1 j n C n α L 0. C k P k \ H H 0 i H 0 i C k generic arrangement A = {H 1, H 2,..., H n } arrangement A = {H,1, H,2,..., H,n } i (i = 1, 2,..., n) H,i = H 0 i H A P k n- S Generic arrangement A Discriminantal arrangement B(n, k, A ) Discrimnantal arrangement combinatorics A [11] B(n, k, A) B(n, k, A ) A generic arrangement B(n, k, A ) combinatorics A A very generic, non very generic 2.3 good 3s-partition A(A ) s 2 n 3s s, n T = {L 1, L 2, L 3 } L i [n] L i = 2s, L i L j = s(i j), L 1 L 2 L 3 = ( L i = 3s) L 1 = {i 1,..., i 2s }, L 2 = {i s+1,..., i 3s }, L 3 = {i 1,..., i s, i 2s+1..., i 3s }. T = {L 1, L 2, L 3 } good 3s-partition. Lemma

5 Lemma 2.2 (Lemma 3.1 [11]). s 2, n = 3s, k = 2s 1 A = {H 0 1, H0 2,..., H0 n} C k generic arrangement [n] = [3s] good 3s-partition T = {L 1, L 2, L 3 } H s H,i, j = t L i L j H,t H,i, j H D L1 D L2 D L3 2. D Li B(n, k, A ) T = {L 1, L 2, L 3 } L 1 = {1, 2, 3, 4}, L 2 = {1, 2, 5, 6}, L 3 = {3, 4, 5, 6} good 6- partition I 1 = L 1 L 2, I 2 = L 2 L 3 I 3 = L 1 L 3 i I 1 H,i, j I 2 H,i k I 3 H,i H [11] dependent s 2, P 2s 2 generic arrangement A = {W,1,..., W,3s } [3s] partition I 1, I 2, I 3 P i = t I i W,t P 2s 2 A dependent {L 1, L 2, L 3 } good 3spartition I 1 = L 1 L 2, I 2 = L 1 L 3, I 3 = L 2 L 3 lemma 2.2 A dependent P k+1 ([n]) = {L [n] L = k + 1} k + 1 [n] A(A ) = (α L ) L Pk+1 ([n]) (2) D L normal vector α L A T (A ) α L, L T, T P k+1 ([n]) A(A ) 2.4 Gr(k, n) Gr(k, n) C n k- γ : Gr(k, n) P( k C n ) < v 1,..., v k > [v 1 v k ], [x] P( k C n ) γ(gr(k, n)) φ x : C n k+1 C n v v x k ker φ x =< v 1,..., v k > e 1,..., e n C n e I = e i1... e ik, I = {i 1,..., i k } [n], i 1 < < i k k C n x k C n x = β I e I = I [n] I =k 1 i 1 < <i k n β i1...i k (e i1 e ik ) (3)

6 β I C n e 1,..., e n P( k C n ) = P (n k) 1 φ x M x = (b i j ) ( n k+1) n I [n], I = k i I b i j = ( 1) i β I { j}\{i} b i j = 0 dim(ker φ x ) = k M x (n k + 1) (n k + 1) 0 ( [6]) (i 1,..., i k 1, j 0,..., j k ) k ( 1) l β i1...i k 1 j l β j0... ĵ l... j k = 0 (4) l=0 Remark 2.3. (1) α L M x I = L D L A(A ) = M x A(A ) det (α s1,..., αˆ si,..., α sk+1 ) β I, I = {s 1, s 2,..., s k+1 }\{s i } 2.5 C 3 generic arrangement A = {H 0 1, H0 2,..., H0 6 } normal vector α i = (a i1, a i2, a i3 ), 1 i 6 H t i i Hi 0 α i H t i i = Hi 0 + t i α i, t i C. T = {L 1, L 2, L 3 } [6] good 6-partition L 1 = {1, 2, 3, 4}, L 2 = {1, 2, 5, 6},L 3 = {3, 4, 5, 6} A T (A ) = α L1 α L2 α L3 β 234 β 134 β 124 β = β 256 β β 126 β 125, 0 0 β 456 β 356 β 346 β 345 a i1 a j1 a k1 β i jk = det a i2 a j2 a k2 a i3 a j3 a k3 A(A ) α 1 α 2 α i α j α i, α j (α i α i+1 ) α 3 α 4 α 5 α 6 α i α j H i H j α i α j H i H j ranka T (A ) = 2 rank(α i α i+1 ) = 2 ranka T (A ) = 2 codim (D L1 D L2 D L3 ) = 2 Lemma 2.2 H ti i H = H t 3 3 H t 4 4 H, H ti i H = H t 1 1 H t 2 2 H, H ti i H = H t 5 5 H t 6 6 H i L 1 L 2 i L 1 L 3 i L 2 L 3 H t i i Ht i+1 i+1 rank(α i α i+1 ) = 2 ( 2).

7 2: Picture of case B(6, 3, A 0 ) A T (A ) 2 A T (A ) 3 0. β i jk β 456 (β 134 β 256 β 234 β 156 ) = 0 β 156 (β 124 β 356 β 123 β 456 ) = 0 β 356 (β 134 β 256 β 234 β 156 ) = 0 (β 134 β 126 β β 124 β 156 β 346 ) = 0 β 346 (β 134 β 256 β 234 β 156 ) = 0 β 134 β 125 β β 124 β 156 β 345 = 0 β 345 (β 134 β 256 β 234 β 156 ) = 0 β 134 β 126 β β 123 β 156 β 346 = 0 β 256 (β 124 β 356 β 123 β 456 ) = 0 β 234 β 126 β β 124 β 256 β 346 = 0 and (β 134 β 125 β β 123 β 156 β 345 ) = 0 β 134 (β 125 β 346 β 126 β 345 ) = 0 (5) (β 234 β 125 β β 124 β 256 β 345 ) = 0 β 126 (β 124 β 356 β 123 β 456 ) = 0 (β 234 β 126 β β 123 β 256 β 346 ) = 0 β 125 (β 124 β 356 β 123 β 456 ) = 0 β 234 β 125 β β 123 β 256 β 345 = 0 β 234 (β 125 β 346 β 126 β 345 ) = 0 β 124 (β 125 β 346 β 126 β 345 ) = 0 β 123 (β 125 β 346 β 126 β 345 ) = 0 3 Gr(3, n) A section 2.5 C 3 6 generic arrangement a 11 a 12 a A =... a 61 a 62 a 63 (6) Hi 0 A normal vector α i A generic A C 6 C 6 3

8 Gr(3, 6) A 0 3 β i jk A(A ) φ x : C 6 4 C 6 v v x, x = 1 i< j<k n β i jk (e i e j e k ). A dependent β i jk (5) : (d) : β 234 β 126 β β 124 β 256 β 346 = 0 (a) : β 134 β 256 β 234 β 156 = 0 (I) : (b) : β 124 β 356 β 123 β 456 = 0 (c) : β 125 β 346 β 126 β 345 = 0 (e) : β 234 β 125 β β 124 β 256 β 345 = 0 ( f ) : β 234 β 126 β β 123 β 256 β 346 = 0 (g) : β 234 β 125 β β 123 β 256 β 345 = 0 and (II) : (h) : β 134 β 126 β β 124 β 156 β 346 = 0 (i) : β 134 β 125 β β 124 β 156 β 345 = 0 ( j) : β 134 β 126 β β 123 β 156 β 346 = 0 (k) : β 134 β 125 β β 123 β 156 β 345 = 0. (a) (k) (a) β 134 β 256 β 234 β 156 = 0. (7) (a) Lemma Lemma 3.1 (Lemma 5.1 [10]). A C 3 n generic arrangement A(A ) {i 1, i 2,..., i 6 } [n] T = {L 1, L 2, L 3 } L 1 = {i 1, i 2, i 3, i 4 }, L 2 = {i 1, i 2, i 5, i 6 },L 3 = {i 3, i 4, i 5, i 6 } good 6-partition A(A ) β I ranka T (A ) = 2 A T (A ) 3 0 Remark 3.2. A C 3 n generic arrangement A(A ) ( n ) 4 n L = {s1 < s 2 < s 3 < s 4 } α L (x 1,..., x n ) x i j = ( 1) j β I j, I j = L \ {s j }, j = 1, 2, 3, s 1 <... < s 6 [n] A(A ) ( 6 4) 6 α L, L {s 1,..., s 6 }, L = 4 s 1,..., s 6 ( (α L ) L {s1,...,s 6 }, L =4 j j {s 1,..., s 6 } 0. C 3 n n = 6 s 1 <... < s 6 [n] 6 T = {{s 1, s 2, s 3, s 4 }, {s 1, s 2, s 5, s 6 }, {s 3, s 4, s 5, s 6 }} good 6-partition {1,... 6}

9 {{1, 2, 3, 4}, {1, 2, 5, 6}, {3, 4, 5, 6}} {s 1,..., s 6 } good 6-partition σ.t = {{i 1, i 2, i 3, i 4 }, {i 1, i 2, i 5, i 6 }, {i 3, i 4, i 5, i 6 }} (8) i j = σ(s j ), σ S 6 S 6 {s 1,..., s 6 } i j i j > i j+1 Lemma Lemma 3.3 (Lemma 5.3 [10]). A C 3 n generic arrangement σ.t = {{i 1, i 2, i 3, i 4 }, {i 1, i 2, i 5, i 6 }, {i 3, i 4, i 5, i 6 }} s 1 <... < s 6 [n] good 6-partition ranka σ.t (A ) = 2 A β i1 i 3 i 4 β i2 i 5 i 6 β i2 i 3 i 4 β i1 i 5 i 6 = 0. (9) Remark 3.2 Lemma 3.3 Theorem 3.4 (Theorem 5.4 [10]). C 3 dependent generic arrangement Gr(3, n) [1] C. A. Athanasiadis. The Largest Intersection Lattice of a Discriminantal Arrangement, Beiträge Algebra Geom., 40 (1999), no. 2, [2] A. Bachemand and W. Kern, Adjoints of oriented matroids, Combinatorica 6 (1986) [3] M. Bayer and K.Brandt, Discriminantal arrangements, fiber polytopes and formality, J. Algebraic Combin. 6 (1997), [4] H.Crapo, Concurrence geometries, Adv. in Math., 54 (1984), no. 3, [5] M. Falk, A note on discriminantal arrangements, Proc. Amer. Math. Soc., 122 (1994), no.4, [6] Joe Harris, Algebraic Geometry: A First Course, Springer-Verlag. [7] M. Kapranov, V.Voevodsky, Braided monoidal 2-categories and Manin-Schechtman higher braid groups, Journal of Pure and Applied Algebra, 92 (1994), no. 3, [8] Y. Kawamata, Shaeikukannokikagaku (Geometry in projective space), Asakura Publishing Co., Ltd. [9] R.J. Lawrence, A presentation for Manin and Schechtman s higher braid groups, MSRI pre-print (1991): ruthel/papers/premsh.html.

10 [10] S. Sawada, S. Settepanella and S. Yamagata, Discriminantal arrangement, 3 3minors of Plücker matrix and hypersurfaces in Grassmannian Gr(3,n), C. R. Acad. Sci. Paris, Ser. I 355 (2017) [11] A. Libgober and S. Settepanella, Strata of discriminantal arrangements, arxiv: [12] Yu. I. Manin and V. V. Schectman, Arrangements of Hyperplanes, Higher Braid Groups and Higher Bruhat Orders, Advanced Studies in Pure Mathematics 17, 1989 Algebraic Number Theory in honor K. Iwasawa pp [13] P. Orlik, Introduction to arrangements, CBMS Regional Conf. Ser. in Math., 72, Amer. Math. Soc., Providence, RI, (1989). [14] P.Orlik,H.Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 300, Springer-Verlag, Berlin, (1992). [15] M. Perling, Divisorial Cohomology Vanishing on Toric Varieties, Documenta Math. 16 (2011),

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