GRAVER BASES, MATCHINGS IN SIMPLICIAL COMPLEXES AND TORIC - PDF document

GRAVER BASES, MATCHINGS IN SIMPLICIAL COMPLEXES AND TORIC VARIETIES Anargyros Katsabekis Apostolos Thoma 1

Let V = { v 1 , v 2 , . . . , v n } be a finite set. An abstract simplicial complex D on the vertex set V is a collection of sub- sets of V satisfying: (i) { v i } ∈ D for every i = 1 , . . . , n , (ii) if T ∈ D and G ⊂ T , then G ∈ D . A set T ⊂ D of cardinality m + 1 has dimension m ≥ − 1 and is called an m - simplex of D . The 0-simplices of D are called vertices , while the 1-simplices are called edges . The dimension dim ( D ) of D is the maximum of the dimensions of its simplices. 2

Let D be an abstract simplicial com- plex on the vertex set V and J be a subset of Ω := { 0 , 1 , . . . , dim ( D ) } . A set M = { T 1 , . . . , T s } of simplices of D is called a J - matching in D if T k ∩ T l = ∅ for every 1 ≤ k, l ≤ s and dim( T k ) ∈ J for every 1 ≤ k ≤ s . Set supp ( M ) = ∪ s i =1 T i ⊂ V . A J -matching M in D is called a max- imal J -matching if supp ( M ) has the maximum possible cardinality among all J -matchings. By δ ( D ) J we denote the minimum card ( M ) among all maximal J -matchings M in D . 3

When D is a simple graph, i.e dim( D ) ≤ 1, the notion of { 1 } -matching in D co- incides with the notion of matching. Also maximal { 1 } -matching coincides with the notion of maximal matching in D . Finally δ ( D ) { 1 } equals the matching num- ber of D . Recall that a subset M of the edges of D is called a matching in D if there are no two edges which are incident with a common vertex. M is a maximal matching if it has the maximum possible cardinality among all matchings. The cardinality of a maxi- mal matching in D is commonly known as its matching number . 4

Example. Consider the simplicial com- plex D drawn in the figure. The set {{ v 1 , v 2 } , { v 3 , v 4 } , { v 5 , v 6 } , { v 7 , v 8 } , { v 9 }} is a { 0 , 1 } -matching in D , while the set {{ v 1 , v 2 } , { v 3 , v 4 } , { v 5 , v 6 } , { v 7 , v 8 , v 9 }} is a { 0 , 1 , 2 } -matching in D . Both of them are maximal, since they cover all the vertices of D . 1 2 ✉ ✉ ✔ ❚ ❚ ✔ ❚ 7 ✔ ❚ ◗ ✉ ✔ ◗ ❚ ✔ ◗ ❚ ◗ ✔ ✔ ◗ ❚ 6 ✑✑✑✑✑ 3 ✉ ✉ ✉ ❚ ✔ ✔ 8 ❚ ✔ ❚ ✔ ✉ ❚ 9 ✔ ❚ ✔ ❚ ❚ ✔ ✉ ✉ 5 4 5

Let k be an algebraically closed field and k [ x 1 , . . . , x m ] be the polynomial ring in the variables x 1 , . . . , x m . A binomial in k [ x 1 , . . . , x m ] is a differ- ence of monomials. Given a lattice L on Z m , the ideal I L := ( { x α + − x α − | α = α + − α − ∈ L } ) in k [ x 1 , . . . , x m ] is called lattice ideal . Where α + ∈ N m and α − ∈ N m denote the positive and negative part of α , respectively, and x b = x b 1 1 · · · x b m for m b = ( b 1 , . . . , b m ) ∈ N m . If rank ( L ) = k , then there exists a ma- trix M ∈ Z ( m − k ) × m of rank m − k such that L ⊂ ker Z ( M ). When L = ker Z ( M ), the ideal I L is prime and called toric ideal . The va- riety V ( I L ) is called toric variety . 6

Let A = { a i | 1 ≤ i ≤ m } be the set of columns of M , we associate to I L the rational polyhedral cone σ = pos Q ( A ) := { d 1 a 1 + · · · + d m a m | d i ∈ Q ≥ 0 } . We assume that σ is strongly convex , i.e. { 0 } is a face of σ . With respect to the grading deg A ( x i ) = a i of the polynomial ring k [ x 1 , . . . , x m ] the ideal I L is A -homogeneous. The binomial arithmetical rank bar( I L ) of I L is the smallest integer s for which there exist binomials F 1 , . . . , F s in I L such that √ I L = √ F 1 , . . . , F s . Hence the binomial arithmetical rank is an upper bound for the arithmeti- cal rank ara( I L ) of I L , which is the smallest integer s for which there exists polynomials F 1 , . . . , F s in I L such that √ I L = √ F 1 , . . . , F s . 7

When all the polynomials F 1 , . . . , F s are A -homogeneous, the smallest integer s is called A -homogeneous arithmetical rank ara A ( I L ) of I L . For a lattice ideal I L the following inequality holds: ht( I L ) ≤ ara( I L ) ≤ ara A ( I L ) ≤ bar( I L ) ≤ µ ( I L ) . Find lower bounds for the Problem. minimal number µ ( I L ) of generators, the binomial arithmetical rank and the A -homogeneous arithmetical rank of a lattice ideal. 8

Let σ = pos Q ( r 1 , . . . , r t ) ⊂ Q n be a strongly convex rational polyhedral cone. Where { r 1 , . . . , r t } is a set of integer vectors, one for each extreme ray of σ . For a subset E of { 1 , . . . , t } we denote by σ E the subcone pos Q ( r i | i ∈ E ) of σ . The relative interior relint Q ( σ E ) of σ E is the set of all positive rational linear combinations of r i , i ∈ E . Suppose that σ is not a simplex cone, i.e. the extreme vectors r 1 , . . . , r t are not linearly independent. 9

The set of cones σ E , which are not faces of the cone σ , is not empty and form a poset ordered by inclusion. Let { σ E 1 , . . . , σ E f } be the minimal elements of this poset, which are called the min- imal non faces of σ . To the cone σ we associate a simpli- cial complex D σ with vertices the set { σ E 1 , . . . , σ E f } and T ⊂ { σ E 1 , . . . , σ E f } be- longs to D σ if � � ∩ σ E i ∈ T relint Q � = ∅ . σ E i 10

To the simplicial complex D σ we can associate the 1- skeleton G ( D σ ) of D σ , formed by the vertices and edges of D σ . The complement G ( D σ ) of G ( D σ ) is the graph with the same vertices as G ( D σ ), and { v i , v j } is an edge of G ( D σ ) if and only if { v i , v j } is not an edge of G ( D σ ). The chromatic number γ ( G ( D σ )) of the graph G ( D σ ) is the smallest integer k for which there is a function c : V ertices ( G ( D σ )) → { 1 , . . . , k } such that c ( v i ) � = c ( v j ) if { v i , v j } is an edge of G ( D σ ). 11

Theorem. For a lattice ideal I L with associated cone σ = pos Q ( A ) we have: (i) µ ( I L ) ≥ bar( I L ) ≥ δ ( D σ ) { 0 , 1 } = δ ( D σ ) { 0 } − δ ( D σ ) { 1 } , (ii) ara A ( I L ) ≥ δ ( D σ ) Ω ≥ γ ( G ( D σ )), (iii) If √ I L = √ F 1 , . . . , F s , then (a) the total number of monomials in the nonzero terms of the polynomials F 1 , . . . , F s is greater than or equal to the number of vertices δ ( D σ ) { 0 } of D σ . (b) the total number of A -homogeneous components in F 1 , . . . , F s is greater than or equal to the chromatic number of G ( D σ ). 12

We consider the polynomial ring k [ y 1 , . . . , y t ], by taking one variable for each extreme vector r i . From the set R σ = { r 1 , . . . , r t } we can construct the toric ideal I R σ , which is the kernel of the k -algebra homomorphism φ : k [ y 1 , . . . , y t ] → k [ z 1 , . . . , z n , z − 1 1 , . . . , z − 1 n ] given by φ ( y i ) = z r i . The toric variety V ( I R σ ) is called ex- tremal toric variety . 13

A binomial F ( u ) := y u + − y u − in I R σ is called primitive if there exists no other binomial y v + − y v − ∈ I R σ such that y v + divides y u + and y v − divides y u − . The set of primitive binomials of I R σ is finite and is called the Graver basis Gr ( R σ ) of I R σ . Theorem. Set E := { E ⊂ { 1 , . . . , t } | ∃ F ( u ) ∈ Gr ( R σ ) with supp ( u + ) = E or supp ( u − ) = E } , where supp ( v ) = { i ∈ { 1 , . . . , t } | v i > 0 } for v = ( v 1 , . . . , v t ) ∈ N t . Then σ E is a minimal non face of σ if and only if E is a minimal element of E . Theorem. A set T = { σ E i , σ E j } is an edge of D σ if and only if there is a prim- itive binomial F ( u ) ∈ I R σ with supp ( u + ) = E i and supp ( u − ) = E j . 14

Consider the lattice L = Example. ker Z ( M ), where M is the 3 × 6 ma- trix with columns the vectors of the set A 3 = { r ij = 2 e i + e j | i, j ∈ { 1 , 2 , 3 } , i � = j } and { e i | 1 ≤ i ≤ 3 } is the canonical base of Q 3 . The cone σ = pos Q ( A 3 ) is associated to the toric ideal I L ⊂ k [ x ij ]. Every vector of A 3 is an extreme vec- tor of σ . The Graver Base of I A 3 = I L consist of 3 binomials of the form x ij x kj − x ji x ki , 3 binomials of the form x 2 ij x ki − x 2 ik x ji , 3 binomials of the form x 2 ij x jk − x 2 ji x ik , 3 binomials of the form x 3 ji x ki − x 2 ij x 2 jk and 3 binomials of the form x 3 ki x ji − x 2 kj x 2 ik . 15

The simplicial complex D σ is drawn in the Figure. We can prove that δ ( D σ ) { 0 , 1 } = 5 and δ ( D σ ) { 0 , 1 , 2 } = γ ( G ( D σ )) = 4. In fact bar( I A 3 ) = 5 and ara A 3 ( I A 3 ) = 4. For ara( I A 3 ) we have that 3 ≤ ara( I A 3 ) ≤ 4, but it is unknown whether it is 3 or 4. { 12 , 23 } { 21 , 32 } ✉ ✉ ✔ ❚ ❚ ✔ ❚ ✔ ✔ ❚ ❚ ✔ { 21 , 31 } ✔ ❚ ❚ ✔ ◗ ✉ ✔ ❚ ◗ ❚ ✔ ✔ ❚ ◗ ❚ ✔ ◗ ✔ ❚ ✔ ◗ ❚ ✔ { 13 , 21 } ✑✑✑✑✑ { 23 , 31 } ✉ ❚ ✉ ✉ ❚ ✔ ✔ ✔ { 32 , 12 } ❚ ❚ ✔ ✔ ❚ ❚ ✔ ✔ ❚ ✉ ❚ { 23 , 13 } ✔ ✔ ❚ ❚ ✔ ✔ ❚ ❚ ❚ ❚ ✔ ✔ ✉ ✉ { 31 , 12 } { 32 , 13 } 16