Faithful tropicalization for the Grassmannian of planes elica Cueto 1 Annette Werner 2 Mar´ ıa Ang´ Mathias H¨ abich 1 Department of Mathematics Columbia University 2 Department of Mathematics Goethe Universit¨ at Frankfurt October 26th 2014 AMS Meeting San Francisco State University Special Session on Combinatorics and Algebraic Geometry Math. Ann. 360 (1-2), 391–437 (2014) Cueto - H¨ abich - Werner (CU-GUF) Faithful tropicalization for Gr(2 , n ) October 26th 2014 1 / 10
Non-Archimedean Berkovich spaces • Fix ( K , | · | ) complete non-Archimedean field, | · | : K → R � 0 (1) | a | = 0 ⇐ ⇒ a = 0 (2) | ab | = | a || b | (multiplicative) (3) | a + b | � max {| a | , | b |} (with = if | a | � = | b | ) (non-Arch. triangle ineq.) � − log( | · | ): K → R := R ∪ {−∞} is a valuation on K . Cueto - H¨ abich - Werner (CU-GUF) Faithful tropicalization for Gr(2 , n ) October 26th 2014 2 / 10
Non-Archimedean Berkovich spaces • Fix ( K , | · | ) complete non-Archimedean field, | · | : K → R � 0 (1) | a | = 0 ⇐ ⇒ a = 0 (2) | ab | = | a || b | (multiplicative) (3) | a + b | � max {| a | , | b |} (with = if | a | � = | b | ) (non-Arch. triangle ineq.) � − log( | · | ): K → R := R ∪ {−∞} is a valuation on K . • X = K -scheme of fin. type � Berkovich space X an (top space + sheaf) (Spec A ) an := {� · � : A → R � 0 mult seminorms extending | · | K } . • Topology: coarsest s.t. all ev f : � · � �→ � f � ( f ∈ A ) are continuous. • Construct X an by gluing of affine pieces � Get X ( K ) ⊂ X an . Cueto - H¨ abich - Werner (CU-GUF) Faithful tropicalization for Gr(2 , n ) October 26th 2014 2 / 10
Non-Archimedean Berkovich spaces • Fix ( K , | · | ) complete non-Archimedean field, | · | : K → R � 0 (1) | a | = 0 ⇐ ⇒ a = 0 (2) | ab | = | a || b | (multiplicative) (3) | a + b | � max {| a | , | b |} (with = if | a | � = | b | ) (non-Arch. triangle ineq.) � − log( | · | ): K → R := R ∪ {−∞} is a valuation on K . • X = K -scheme of fin. type � Berkovich space X an (top space + sheaf) (Spec A ) an := {� · � : A → R � 0 mult seminorms extending | · | K } . • Topology: coarsest s.t. all ev f : � · � �→ � f � ( f ∈ A ) are continuous. • Construct X an by gluing of affine pieces � Get X ( K ) ⊂ X an . n . Example: Skeleton (semi) norm on ( A n ) an for each ρ ∈ R n c α x α �− � � δ ( ρ ): K [ x 1 , . . . , x n ] → R � 0 → max α {| c α | exp( α i ρ i ) } . α i =1 δ ( ρ )( x i ) = exp( ρ i ) and it is maximal with this property. Note: If ρ i � = −∞ , we can extend δ ( ρ ) to K [ x 1 , . . . , x ± i , . . . , x n ]. Cueto - H¨ abich - Werner (CU-GUF) Faithful tropicalization for Gr(2 , n ) October 26th 2014 2 / 10
Analytification is the limit of all tropicalizations [Payne] i Fix X = K -scheme of fin. type and X � � � Y Σ (TV with dense torus G n m ). cl . Assume i ( X ) meets G n m and write { y 1 , . . . , y n } basis of characters. Cueto - H¨ abich - Werner (CU-GUF) Faithful tropicalization for Gr(2 , n ) October 26th 2014 3 / 10
� � � � Analytification is the limit of all tropicalizations [Payne] i Fix X = K -scheme of fin. type and X � � � Y Σ (TV with dense torus G n m ). cl . Assume i ( X ) meets G n m and write { y 1 , . . . , y n } basis of characters. X an � � � � (trop , i ) �·� �→ ( − log( � y 1 � ) ,..., − log( � y n � )) � � � � � � � � cont . and surj . � � � � � � � � � � � � X ( K ) Trop( X , i ) ⊂ Trop( Y Σ ) − val( · ) = log( |·| ) Cueto - H¨ abich - Werner (CU-GUF) Faithful tropicalization for Gr(2 , n ) October 26th 2014 3 / 10
� � � � Analytification is the limit of all tropicalizations [Payne] i Fix X = K -scheme of fin. type and X � � � Y Σ (TV with dense torus G n m ). cl . Assume i ( X ) meets G n m and write { y 1 , . . . , y n } basis of characters. X an � � � � (trop , i ) �·� �→ ( − log( � y 1 � ) ,..., − log( � y n � )) � � � � � � � � cont . and surj . � � � � � � � � � � � � X ( K ) Trop( X , i ) ⊂ Trop( Y Σ ) − val( · ) = log( |·| ) Question (after [Payne]): Does there exist a continuous section σ : Trop( X , i ) → X an to (trop , i )? If so, i induces a faithful tropicalization. Cueto - H¨ abich - Werner (CU-GUF) Faithful tropicalization for Gr(2 , n ) October 26th 2014 3 / 10
� � � � Analytification is the limit of all tropicalizations [Payne] i Fix X = K -scheme of fin. type and X � � � Y Σ (TV with dense torus G n m ). cl . Assume i ( X ) meets G n m and write { y 1 , . . . , y n } basis of characters. X an � � � � (trop , i ) �·� �→ ( − log( � y 1 � ) ,..., − log( � y n � )) � � � � � � � � cont . and surj . � � � � � � � � � � � � X ( K ) Trop( X , i ) ⊂ Trop( Y Σ ) − val( · ) = log( |·| ) Question (after [Payne]): Does there exist a continuous section σ : Trop( X , i ) → X an to (trop , i )? If so, i induces a faithful tropicalization. • Curves: if all tropical multiplicities are one (initial degen. are irred. and gen. reduced), then the tropicalization is faithful [Baker-Payne-Rabinoff]. Cueto - H¨ abich - Werner (CU-GUF) Faithful tropicalization for Gr(2 , n ) October 26th 2014 3 / 10
� � � � Analytification is the limit of all tropicalizations [Payne] i Fix X = K -scheme of fin. type and X � � � Y Σ (TV with dense torus G n m ). cl . Assume i ( X ) meets G n m and write { y 1 , . . . , y n } basis of characters. X an � � � � (trop , i ) �·� �→ ( − log( � y 1 � ) ,..., − log( � y n � )) � � � � � � � � cont . and surj . � � � � � � � � � � � � X ( K ) Trop( X , i ) ⊂ Trop( Y Σ ) − val( · ) = log( |·| ) Question (after [Payne]): Does there exist a continuous section σ : Trop( X , i ) → X an to (trop , i )? If so, i induces a faithful tropicalization. • Curves: if all tropical multiplicities are one (initial degen. are irred. and gen. reduced), then the tropicalization is faithful [Baker-Payne-Rabinoff]. Theorem (C.-H¨ abich-Werner) The Grassmannian Gr(2 , n ) of 2-planes in A n is tropicalized faithfully by ucker map. The cont. section σ : Trop Gr(2 , n ) → Gr(2 , n ) an to trop the Pl¨ maps a pt. x to the unique Shilov boundary point in trop − 1 ( x ) and all trop. mult. are 1. The image of σ is a candidate canonical polyhedron. Cueto - H¨ abich - Werner (CU-GUF) Faithful tropicalization for Gr(2 , n ) October 26th 2014 3 / 10
Grassmannian of 2-planes in A n and the space of trees → P ( n 2 ) − 1 by the list of 2 × 2-minors: • The Pl¨ ucker map ϕ embeds Gr(2 , n ) ֒ ϕ ( X ) = [ p ij := det( X ( i , j ) )] i < j ∀ X ∈ Gr(2 , n ) := A 2 × n rk 2 / GL(2) . 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: G n m / G m acts on Gr(2 , n ) via t ∗ ( p ij ) = t i t j p ij . • Write Gr 0 (2 , n ) := ϕ − 1 ( G ( n 2 ) m / G m ) (proj. dim = 2( n − 2)). Cueto - H¨ abich - Werner (CU-GUF) Faithful tropicalization for Gr(2 , n ) October 26th 2014 4 / 10
Grassmannian of 2-planes in A n and the space of trees → P ( n 2 ) − 1 by the list of 2 × 2-minors: • The Pl¨ ucker map ϕ embeds Gr(2 , n ) ֒ ϕ ( X ) = [ p ij := det( X ( i , j ) )] i < j ∀ X ∈ Gr(2 , n ) := A 2 × n rk 2 / GL(2) . 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: G n m / G m acts on Gr(2 , n ) via t ∗ ( p ij ) = t i t j p ij . • Write Gr 0 (2 , n ) := ϕ − 1 ( G ( n 2 ) m / G m ) (proj. dim = 2( n − 2)). Theorem (Speyer-Sturmfels) The (open) tropical Grassmannian Trop(Gr 0 (2 , n )) 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 . Cueto - H¨ abich - Werner (CU-GUF) Faithful tropicalization for Gr(2 , n ) October 26th 2014 4 / 10
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 Cueto - H¨ abich - Werner (CU-GUF) Faithful tropicalization for Gr(2 , n ) October 26th 2014 5 / 10
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 ) ( ik | jl ) ∩ ( im | jl ) ∩ ( km | jl ) ∩ . . . � x ik = ω i + ω k , x ij = ω i + ω 0 + ω j , . . . Cueto - H¨ abich - Werner (CU-GUF) Faithful tropicalization for Gr(2 , n ) October 26th 2014 5 / 10
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