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 Email: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 )
very generic discriminantal arrangement 1999 Athanasiadeis [1] non very generic combinatorics 2016 Libgober Settepanella [11] [11] [10] combinatorics 2 2.1 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 =.
p 1234 p 1256 p 3456 8 8888888 8 l 12 8888888 l 56 l 34 8 8888888 8 8888888 H 1 H 2 H 3 88888888 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)}.
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
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)
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 123 0 0 = β 256 β 156 0 0 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).
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 456 + β 124 β 156 β 346 ) = 0 β 346 (β 134 β 256 β 234 β 156 ) = 0 β 134 β 125 β 456 + β 124 β 156 β 345 = 0 β 345 (β 134 β 256 β 234 β 156 ) = 0 β 134 β 126 β 356 + β 123 β 156 β 346 = 0 β 256 (β 124 β 356 β 123 β 456 ) = 0 β 234 β 126 β 456 + β 124 β 256 β 346 = 0 and (β 134 β 125 β 356 + β 123 β 156 β 345 ) = 0 β 134 (β 125 β 346 β 126 β 345 ) = 0 (5) (β 234 β 125 β 456 + β 124 β 256 β 345 ) = 0 β 126 (β 124 β 356 β 123 β 456 ) = 0 (β 234 β 126 β 356 + β 123 β 256 β 346 ) = 0 β 125 (β 124 β 356 β 123 β 456 ) = 0 β 234 β 125 β 356 + 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 13... A =... a 61 a 62 a 63 (6) Hi 0 A normal vector α i A generic A C 6 C 6 3
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 456 + 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 456 + β 124 β 256 β 345 = 0 ( f ) : β 234 β 126 β 356 + β 123 β 256 β 346 = 0 (g) : β 234 β 125 β 356 + β 123 β 256 β 345 = 0 and (II) : (h) : β 134 β 126 β 456 + β 124 β 156 β 346 = 0 (i) : β 134 β 125 β 456 + β 124 β 156 β 345 = 0 ( j) : β 134 β 126 β 356 + β 123 β 156 β 346 = 0 (k) : β 134 β 125 β 356 + 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, 4 0 6 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}
{{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, 283 289. [2] A. Bachemand and W. Kern, Adjoints of oriented matroids, Combinatorica 6 (1986) 299 308. [3] M. Bayer and K.Brandt, Discriminantal arrangements, fiber polytopes and formality, J. Algebraic Combin. 6 (1997), 229 246. [4] H.Crapo, Concurrence geometries, Adv. in Math., 54 (1984), no. 3, 278 301. [5] M. Falk, A note on discriminantal arrangements, Proc. Amer. Math. Soc., 122 (1994), no.4, 1221-1227. [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, 241 267. [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): http://www.ma.huji.ac.il/ ruthel/papers/premsh.html.
[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) 1111-1120. [11] A. Libgober and S. Settepanella, Strata of discriminantal arrangements, arxiv:1601.06475. [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. 289-308. [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), 209 251.