Secant Varieties and Equations of Abo-Wan Hypersurfaces

Transcription

1 Secant Varieties and Equations of Abo-Wan Hypersurfaces Luke Oeding Auburn University Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

2 Secant varieties Suppose X is an algebraic variety in P N. The X-rank of [p] P N is the min. r such that p = r i=1 x i with [x i ] X. The Zariski closure of the points of X-rank r is the r-secant variety to X, denoted σ r (X), and consists of the points in P N of X-border rank r. Taking the Zariski closure often causes problems. If p = b a a + a b a + a a b, with a, b independent, Rank(p) = 3 but Brank(p) = 2! Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

3 Secant varieties and tensors Let A = {a i }, B = {b j }, C = {c k }, be C-vector spaces, then the tensor product A B C has basis elements of the form a i b j c k, with coordinates p ijk. Segre variety (rank 1 tensors): (Independence model) Defined by Seg : PA PB PC P ( A B C ) In coordinates: p i,j,k = u i v j w k. ([u], [v], [w]) [u v w]. The r th secant variety of a variety X P n : (Mixture model) σ r (X) := x 1,...,x r X P(span{x 1,..., x r }) P n. General points of σ r (Seg(PA PB PC)) have the form [ r s=1 us v s w s ], or in coordinates: p i,j,k = r s=1 us i vs j ws k. (*Might also work over R or -probability simplex, but not today.) Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

4 Some Applications of Secant Varieties Classical Algebraic Geometry: When can a given projective variety X P n be isomorphically projected into P n 1? Determined by the dimension of the secant variety σ 2 (X). Algebraic Complexity Theory: Bound the border rank of algorithms via equations of secant varieties. Berkeley-Simons program Fall 14 Algebraic Statistics and Phylogenetics: Given contingency tables for DNA of several species, determine the correct statistical model for their evolution. Find invariants (equations) of mixture models (secant varieties). For star trees / bifurcating trees this is the salmon conjecture. Signal Processing: Blind identification of under-determined mixtures, analogous to CDMA technology for cell phones. A given signal is the sum of many signals, one for each user. Decompose the signal uniquely to recover each user s signal. Computer Vision, Neuroscience, Quantum Information Theory, Chemistry... Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

5 First questions for secant varieties Given X P N we ask: 1 [Dimensions] What is the dimension of σ r (X)? When does σ r (X) fill the ambient P N? (defectivity) 2 [Equations] What are the polynomial defining equations of σ r (X)? 3 [Generic Identifiability] For generic x P N, does x have a unique expression as a sum of points from X? (ignoring trivialities) 4 [Decomposition] For my favorite x P N, can you find an expression of x as a sum of points from X? Sometimes Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

6 Tensors with different types of symmetry Suppose A, B, C are vector spaces over C. Hypermatrices, symmetric, partially symmetric, skew-symmetric, and partially skew-symmetric tensors: Space Hypermatrix Symmetry A B C (T i,j,k ) S d A (T i1,...,i d ) T i1,...,i d = T σ(i1 ),...,σ(i d ) for all σ S d A S 2 B (T i,j,k ) T i,j,k = T i,k,j for all i, j, k k+1 A (T i,...,i k ) T i,...,i k = sgn(σ)t σ(i ),...,σ(i k ) for all σ S k+1 A k+1 B (T i,j,...,j k ) T i,j,...,j k = sgn(σ)t i,σ(j ),...,σ(j k ) for all σ S k+1 Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

7 Partially skew-symmetric tensors (in coordinates) Choose bases u 1,..., u m of A = C m and v 1,..., v n of B = C n. In coordinates T A 2 B is T = i,j,k T ijk u i v j v k with symmetry: T ijk = T ikj. Collect terms: T = u i i j,k T ijk v j v k = i u i T i with T i 2 B, T is a collection of m skew-symmetric n n matrices. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

8 Tensors and examples of classical algebraic varieties Consider each symmetry type and the (classical) variety of rank-1 tensors: Name Notation Points Ambient Space Segre Seg (PA PB PC) [a b c] P(A B C) Veronese ν d (PA) [a a] P(S d A) Segre-Veronese Seg (PA ν 2(PV )) [a b b] P(A S 2 B) ( k+1 ) Grassmannian G(k, A) [a a k ] P A Segre-Grassmann Seg(PA G(k, B)) [a (b b k )] P(A k+1 B) In each case a rank 1 tensor can be put in the form (in an appropriate basis): 1 Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

9 How to the Segre-Grassmann Seg(PA G(k, B)) KK = QQ; makerank1 = (m,k,n)->( E = random(kk^(m+1),kk^(n+1)); e = gens minors(k+1,e); v = random(kk^(m+1),kk^1); matrix {flatten entries(v*e)} ) makerank1(2,2,5) -- a point on Seg(P^2 x G(2,5)) sum(5,i->makerank1(2,2,5)) -- a point on sig_5(seg(p^2 x G(2,5))) (In practice I m more careful to name the coordinates and avoid collisions.) Let v = (v,..., v m ), and let E = (e i,j ) be a (k + 1) (n + 1) matrix. Get a (m + 1) ( n+1 k+1) vector for a point on Seg(P m G(k, n)) as (v i I (E)) i,i, where I maximal minor of E with columns I = (i 1,..., i k+1 ). Pseudo-random points on σ r (Seg(P m G(k, n))): let v and E have random entries, and summing r pseudorandom points of Seg(P m G(k, n)). Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

10 What is a flattening? Express a tensor T = i,j,k p ijk a i b j c k A B C as a matrix (in 3 ways): T = i T = j T = k a i p ijk b j c k A (B C), j,k b j p ijk a i c k i,k B (A C), p ijk a i b j c k i,j (A B) C. Example: T = [p ijk ] C 3 C 3 C 3 to C 3 (C 3 C 3 ) = C 3 C 9 flattens to: p 111 p 121 p 131 p 112 p 122 p 132 p 113 p 123 p 133 ψ,t = p 211 p 221 p 231 p 212 p 222 p 232 p 213 p 223 p 233 p 311 p 321 p 331 p 312 p 322 p 332 p 313 p 323 p 333 When they exist, (r + 1) (r + 1) minors of ψ,t are (some) equations of σ r (PA PB PC). Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

11 Equations from flattenings Realize T A B C = A (B C) as a linear map (a matrix) Rank 1 case: Let T 1 = ψ,t : A ( 1 ψ,t =... ), T i = ( T1 T2... T l ) t B C ( ), i = 2... m ( ) t If Rank(T ) = 1, then Rank(ψ,T ) = 1. Construction is linear in T : T 1 T 2. T l ψ,t + ψ,t = ψ,t +T + T 1 T 2. T l = T 1+T 1 T 2+T 2. T l +T l Rank(ψ,T ) + Rank(ψ,T ) Rank(ψ,T + ψ,t ) = ψ,t +T By subadditivity of rank, if Rank(T ) = r, then Rank(ψ,T ) r. The (r + 1) (r + 1) minors of ψ,t are necessary conditions for Brank(T ) r. However, flattenings are trivial when Rank(T ) min{dim A, dim B, dim C}. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

12 Symmetric Flattenings Consider φ Sym d V as a symmetric multilinear form: φ eats d vectors (symmetrically) and spits out a number. If we only feed s vectors to φ, it still wants to eat d s more. So we can construct a linear map φ s,d s : Sym s (V ) Sym d s V [v 1,..., v s ] φ(v 1,..., v s,,..., ) Macaulay (1916) showed that Brank φ Rank φ s,d s for all 1 s d. The minors of φ s,d s are called minors of Catalecticants. (Also called symmetric flattenings or special Hankel matrices). Give some equations for the secant varieties to Veronese varieties. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

13 Flattenings and the Segre variety Classical: the ideal of any Segre is generated by all 2 2 minors of flattenings. Theorem (Raicu (212), (Garcia, Stillman and Sturmfels Conj.)) The prime ideal of σ 2 (Seg(PV 1 PV n )) is generated by the 3 3 minors of flattenings. Built on work of Landsberg-Manivel, Landsberg-Weyman, Geramita et al., Allman-Rhodes. Raicu also proved the stronger analogous result for the secant variety of any Segre-Veronese. [Michalek-O.-Zwiernik (214)] gave a toric proof of the scheme-theoretic version that works in any characteristic using secant cumulants. Flattenings run out quickly: σ 3 (Seg(P 2 P 2 P 2 )) has no equations from flattenings since there are no 4 4 minors of 3 9, ([Strassen 83, Y. Qi 14].) Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

14 Exterior flattenings Suppose T A B C. Have a natural inclusion A 2 A A. Construct a new linear map via the inclusion A B C 2 ( 2 ) A A B C = (A B) A C. Fix dim(a) = 3. With T = i u i T i, we choose a good basis and write Basic idea: ( T3 T 2 T 3 T 1 T 2 T 1 ψ 1,T : A B 2 A C. ψ 1,T +T = ψ T + ψ T construction is linear in T Rank(T ) = 1 Rank(ψ T ) = 2 Rank(T ) = r Rank(ψ T ) 2r ) base case subadditivity of matrix rank The (2r + 1) (2r + 1) minors of ψ 1,T are necessary conditions for Brank(T ) r. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

15 Exterior Flattenings: Rank(T ) = 1 Rank ψ 1,T = 2 ( 1 (... Let T 1 =... ), T i =... ), i = 2..m. ( ) ( ) ( ). T3 T ( ) ( )... 1 ψ 1,T = T 3 T 1 = T 2 T 1...,... ( ) ( ) ,.. Result is invariant under natural changes of coordinates. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

16 Exterior flattenings (partially symmetric case) Have natural inclusions A 2 A A, S 2 B B B. Construct a new linear map via the inclusion A S 2 B (B A ) Fix m = 3. With T = i u i T i, we choose a good basis and write ( T3 T 2 T 3 T 1 T 2 T 1 ψ 1,T : B A B 2 A. Note T i S 2 B ψ 1,T is skew-symmetric rank is even. Construction is linear in T : ) ( B 2 ) A. ( ) T3 T 2 T 3 T 1 + T 2 T 1 ψ 1,T + ψ 1,T = ψ 1,T +T ) = ( T 3 T 2 T 3 T 1 T 2 T 1 ( T3+T 3 T2 T 2 T 3 T 3 T 1+T 1 T 2+T 2 T1 T 1 ) 2(r + 1) 2(r + 1) Pfaffians of ψ 1,T are necessary conditions for Brank(T ) r. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

17 Exterior flattenings more generally Include A B C (B j ) ( A C ) j+1 A. Example dim(a) = l = 4: Let T = 4 i=1 u i T i. In suitable coordinates: ( T1 T2 T3 T4 ) t ψ,t : B A C 1 A, T 3 T 2 T 3 T 1 T 2 T 1 T 4 T 1 T 4 T 2 T 4 T 3 ψ 1,T : B 1 A C 2 A, T 4 T 3 T 2 T 4 T 3 T 1 T 4 T 2 T 1 T 1 T 2 T 3 ψ 2,T : B 2 A C 3 A ψ 3,T : B 3 A ( T1 T2 T3 T4 ) C 4 A. In general one finds Rank(T ) r Rank(ψ j,t ) r ( ) l 1 j. We want to know when the necessary conditions are also sufficient. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

18 Subspace varieties Definition The subspace variety Sub p,q,r (A B C) is the variety of tensors [T ] P(A B C) such that there exist subspaces C p A, C q B, C r C, and [T ] P(C p C q C r ). Theorem (Thm. 3.1, Landsberg Weyman 7) Sub p,q,r (A B C) is normal with rational singularities. Its ideal is generated by the minors of flattenings; ( p+1 A ( p+1 (B C )) q+1 B ) q+1 (A C ) ( r+1 (A B ) r+1 C ) Key Point: Sub r,r,r σ r (PA PB PC) and therefore get some (determinantal) equations of the secant varieties. Note: C. Raicu has recently proved that for any Segre-Veronese variety X, and r 2, the ideal I(σ 2 (X)) is generated by 3 3 minors of flattenings. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

19 Partially symmetric subspace and secant varieties Definition The subspace variety Sub p,q (A S 2 B) is the variety of tensors [T ] P(A S 2 B) such that there exist subspaces C p A, C q B, and [T ] P(C p S 2 C q ). T Sub p,q (A S 2 B) implies that after changing coordinates, ψ,t = ( T 1 T 2 T m ) t, with T i = ( B i ) and Bi a symmetric q q matrix. Proposition (Cartwright-Erman-O. (212)) The defining ideal of Sub p,q (A S 2 B) is generated by the (p + 1) (p + 1) minors of the flattening B A B and the (q + 1) (q + 1) minors of the flattening A S 2 B. Theorem (Cartwright-Erman-O. (212) ) For r 5, the ideal of σ r (Seg(P 2 ν 2 (PB))) is generated by (r + 1) (r + 1) minors of flattenings, and (2r + 2) (2r + 2) Pfaffians of exterior flattenings. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

20 Skew-symmetric subspace varieties Definition The subspace variety Sub p ( ( k+1 k+1 ) A) is the variety of tensors [T ] P A ( k+1 ) such that there exist a subspace C p A, and [T ] P C p. The equations of this variety are mysterious when k 2: Proposition (Boralevi-O.(212) 1 ) The ideal of the subspace variety Sub 5 ( 3 C 7 ) P 34 is generated in degree 3 by the GL 7 -modules S (3,1,1,1,1,1,1) C 7 (28 cubics) and S (2,2,2,1,1,1) C 7 (224 cubics). The space of 28 cubics is vector space isomorphic to quadrics on 7 variables. The space of 224 cubics is inherited from a space of 2 cubics on 6 variables, which is vector space isomorphic the span of the 3 3 minors of a 3 6 matrix. Used Weyman s Geometric Technique, Representation Theory, Bott s algorithm, and careful combinatorial book-keeping. 1 Appeared in [J.M. Landsberg, Tensors: Geometry and Applications, (p45-47), AMS GSM, vol. 128, (212)]. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

21 Equations of secant varieties via more general flattenings Symmetric: Aronhold s invariant (1849): σ 3 (v 3 (P 2 )) P 9. Partially symmetric: Toeplitz (1877): σ 5 ( P 2 ν 2 (P 3 ) ) P 29. Cartwright-Erman-O. (212): σ r ( P 2 ν 2 (P n ) ) P 3(n+2 2 ) 1, r 5. Partially skew-symmetric: Abo-Wan (213): σ 3l (Seg(P 2 G(1, 4l + 2))) P 3(4l+3 2 ) 1. Unrestricted: Strassen (1983): σ 4 ( P 2 P 2 P 2 ) ) P 26 Ubiquitous: Ottaviani (27) united and generalized all of these equations with a uniform construction we call exterior flattenings. Landsberg-Ottaviani (212): Many more cases, much more general. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

22 Abo-Wan Hypersurfaces: σ 3l+2 (Seg(P 2 G(1, 4l + 2))) Consider σ 5 (P 2 G(1, 6)) (PC 3 2 ) C 7 = P(V 2 W ). View T V 2 W ) as an element in 2 V 2 W, which induces ϕ T : V W V W, Explicitly ϕ T is the (21 21) Kronecker product of two matrices: e 12 e 13 e 14 e 15 e 16 e 17 e 12 e 23 e 24 e 25 e 26 e 27 v 1 v 2 e ϕ T = v 1 v 3 13 e 23 e 34 e 35 e 36 e 37 e 14 e 24 e 34 e 45 e 46 e 47 v 2 v 3 e 15 e 25 e 35 e 45 e 56 e 57 e 16 e 26 e 36 e 46 e 56 e 67 e 17 e 27 e 37 e 47 e 57 e 67 Replace v i e jk with x ijk we obtain the matrix ϕ T = (too big for screen). Check: Rank(T ) = 1 Rank ϕ T = 4 so Rank(T ) = r Rank ϕ T 4r. In particular if Rank T = 5 then Rank ϕ T 2, so det ϕ T vanishes. Check: Rank(ϕ T ) = 21 for random T, so det ϕ T is non-trivial. det ϕ T is an equation of σ 5 (Seg(P 2 G(1, 6))). (Verified in Macaulay 2.) Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

23 How to an exterior flattening KK = ZZ/11 T = KK[v1, v2, v3]**kk[w12, w13, w14, w15, w16, w17, w23, w24, w25, Mv = matrix {{, v1, -v2}, {-v1,, v3}, {v2, -v3,}} Mw = matrix {{, w12, w13, w14, w15, w16, w17}, {-w12,, w23, w24, w25, w26, w27}, {-w13, -w23,, w34, w35, w36, w37}, {-w14, -w24, -w34,, w45, w46, w47}, {-w15, -w25, -w35, -w45,, w56, w57}, {-w16, -w26, -w36, -w46, -w56,, w67}, {-w17, -w27, -w37, -w47, -w57, -w67, } K = Mw**Mv for i from 1 to 6 do print(i, rank diff(k, sum(i, j-> makerank1()))) R = KK[x112, x212, x312, x113, x213, x313, x114, x214, x314, x115, x P = T**R vwtox = v1*w12*x112 + v1*w13*x113 + v1*w14*x114 + v1*w15*x115 + v1*w mymat = diff(sub(k,p),vwtox) mypoly = det(mymat, Strategy => Cofactor); Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

24 Kronecker product x 112 x 212 x 113 x 213 x 114 x 214 x 115 x 215 x 112 x 312 x 113 x 313 x 114 x 314 x 115 x 315 x 212 x 312 x 213 x 313 x 214 x 314 x 215 x 315 x 112 x 212 x 123 x 223 x 124 x 224 x 125 x 225 x 112 x 312 x 123 x 323 x 124 x 324 x 125 x 325 x 212 x 312 x 223 x 323 x 224 x 324 x 225 x 325 x 113 x 213 x 123 x 223 x 134 x 234 x 135 x 235 x 113 x 313 x 123 x 323 x 134 x 334 x 135 x 335 x 213 x 313 x 223 x 323 x 234 x 334 x 235 x 335 x 114 x 214 x 124 x 224 x 134 x 234 x 145 x 245 x 114 x 314 x 124 x 324 x 134 x 334 x 145 x 345 x 214 x 314 x 224 x 324 x 234 x 334 x 245 x 345 x 115 x 215 x 125 x 225 x 135 x 235 x 145 x 245 x 115 x 315 x 125 x 325 x 135 x 335 x 145 x 345 x 215 x 315 x 225 x 325 x 235 x 335 x 245 x 345 x 116 x 216 x 126 x 226 x 136 x 236 x 146 x 246 x 156 x 256 x 116 x 316 x 126 x 326 x 136 x 336 x 146 x 346 x 156 x 356 x 216 x 316 x 226 x 326 x 236 x 336 x 246 x 346 x 256 x 356 x 117 x 217 x 127 x 227 x 137 x 237 x 147 x 247 x 157 x 257 x 117 x 317 x 127 x 327 x 137 x 337 x 147 x 347 x 157 x 357 x 217 x 317 x 227 x 327 x 237 x 337 x 247 x 347 x 257 x 357 Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

25 Bertini team [N. Daleo & J. Hauenstein] to the rescue! How do we know that the determinant det ϕ T is irreducible (and hence minimally defines the ideal of σ 5 (Seg(P 2 G(1, 6))))? A computation in Bertini provides missing ingredient: Proposition The hypersurface σ 5 (Seg(P 2 G(1, 6))) P 62 has degree 21. We have an irreducible hypersurface of degree 21 inside of the zero-locus of a degree 21 equation! Proposition The hypersurface σ 8 (Seg(P 2 G(1, 1))) P 164 has degree 33. A similar exterior flattening provides the equation. Proposition The hypersurface σ 5 (Seg(P 2 G(2, 5))) P 59 has degree 6. No 6 6 flattening... So now what? Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

26 Find equations in the ideal Consider σ 5 (Seg(P 2 G(2, 5))) P 59. Look for equations in R = C[x,..., x 59]. The Hilbert function of R starts like this: d = HF R (d) = HF R/I (d) = dim(i d ) = 1... To find the space of linear forms in the ideal, compute: M = matrix apply(6,i-> {makerank5(2,2,5)} ); rank M Naively, to find the space of sextics in the ideal: compute points on the variety, evaluate them on the monomials of degree 6 and compute the kernel of the resulting matrix. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

27 Symmetry The symmetry group of σ 5 (Seg(P 2 G(2, 5))) P 59 = P is change of coordinates in each factor, GL(A) GL(B) ( A 2 ) B A large group acts so we can use tools from Representation Theory! This symmetry is a powerful tool and we should exploit it! Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

28 Find equations in the ideal using representation theory Module notation: S (A d 2 ) B = C[p ijk p ijk = p ikj ] d. Fact: S (A d 2 ) B is a GL(A) GL(B)-module. The coordinate ring C[A 2 B] has an isotypic decomposition: d C[A 2 B] d = ( ) d λ d,π 2d (S λa S πb) C m λ,π ( ) d I d(x) = λ d,π d d I d(x) λ,π Sλ A S πb: Schur modules indexed by partitions λ and π, C m λ,π : multiplicity space. Given λ, π and the multiplicity m λ,π, there is a combinatorial algorithm for constructing polynomials! ( Modified version of Landsberg-Manivel 4 algorithm) Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

29 Invariants via Young Symmetrizers Use LiE: sym_tensor(6,[1,]^[,,1,,],a2a5) only one SL(3) SL(6) invariant of degree 6 in C[A 2 B]. Start with partitions (2, 2, 2) and (3, 3, 3, 3, 3, 3) associated (respectively) to the trivial representations of GL(3) and GL(6) in degrees 6 and 18 respectively. Find fillings: a c b e d f a b c a b d a d e b d f c e f c e f Use fillings as input to the Young Symmetrizer algorithm. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

30 Invariants via Young Symmetrizers Construct a generic polynomial (in auxiliary variables): 1 To the filling 2 To the filling a c b e d f associate p V = a b c a b d a d e b d f c e f c e f associate p W = a 11 a 12 a 13 a 14 a 15 a 16 a 21 a 22 a 23 a 24 a 25 a 26 a 31 a 32 a 33 a 34 a 35 a 36 b 31 b 32 b 33 b 34 b 35 b 36 c 21 c 22 c 23 c 24 c 25 c 26 c 31 c 32 c 33 c 34 c 35 c 36 a1 a2 a3 b 1 b 2 b c 1 c 2 c 3 3 e 1 e 2 e 3 d 1 d 2 d 3 f 1 f 2 f 3 b 11 b 12 b 13 b 14 b 15 b 16 b 21 b 22 b 23 b 24 b 25 b 26 d 21 d 22 d 23 d 24 d 25 d 26 d 31 d 32 d 33 d 34 d 35 d 36 e 21 e 22 e 23 e 24 e 25 e 26 e 31 e 32 e 33 e 34 e 35 e 36 c 11 c 12 c 13 c 14 c 15 c 16 d 11 d 12 d 13 d 14 d 15 d 16 e 11 e 12 e 13 e 14 e 15 e 16 f 11 f 12 f 13 f 14 f 15 f 16 f 21 f 22 f 23 f 24 f 25 f 26 f 31 f 32 f 33 f 34 f 35 f 36 3 Extract the terms p V p W : replace a i (a 1j a 2k a 3l ) with x i,j,k,l 4 Repeat previous step for b, c,..., f. The resulting polynomial will be a polynomial in the image of the Young Symmetrizer associated to our initial fillings. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34.

31 Solution to Problem 6.5 in [Abo-Wan 212] Theorem (Abo-Daleo-Hauenstein-O.) The hypersurface σ 5 (P 2 G(2, 5)) P 59 is defined by the image of the Young symmetrizer produced by the recipe given by the filling a c b e d f a b c a b d a d e b d f c e f c e f. The resulting polynomial has precisely 18 monomials, 54 of which have coefficient +1 and 54 of which have coefficient 1. Download from ancillary files associated to the arxiv version of our paper! Theorem not Theorem because there is no lower degree invariant equation. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

32 An Ottaviani-type expression? We suppose that this equation may have an expression as a root of a determinant of a special matrix, similar to Ottaviani s degree 15 equation in [Ottaviani 9], however our initial attempts at finding such an expression were unsuccessful. A natural guess is to start from T A 3 B and produce the matrix A T : (B B) (A B), which as rank 3 when T has rank 1 and rank 3r when T has rank r. However, this map actually factors through a map 2 B (A B) but this matrix is 18 15, and it s maximum rank is 15. This means that this construction cannot distinguish rank 5 tensors from rank 6 tensors. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

33 Numerical Algebraic Geometry: Bertini Let H be an irreducible hypersurface and L be a line so that deg H = H L. 1 Generate a point x H L. Initialize W := {x}. 2 Perform a random monodromy loop starting at the points in W: (a) Pick a random loop M(t) in the space of lines so that M() = M(1) = L. (b) Trace the curves H M(t) starting at the points in W at t = to compute the endpoints E at t = 1. (Hence, E H L). (c) Update W := W E. 3 Repeat (2) until the trace test verifies that W = H L. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34

34 Proposition The hypersurface σ 5 (Seg(P 2 G(2, 5))) P 59 has degree 6. In our execution of the procedure for the hypersurface H = σ 5 (Seg(P 2 G(2, 5))), it took 6 random monodromy loops to compute the six points in H L. The total procedure lasted 5 seconds using a single 2.3 GHz core of an AMD Opteron 6376 processor. Proposition The hypersurface σ 5 (Seg(P 2 G(1, 6))) P 62 has degree 21. Proposition The hypersurface σ 8 (Seg(P 2 G(1, 1))) P 164 has degree 33. In our execution, it took 13 and 12 random monodromy loops to yield the degree many points for these cases, respectively. Using a total of sixteen 2.3 GHz cores, the total procedure lasted 2.5 and 32 minutes, respectively. Oeding (Auburn, NIMS) Secants, Equations and Applications August 9, / 34