Combinatorial Aspects of Tropical Geometry Mar´ ıa Ang´ elica Cueto Department of Mathematics Columbia University Annual SACNAS National Meeting Combinatorial Algebraic Geometry Session Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 1 / 15
What is tropical geometry? • Trop. semiring R tr :=( R ∪ {−∞} , ⊕ , ⊙ ) , a ⊕ b =max { a , b } , a ⊙ b = a + b . • Fix K = C { { t } } field of Puiseux series, with valuation given by lowest exponent , e.g. val( t − 4 / 3 + 1 + t + . . . ) = − 4 / 3, val(0) = ∞ . Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 2 / 15
What is tropical geometry? • Trop. semiring R tr :=( R ∪ {−∞} , ⊕ , ⊙ ) , a ⊕ b =max { a , b } , a ⊙ b = a + b . • Fix K = C { { t } } field of Puiseux series, with valuation given by lowest exponent , e.g. val( t − 4 / 3 + 1 + t + . . . ) = − 4 / 3, val(0) = ∞ . F ( x ) in K [ x ± 1 , . . . , x ± n ] � Trop( F )( ω ) in R tr [ ω ⊙± , . . . , ω ⊙± ] 1 n c α x α �→ Trop( F )( ω ):= − val( c α ) ⊙ ω ⊙ α = max � � F := α {− val( c α )+ � α, ω �} α α ( F = 0) in ( K ∗ ) n � Trop( F ) = { ω ∈ R n : max in Trop( F )( ω ) is not unique } Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 2 / 15
What is tropical geometry? • Trop. semiring R tr :=( R ∪ {−∞} , ⊕ , ⊙ ) , a ⊕ b =max { a , b } , a ⊙ b = a + b . • Fix K = C { { t } } field of Puiseux series, with valuation given by lowest exponent , e.g. val( t − 4 / 3 + 1 + t + . . . ) = − 4 / 3, val(0) = ∞ . F ( x ) in K [ x ± 1 , . . . , x ± n ] � Trop( F )( ω ) in R tr [ ω ⊙± , . . . , ω ⊙± ] 1 n c α x α �→ Trop( F )( ω ):= − val( c α ) ⊙ ω ⊙ α = max � � F := α {− val( c α )+ � α, ω �} α α ( F = 0) in ( K ∗ ) n � Trop( F ) = { ω ∈ R n : max in Trop( F )( ω ) is not unique } g = − t 3 x 3 + t 3 y 3 + t 2 y 2 + (4 + t 5 ) xy + 2 x + 7 y + (1 + t ). Example: Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 2 / 15
Tropical Geometry is a combinatorial shadow of algebraic geometry Input: X ⊂ ( K ∗ ) n irred. of dim d defined by an ideal I ⊂ K [ x ± 1 , . . . , x ± n ]. f ∈ I Trop( f ) ⊂ R n Output: Its tropicalization Trop( I ) = � Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 3 / 15
Tropical Geometry is a combinatorial shadow of algebraic geometry Input: X ⊂ ( K ∗ ) n irred. of dim d defined by an ideal I ⊂ K [ x ± 1 , . . . , x ± n ]. f ∈ I Trop( f ) ⊂ R n Output: Its tropicalization Trop( I ) = � • Trop( I ) is a polyhedral complex of pure dim. d & connected in codim. 1. obner theory: Trop( I ) = { ω ∈ R n | in ω ( I ) � = 1 } . • Gr¨ Weight of ω ∈ mxl cone = # { components of in ω ( I ) } (with mult.) With these weights, Trop( I ) is a balanced complex (0-tension condition) • Fund. Thm. Trop. Geom.: Trop( I ) = { ( − val( x i )) n i =1 : x ∈ X } . Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 3 / 15
Tropical Geometry is a combinatorial shadow of algebraic geometry Input: X ⊂ ( K ∗ ) n irred. of dim d defined by an ideal I ⊂ K [ x ± 1 , . . . , x ± n ]. f ∈ I Trop( f ) ⊂ R n Output: Its tropicalization Trop( I ) = � • Trop( I ) is a polyhedral complex of pure dim. d & connected in codim. 1. obner theory: Trop( I ) = { ω ∈ R n | in ω ( I ) � = 1 } . • Gr¨ Weight of ω ∈ mxl cone = # { components of in ω ( I ) } (with mult.) With these weights, Trop( I ) is a balanced complex (0-tension condition) • Fund. Thm. Trop. Geom.: Trop( I ) = { ( − val( x i )) n i =1 : x ∈ X } . • ( K ∗ ) r action on X via A ∈ Z r × n � Row span ( A ) in all cones of Trop( I ). � Mod. out Trop( I ) by this lineality space preserves the combinatorics. • The ends of a curve Trop( X ) in R 2 give an ambient toric variety ⊃ X . Conclusion: Trop( I ) sees dimension, torus actions, initial degenerations, compactifications and other geometric invariants of X (e.g. degree) Notice: Trop( X ) is highly sensitive to the embedding of X Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 3 / 15
Grassmannian of lines in P n − 1 and the space of trees Definition: Gr(2 , n ) = { lines in P n − 1 } := K 2 × n rk 2 / GL 2 (dim = 2( n − 2)). → P ( n 2 ) − 1 by the list of 2 × 2-minors: The Pl¨ ucker map embeds Gr(2 , n ) ֒ ϕ ( X ) = [ p ij := det( X ( i , j ) )] i < j ∀ X ∈ K 2 × n . Its Pl¨ ucker ideal I 2 , n is generated by the 3-term (quadratic) Pl¨ ucker eqns: p ij p kl − p ik p jl + p il p jk (1 � i < j < k < l � n ) . Note: ( K ∗ ) n / K ∗ acts on Gr(2 , n ) via t ∗ ( p ij ) = t i t j p ij . Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 4 / 15
Grassmannian of lines in P n − 1 and the space of trees Definition: Gr(2 , n ) = { lines in P n − 1 } := K 2 × n rk 2 / GL 2 (dim = 2( n − 2)). → P ( n 2 ) − 1 by the list of 2 × 2-minors: The Pl¨ ucker map embeds Gr(2 , n ) ֒ ϕ ( X ) = [ p ij := det( X ( i , j ) )] i < j ∀ X ∈ K 2 × n . Its Pl¨ ucker ideal I 2 , n is generated by the 3-term (quadratic) Pl¨ ucker eqns: p ij p kl − p ik p jl + p il p jk (1 � i < j < k < l � n ) . Note: ( K ∗ ) n / K ∗ acts on Gr(2 , n ) via t ∗ ( p ij ) = t i t j p ij . � Tropical Pl¨ ucker eqns: max { x ij + x kl , x ik + x jl , x il + x jl } . Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 4 / 15
Grassmannian of lines in P n − 1 and the space of trees Definition: Gr(2 , n ) = { lines in P n − 1 } := K 2 × n rk 2 / GL 2 (dim = 2( n − 2)). → P ( n 2 ) − 1 by the list of 2 × 2-minors: The Pl¨ ucker map embeds Gr(2 , n ) ֒ ϕ ( X ) = [ p ij := det( X ( i , j ) )] i < j ∀ X ∈ K 2 × n . Its Pl¨ ucker ideal I 2 , n is generated by the 3-term (quadratic) Pl¨ ucker eqns: p ij p kl − p ik p jl + p il p jk (1 � i < j < k < l � n ) . Note: ( K ∗ ) n / K ∗ acts on Gr(2 , n ) via t ∗ ( p ij ) = t i t j p ij . � Tropical Pl¨ ucker eqns: max { x ij + x kl , x ik + x jl , x il + x jl } . Theorem (Speyer-Sturmfels) The tropical Grassmannian Trop(Gr(2 , n ) ∩ (( K ∗ )( n 2 ) / K ∗ )) in R ( n 2 ) / R · 1 is the space of phylogenetic trees on n leaves: • all leaves are labeled 1 through n (no repetitions); • weights on all edges (non-negative weights for internal edges). It is cut out by the tropical Pl¨ ucker equations. The lineality space is generated by the n cut-metrics ℓ i = � j � = i e ij , modulo R · 1 . Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 4 / 15
The space of phylogenetic trees T n on n leaves • all leaves are labeled 1 through n (no repetitions); • weights on all edges (non-negative weights for internal edges). From the data ( T , ω ), we construct x ∈ R ( n 2 ) by x pq = � ω ( e ): e ∈ p → q Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 5 / 15
The space of phylogenetic trees T n on n leaves • all leaves are labeled 1 through n (no repetitions); • weights on all edges (non-negative weights for internal edges). From the data ( T , ω ), we construct x ∈ R ( n 2 ) by x pq = � ω ( e ): e ∈ p → q ( ij | kl ) ( ij | kl ) ∩ ( im | kl ) ∩ ( jm | kl ) ∩ . . . � x ij = ω i + ω j , x ik = ω i + ω 0 + ω k , . . . Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 5 / 15
The space of phylogenetic trees T n on n leaves • all leaves are labeled 1 through n (no repetitions); • weights on all edges (non-negative weights for internal edges). From the data ( T , ω ), we construct x ∈ R ( n 2 ) by x pq = � ω ( e ): e ∈ p → q ( ij | kl ) ( ij | kl ) ∩ ( im | kl ) ∩ ( jm | kl ) ∩ . . . � x ij = ω i + ω j , x ik = ω i + ω 0 + ω k , . . . 1 − to − 1 � x satisfying Tropical Pl¨ Claim: ( T , ω ) � ucker eqns. Why? (1) max { x ij + x kl , x ik + x jl , x il + x jk } ⇐ ⇒ quartet ( ij | kl ) . (2) tree T is reconstructed form the list of quartets, (3) linear algebra recovers the weight function ω from T and x . Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 5 / 15
Examples: T 4 / R 3 has f -vector (1 , 3). T 5 / R 4 is the cone over the Petersen graph. f -vector = (1 , 10 , 15). dim Gr(2 , n ) = dim(Trop( Gr (2 , n ) ∩ R ( n 2 ) − 1 ) = 2( n − 2) . Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 6 / 15
How to compactify T n ? 2 ) − 1 := ( R ∪ {−∞} )( n • Write TP ( n 2 ) � ( −∞ , . . . , −∞ )) / R · (1 , . . . , 1) • Compactify T n using Trop(Gr(2 , n )) ⊂ TP ( n 2 ) − 1 . • Cell structure? Generalized space of phylogenetic trees [C.]. Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 7 / 15
How to compactify T n ? 2 ) − 1 := ( R ∪ {−∞} )( n • Write TP ( n 2 ) � ( −∞ , . . . , −∞ )) / R · (1 , . . . , 1) • Compactify T n using Trop(Gr(2 , n )) ⊂ TP ( n 2 ) − 1 . • Cell structure? Generalized space of phylogenetic trees [C.]. Angie Cueto (Columbia U) Combinatorics in Tropical Geometry October 16th 2014 7 / 15
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