Secant Varieties, Symbolic Powers, Statistical Models

Transcription

1 Secant Varieties, Symbolic Powers, Statistical Models Seth Sullivant North Carolina State University November 19, 2012 Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

2 Joins and Secant Varieties Given V, W P n 1, the join is V W := v, w P n 1 v V,w W W V w v Given V P n 1, the r-th secant variety of V is given by Sec 1 (V ) = V, and Sec r (V ) := V Sec r 1 (V ) = v 1,...,v r V < v 1, v 2,..., v r > P n 1 Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

3 Joins and Secant Ideals Definition Let I, J K[x 1,..., x n ] homogeneous ideals. The join of I and J is I J := (I(y) + J(z) + x i y i z i : i = 1,..., n ) K[x 1,..., x n ]. i.e. f I J f (y + z) I(y) + J(z). Definition The r-th secant ideal I K[x 1,..., x n ] is defined by 1 I {1} = I 2 I {r} = I I {r 1}, r > 1. Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

4 Matrices of Bounded Rank Consider K[X] := K[x ij : 1 i m, 1 j n] Let I be the ideal of 2 2 minors of generic matrix x 11 x 12 x 1n X = x m1 x m2 x mn V (I) P mn 1 is the set of rank 1 matrices. (aka Seg(P m 1 P n 1 )) Sec r (V (I)) is the set of all matrices of rank r. I {r} = I(Sec r (V (I))) is the ideal of (r + 1) (r + 1) minors of X. Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

5 When is the Secant Ideal Nice? Question For which ideals I does I {2} = 0? i.e. When is V (I) {2} = P n 1? Theorem (Zak 1994) If V (I) is a linearly nondegenerate, smooth irreducible variety of codimension 1 3 n + 1, then I{2} = 0 unless V (I) is one of ν 2 (P 2 ), Seg(P 2 P 2 ), Gr(2, 6), or the Cartan variety. Question For which ideals I is I {r} generated by polynomials of degree r + 1? Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

6 Symbolic Powers Definition The rth symbolic power of I R is the ideal I (r) = (R 1 I I r ) R where R I is the complement of the union of the minimal primes of I. If P is a prime ideal, P (r) is the P-primary component of P r. Theorem ((Special case of) Zariski-Nagata) If I C[x 1,..., x n ] is radical then I (r) = I <r> := f a f x a I for all a Nn with n a i r 1. i=1 Symbolic power gives equations vanishing to high order on V (I). Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

7 Ideal of 2 2 minors Consider K[X] := K[x ij : 1 i m, 1 j n] Let I be the ideal of 2 2 minors of generic matrix X. The partial derivatives of minors of a generic matrix are themselves minors of a generic matrix. e.g. x 11 x 12 x 13 x x 21 x 22 x 23 = x 22 x x 31 x 32 x 33 x 32 x 33 I (2) generated by 3 3 minors and products of minors of X. Note, if m = 2, then I (2) = I 2. Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

8 When are the Symbolic Powers Nice? I Definition A radical ideal I is normally torsion free if I (r) = I r for all r 1. Complete intersections, Maximal minors Theorem (Ein-Lazarsfeld-Smith, Hochster-Huneke) I (codim(i)r) I r for all r in an equal characteristic regular ring. Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

9 Secant Ideals and Symbolic Powers Proposition For I C[x 1,..., x n ], I <r> = I x 1,..., x n r. Proposition (Catalano-Johnson) If I C[x 1,..., x n ], homogeneous and I x 1,..., x n 2, then Proof. I {r} I <r>. I {r} = I I {r 1} I x 1,..., x n r = I <r>. Proposition (Landsberg-Manivel, Sidman-Sullivant) If I C[x 1,..., x n ], homogeneous and I x 1,..., x n d, then I <(d 1)(r 1)+1> (d 1)r+1 = I {r} (d 1)r+1 Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

10 When are the Symbolic Powers nice? II Catalano-Johnson says I {r} I <r>. Product rule gives I <r i> I <i> I <r>, i = 1,..., r 1. Putting together gives I {λ 1} I {λ l } I <r> λ r where the sum is over all partitions of r. Definition An ideal I C[x 1,..., x n ] is differentially perfect if and only if I <r> = λ r I {λ 1} I {λ l } for all r 1. Normally torsion free ideal, determinantal and Pfaffian ideals Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

11 Secants and Symbolic Powers of Monomial Ideals Definition Let G be an undirected graph, with vertex set [n] = {1, 2,..., n}. The edge ideal of G is I(G) = x i x j : ij E(G) I(G) = x 1 x 2, x 2 x 3, x 3 x 4, x 4 x 5, x 5 x 6, x 1 x 6, x 2 x 4 5 Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

12 Secants of Edge Ideals and Colorings Given a graph G and subset of vertices V, G V is the induced subgraph of G with vertex set V. The chromatic number χ(g) is the smallest number of colors to properly color the vertices of a graph. Theorem (Sturmfels-Sullivant) The secant ideals of an edge ideal I(G) are generated by I(G) {r} = i V x i : χ(g V ) > r. Minimal generators of I(G) {r} are minimal obstructions to r-coloring G I(G) {2} = x 2 x 3 x 4, x 1 x 2 x 4 x 5 x 6 5 Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

13 Nice Secant Ideals of Edge Ideals Proposition I(G) {2} = 0 if and only if G is a bipartite graph. Proposition I(G) {r} is generated in degree r + 1 for all r if and only if G is a perfect graph Not perfect I(G) {2} = x 2 x 3 x 4, x 1 x 2 x 4 x 5 x 6 5 Perfect I(G) {2} = x 1 x 2 x 5, x 1 x 5 x 6, x 2 x 3 x 4, x 2 x 4 x 5 Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

14 Symbolic Powers of Edge Ideals and Coverings The covering number τ(g) is the size of the smallest vertex cover of G. 5 a = (1, 2, 1, 1, 0, 0) For a vector a N n the parallelization G a is the graph with each vertex i replicated a i times. Theorem I(G) <r> = x a : τ(g a ) r Generators of symbolic powers are minimal obstructions to covering. Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

15 Nice Symbolic Powers of Edge Ideals Theorem (Simis-Vasconcelos-Villarreal) I(G) <r> = I(G) r for all r if and only if G is a bipartite graph. Theorem (Villarreal, Sullivant) I(G) is differentially perfect, i.e. I(G) <r> = I(G) {λ 1} I(G) {λ l } λ r for all r if and only if G is a perfect graph. Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

16 Statistical Models Definition A statistical model is a set of probability distributions or densities. Independence model for two discrete random variables: M X Y = P = (p ij) R m n : p ij = 1, p ij 0, rank P = 1 i,j (Zariski closure of ) many statistical models have structure of algebraic varieties Many of those varieties are secant varieties or joins Mixture models Factor analysis Can use that algebraic structure to answer statistical questions. Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

17 Join vs. Mixture Join W V w v Mixture W V w v Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

18 Identifiability of Phylogenetic Mixture Models Theorem (Rhodes-Sullivant) The unrooted tree and numerical parameters in a r-class, same tree phylogenetic mixture model on n-leaf trivalent trees are generically identifiable, if r < 4 n/4. Identifiability for Secant Varieties: Does a generic point on the r th secant variety Sec r (V ) lie on a unique secant r 1 plane to V? Proof uses: Relation to identifiability problem for Sec 4r (Seg(P a P b P c )) Knowledge of generators of the secant ideals I {r} T Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

19 Factor Analysis A centered Gaussian random vector X R n has density function ( 1 f (x) = (2π) n/2 exp 1 ) Σ 1/2 2 x T Σ 1 x with Σ PD n R n(n+1)/2. Set of all centered Gaussian r.v. s parametrized by PD n. A Gaussian statistical model is a subset Θ PD n. Definition (Factor Analysis Model) Θ r,n = {Ψ + ΛΛ T Ψ > 0 diagonal, Λ R r n } PD n Diagonal plus rank r PSD matrix. Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

20 Factor Analysis Model assumptions: X 1,..., X n are observable covariates. Y 1,..., Y r are fewer unobservable (hidden) factors, which explain the correlation among X 1,..., X n. For all i, j, X i X j Y 1,..., Y r For all i, j Y i Y j Y 1 Y 2 Example (g-theory) X i are test scores, and Y j are underlying types of intelligence. X 1 X 2 X 3 X 4 X 5 Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

21 Model Selection: How Many Factors? What is r? Statistical Approach Mathematical Problem Wald-Type Test Find generators of I(Θ r,n ) Likelihood Ratio Test Compute tangent cone TC p (Θ r,n ) at p Θ r 1,n. Information Criteria (AIC, BIC, WAIC) Determine the Real Log-Canonical Threshold of singular fibers of Θ r,n. Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

22 The Second Hypersimplex Observation Θ r,n = Sec r (Θ 1,n ) Θ 1,n is a toric variety called the second hypersimplex. φ : P 2n 1 P n(n+1)/2 1 Definition { λi λ φ ij (ψ, λ) = j if i j ψ i + λ 2 i if i = j I 1,n = I(im φ) toric ideal of second hypersimplex I r,n = I {r} 1,n = vanishing ideal of r-factor model Quadratic Gröbner basis of I 1,n computed by De Loera-Sturmfels-Thomas Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

23 Theorem (Sullivant) The secant ideal I 2,n = I {2} 1,n has a Gröbner basis, with squarefree initial terms, consisting of polynomials of all odd degrees between 3 and n, with respect to a circular term order. Proof idea: Degenerative strategy: Hope that in (I {2} 1,n ) = (in (I 1,n )) {2} Lemma (Simis-Ulrich) For any I and any term order in (I {r} ) (in (I)) {r} Just need to construct polynomials in I {r} whose initial terms generate (in (I)) {r}. Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

24 Initial Ideal Structure and Symbolic Powers The initial ideal in (I 1,n ) is an edge ideal. The graph G n such that in (I 1,n ) = I(G n ) is the non-crossing pair graph of the cyclically embedded complete graph K n. [De Loera-Sturmfels-Thomas] in (I 1,n ) {2} is generated by the odd cycles in G n. Construct polynomials in I 2,n having those cycles as initial terms. σ 12 σ 15 σ 23 σ 34 σ 45 σ 12 σ 13 σ 25 σ 34 σ 45 σ 12 σ 14 σ 23 σ 35 σ 45 +σ 12 σ 14 σ 25 σ 34 σ 35 +σ 12 σ 13 σ 24 σ 35 σ 45 σ 12 σ 15 σ 24 σ 34 σ 35 +σ 13 σ 14 σ 23 σ 25 σ 45 σ 13 σ 14 σ 24 σ 25 σ 35 σ 13 σ 15 σ 23 σ 24 σ 45 +σ 13 σ 15 σ 24 σ 25 σ 34 σ 14 σ 15 σ 23 σ 25 σ 34 +σ 14 σ 15 σ 23 σ 24 σ 35 Prove that these generalized pentads are in I {2} 1,n using symbolic powers. Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

25 Summary Secant varieties and symbolic powers are intimately related. Symbolic powers shed light on equations for secant varieties. In special cases, symbolic powers structure complete determined by secant ideals. Nice combinatorial and computational structure. Secant varieties make frequent appearance in statistics. Mixture models Factor analysis model Theoretical advances on secant varieties lead to advances in statistics. Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

26 References M. Catalano-Johnson. The homogeneous ideals of higher secant varieties. J. Pure Appl. Algebra 158 (2001), no. 2-3, J. de Loera, B. Sturmfels, R. Thomas. Gröbner bases and triangulations of the second hypersimplex. Combinatorica 15 (1995), no. 3, L. Ein, R. Lazarsfeld, K. Smith. Uniform bounds and symbolic powers on smooth varieties. Invent. Math. 144 (2001), no. 2, D. Eisenbud and M. Hochster. A Nullstellensatz with nilpotents and Zariski s main lemma on holomorphic functions. J. Algebra 58 (1979), no. 1, M. Haiman. Commutative algebra of n points in the plane. With an appendix by Ezra Miller. Math. Sci. Res. Inst. Publ., 51, Trends in commutative algebra, , Cambridge Univ. Press, Cambridge, M. Hochster, C. Huneke. Comparison of symbolic and ordinary powers of ideals. Invent. Math. 147 (2002), no. 2, J. Landsberg, L. Manivel. On the projective geometry of rational homogeneous varieties. Comment. Math. Helv. 78 (2003), no. 1, J. A. Rhodes and S. Sullivant, Identifiability of large phylogenetic mixture models. Bull. Math. Biol. 74 (2012), no. 1, J. Sidman and S. Sullivant. Prolongations and computational algebra. Canadian Journal of Mathematics 61 no. 4 (2009) A. Simis, B. Ulrich. On the ideal of an embedded join. J. Algebra 226 (2000), no. 1, Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27

27 A. Simis, W. Vasconcelos. R. Villarreal. On the ideal theory of graphs. J. Algebra 167 (1994), no. 2, B. Sturmfels and S. Sullivant. Combinatorial secant varieties. Quarterly Journal of Pure and Applied Mathematics 2 (2006) S. Sullivant Combinatorial symbolic powers. J. Algebra 319 (2008), no. 1, S. Sullivant. A Gröbner basis for the secant ideal of the second hypersimplex. Journal of Commutative Algebra 1 no.2 (2009) R. H. Villarreal, Rees algebras and polyhedral cones of ideals of vertex covers of perfect graphs, J. Algebraic Combin. 27 (2008), no. 3, F. L. Zak. Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, 127, AMS Press, Seth Sullivant (NCSU) Secant Varieties, etc. November 19, / 27