Toric Fiber Products Seth Sullivant North Carolina State University June 8, 2011 Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 1 / 26
Families of Ideals Parametrized by Graphs Let G be a finite graph Let R G a polynomial ring associated to G Let I G ⊆ R G an ideal associated to G Problem Classify the graphs G such that I G satisfies some “nice” property. Often I G := ker φ G for some ring homomorphism φ G . Problem Determine generators for I G . How do they depend on the graph G ? Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 2 / 26
Graph Decompositions Question Are decompositions of the graphs G = G 1 # G 2 reflected in the ideals I G = I G 1 # I G 2 ? How to study ideals of large graphs by breaking into simple pieces? + = = + Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 3 / 26
Toric Fiber Product Let A = { a 1 , . . . , a r } ⊂ Z d . Let K [ x ] := K [ x i j : i ∈ [ r ] , j ∈ [ s ]] , with deg x i j = a i . Let K [ y ] := K [ y i k : i ∈ [ r ] , k ∈ [ t ]] , with deg y i k = a i . Let K [ z ] := K [ z i jk : i ∈ [ r ] , j ∈ [ s ] , k ∈ [ t ]] , with deg z i jk = a i . Let φ : K [ z ] → K [ x ] ⊗ K K [ y ] = K [ x , y ] defined by z i jk �→ x i j y i for all i , j , k . k Definition Let I ⊆ K [ x ] , J ⊆ K [ y ] ideals homogeneous w.r.t. grading by A . The toric fiber product of I and J is the ideal I × A J = φ − 1 ( I + J ) . Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 4 / 26
Two Important Examples Example (Coarse Grading) Let K [ x ] , K [ y ] have common grading deg x i = deg y j = 1 for all i , j . Then I ⊆ K [ x ] , J ⊆ K [ y ] homogeneous, are homogeneous in the standard/coarse grading. I × A J ⊆ K [ z ] is the ordinary Segre product ideal. Example (Fine Grading) Let K [ x 1 , . . . , x r ] , K [ y 1 , . . . , y r ] have common grading deg x i = deg y i = e i for all i . Then I ⊆ K [ x ] , J ⊆ K [ y ] homogeneous, are monomial ideals. I × A J = I ( z ) + J ( z ) ⊆ K [ z 1 , . . . , z r ] is the sum of monomial ideals. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 5 / 26
A More Complex Example Let I = � x kl 1 m 1 x kl 2 m 2 − x kl 1 m 2 x kl 2 m 1 : k ∈ [ r 1 ] , l 1 , l 2 ∈ [ r 2 ] , m 1 , m 2 ∈ [ r 3 ] � Let J = � y l 1 m 1 n y l 2 m 2 n − y l 1 m 2 n y l 2 m 1 n : , l 1 , l 2 ∈ [ r 2 ] , m 1 , m 2 ∈ [ r 3 ] , n ∈ [ r 4 ] � Let deg x klm = deg y lmn = e l ⊕ e m Define φ : K [ z klmn : k ∈ [ r 1 ] , . . . ] → K [ a kl , b km , c ln , d mn : k ∈ [ r 1 ] , . . . ] by z klmn �→ a kl b km c ln d mn Then I × A J = ker φ . Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 6 / 26
Where’s the “Fiber Product”? Suppose I , J are toric ideal I = I B , J = I C , B ∈ Z e 1 , C ∈ Z e 2 B = { b i C = { c i j : i ∈ [ r ] , j ∈ [ s ] } k : i ∈ [ r ] , k ∈ [ t ] } If I B homogeneous with respect to A , then there is a linear map π 1 : Z e 1 → Z d , π 1 ( b i j ) = a i . If I C homogeneous with respect to A , then there is a linear map π 2 : Z e 2 → Z d , π 2 ( c i k ) = a i . Then I × A J is a toric ideal, whose vector configuration is a fiber product B × A C . Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 7 / 26
Codimension-0 Toric Fiber Products Definition The codimension of a TFP is the codimension of the toric ideal I A . I × A J has codimension 0 iff A is linearly independent. Proposition Suppose A linearly independent. Let i ′ i ′ j n and m ′ = x m = x i 1 j 1 · · · x i n n ′ 1 · · · x 1 n ′ . j ′ j ′ If deg m = deg m ′ then n = n ′ and i 1 = i ′ 1 , . . . , i n = i ′ n . Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 8 / 26
Persistence of Normality and Generating Sets Theorem (Ohsugi, Michałlek, etc.) Let I ⊆ K [ x ] , and J ⊆ K [ y ] be toric ideals, and A linearly independent. Then K [ z ] / ( I × A J ) normal ⇔ K [ x ] / I and K [ y ] / J normal. If f ∈ K [ x ] , homogeneous w.r.t. A , write � c u x i 1 � c u z i 1 j u 1 · · · x i n j u 1 k 1 · · · z i n f = j un . Lift to j un k n . Theorem Let A linearly independent. Then I × A J generated by Lifts of generators of I and J 1 z i j 1 k 1 z i j 2 k 2 − z i j 1 k 2 z i “Obvious” quadrics 2 j 2 k 1 Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 9 / 26
GIT Quotients N A = { λ 1 a 1 + . . . + λ r a r : λ i ∈ N } R = K [ x ] / I is an N A graded ring. R = � a ∈ N A R a S = K [ y ] / J is an N A graded ring. S = � a ∈ N A S a Proposition Suppose A linearly independent. Then � K [ z ] / ( I × A J ) = R a ⊗ K S a . a ∈ N A If K = K , then Spec ( K [ z ] / ( I × A J )) ∼ = ( Spec ( K [ x ] / I ) × Spec ( K [ y ] / J )) // T , where T acts on Spec ( K [ x ] / I ) × Spec ( K [ y ] / J ) via t · ( x , y ) = ( t · x , t − 1 · y ) . Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 10 / 26
Higher Codimension Toric Fiber Products Not a GIT quotient No hope for general construction of generators When is normality preserved? Proposition Let K = K . Suppose that I = � i P i , and J = � j Q j are primary decompositions. Then � I × A J = ( P i × A Q j ) i , j is a primary decomposition of I × A J. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 11 / 26
Generators of Higher Codim TFPs Suppose I ∈ K [ x ] , J ∈ K [ y ] , are toric ideals. A = { a 1 , . . . , a r } , Let K [ w ] := K [ w 1 , . . . , w r ] . Let ψ xw : K [ x ] → K [ w ] , x i j �→ w i (similarly ψ yw ) Definition Let I = � x u − x v ∈ I : φ ( x u − x v ) = 0 � ˜ J = � y u − y v ∈ J : φ ( y u − y v ) = 0 � ˜ The ideal ˜ A ˜ I × ˜ J is the associated codimension 0 TFP. ˜ A ˜ I × ˜ J ⊆ I × A J ˜ A ˜ I × ˜ J is usually related (via graph theory) in a nice way to I × A J . Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 12 / 26
Gluing Generators Let i ′ 1 · · · x i ′ f = x i 1 j 1 · · · x i n j n − x 1 n ∈ I n j ′ j ′ and i ′ 1 · · · y i ′ g = y i 1 k 1 · · · y i n k n − y 1 n n ∈ J k ′ k ′ that is, φ xw ( f ) = φ yw ( g ) . Then i ′ 1 · · · z i ′ glue ( f , g ) = z i 1 j 1 k 1 · · · z i n j n k n − z 1 n n ∈ I × A J j ′ 1 k ′ j ′ n k ′ Question Two natural classes of generators of I × A J : when do they suffice? Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 13 / 26
Awesome Pictures that Explain Everything Answer Gluing and the associated codim 0 tfp always suffice to generate I × A J . But.... how to find the right binomials to glue? Projecting a fiber onto ker Z A . U 3 0 2 3 0 1 Projected and connected fibers = need not be compatible 0 3 Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 14 / 26
Summary of General Results on Toric Fiber Products Codim 0 TFPs Can Explicitly Describe Generators/ Gröbner bases from I and J Normality Preserved for Toric Ideals Geometric Interpretation as GIT Quotient Arbitrary Codim TFPs Primary Decompositions “Multiply” Can Explicitly Describe Generators given generators of I and J with special properties (toric case only) Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 15 / 26
Markov Bases Definition Let A : Z n → Z d be a linear transformation. A Markov Basis for A is a finite subset B ⊂ ker Z ( A ) such that for all u , v ∈ N n with A ( u ) = A ( v ) there is a sequence b 1 , . . . , b L ∈ B such that u = v + � L i = 1 b i , and 1 v + � l i = 1 b i ≥ 0 for l = 1 , . . . , L . 2 Markov bases allow us to take random walks over the set of nonnegative integral points inside of polyhedra. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 16 / 26
Example: 2-way tables Let A : Z k 1 × k 2 → Z k 1 + k 2 such that m m k k � � � � A ( u ) = u 1 j , . . . , u k 1 j ; u i 1 , . . . , u ik 2 j = 1 j = 1 i = 1 i = 1 = vector of row and column sums of u ker Z ( A ) = { u ∈ Z k 1 × k 2 : row and columns sums of u are 0 } � k 1 �� k 2 � Markov basis consists of the 2 moves like: 2 2 0 0 0 0 1 0 − 1 0 − 1 0 1 0 Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 17 / 26
Fundamental Theorem of Markov Bases Definition Let A : Z n → Z d . The toric ideal I A is the ideal � p u − p v : u , v ∈ N n , Au = Av � ⊂ K [ p 1 , . . . , p n ] , where p u = p u 1 1 p u 2 2 · · · p u n n . Theorem (Diaconis-Sturmfels 1998) The set of moves B ⊆ ker Z A is a Markov basis for A if and only if the set of binomials { p b + − p b − : b ∈ B} generates I A . 0 0 0 0 1 0 − 1 0 − → p 21 p 33 − p 23 p 31 − 1 0 1 0 Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 18 / 26
More Complicated Marginals Let Γ = { F 1 , . . . , F r } , each F i ⊆ { 1 , 2 , . . . , n } . Let d = ( d 1 , . . . , d n ) and u ∈ Z d 1 ×···× d n . ≥ 0 Let A Γ , d ( u ) = ( u | F 1 , . . . , u | F r ) , lower order marginals. 1 2 2-way margins of 4-way table: { 1 , 2 } , { 2 , 3 } , { 3 , 4 } , { 1 , 4 } -margins 3 4 Question How does the Markov basis of A Γ , d depend on Γ and d ? How do the generators of I Γ , d depend on Γ and d ? Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 19 / 26
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