Tropical Secant Graphs of Monomial Curves Mar a Ang elica Cueto - PowerPoint PPT Presentation

Tropical Secant Graphs of Monomial Curves Mar´ ıa Ang´ elica Cueto UC Berkeley Joint work with Shaowei Lin arXiv:1005.3364v1 AMS-SMM Eighth International Meeting Special Session on Singularity Theory and Algebraic Geometry June 4th, 2010 M.A. Cueto (UC Berkeley) Tropical Secant Graphs 1 / 10

Summary GOAL: Study the affine cone over the first secant variety of a monomial curve t �→ (1 : t i 1 : t i 2 : . . . : t i n ) . STRATEGY: Compute its tropicalization, which is a pure, balanced weighted polyhedral fan of dim. 4 in R n +1 , with a 2-dimensional lineality space R � 1 , (1 , i 1 , i 2 , . . . , i n ) � . We encode it as a weighted graph in an ( n − 2) -dim’l sphere. M.A. Cueto (UC Berkeley) Tropical Secant Graphs 2 / 10

Summary GOAL: Study the affine cone over the first secant variety of a monomial curve t �→ (1 : t i 1 : t i 2 : . . . : t i n ) . STRATEGY: Compute its tropicalization, which is a pure, balanced weighted polyhedral fan of dim. 4 in R n +1 , with a 2-dimensional lineality space R � 1 , (1 , i 1 , i 2 , . . . , i n ) � . We encode it as a weighted graph in an ( n − 2) -dim’l sphere. Why? Given the tropicalization T X of a projective variety X , we can recover its Chow polytope (hence, its multidegree , etc.) by known algorithms (a.k.a. “Tropical implicitization.” ) M.A. Cueto (UC Berkeley) Tropical Secant Graphs 2 / 10

Summary GOAL: Study the affine cone over the first secant variety of a monomial curve t �→ (1 : t i 1 : t i 2 : . . . : t i n ) . STRATEGY: Compute its tropicalization, which is a pure, balanced weighted polyhedral fan of dim. 4 in R n +1 , with a 2-dimensional lineality space R � 1 , (1 , i 1 , i 2 , . . . , i n ) � . We encode it as a weighted graph in an ( n − 2) -dim’l sphere. Why? Given the tropicalization T X of a projective variety X , we can recover its Chow polytope (hence, its multidegree , etc.) by known algorithms (a.k.a. “Tropical implicitization.” ) Main examples: monomial curves C in P 4 . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 2 / 10

Summary GOAL: Study the affine cone over the first secant variety of a monomial curve t �→ (1 : t i 1 : t i 2 : . . . : t i n ) . STRATEGY: Compute its tropicalization, which is a pure, balanced weighted polyhedral fan of dim. 4 in R n +1 , with a 2-dimensional lineality space R � 1 , (1 , i 1 , i 2 , . . . , i n ) � . We encode it as a weighted graph in an ( n − 2) -dim’l sphere. Why? Given the tropicalization T X of a projective variety X , we can recover its Chow polytope (hence, its multidegree , etc.) by known algorithms (a.k.a. “Tropical implicitization.” ) Main examples: monomial curves C in P 4 . � Compute Newton polytope of the defining equation of Sec 1 ( C ) . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 2 / 10

A tropical approach to the first secant of monomial curves Let C be the monomial projective curve (1 : t i 1 : . . . : t i n ) parameterized by n coprime integers 0 < i 1 < . . . < i n . By definition, Sec 1 ( C ) = { a · p + b · q | ( a : b ) ∈ P 1 , p, q ∈ C } ⊂ T n +1 . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 3 / 10

A tropical approach to the first secant of monomial curves Let C be the monomial projective curve (1 : t i 1 : . . . : t i n ) parameterized by n coprime integers 0 < i 1 < . . . < i n . By definition, Sec 1 ( C ) = { a · p + b · q | ( a : b ) ∈ P 1 , p, q ∈ C } ⊂ T n +1 . • Pick points p = (1 : t i 1 : . . . : t i n ) , q = (1 : s i 1 : . . . : s i n ) in C . Use the monomial change of coordinates b = − λa , t = ωs , and rewrite v = a · p + b · q , as · ( ω i k − λ ) v k = as i k for all k = 0 , . . . , n, ���� � �� � ∈ ˜ C ∈ Z C is the cone in T n +1 over the curve C . where ˜ M.A. Cueto (UC Berkeley) Tropical Secant Graphs 3 / 10

A tropical approach to the first secant of monomial curves Let C be the monomial projective curve (1 : t i 1 : . . . : t i n ) parameterized by n coprime integers 0 < i 1 < . . . < i n . By definition, Sec 1 ( C ) = { a · p + b · q | ( a : b ) ∈ P 1 , p, q ∈ C } ⊂ T n +1 . • Pick points p = (1 : t i 1 : . . . : t i n ) , q = (1 : s i 1 : . . . : s i n ) in C . Use the monomial change of coordinates b = − λa , t = ωs , and rewrite v = a · p + b · q , as · ( ω i k − λ ) v k = as i k for all k = 0 , . . . , n, ���� � �� � ∈ ˜ C ∈ Z C is the cone in T n +1 over the curve C . where ˜ Definition Let X, Y ⊂ T N be two subvarieties of tori. The Hadamard product of X and Y equals X � Y = { ( x 0 y 0 , . . . , x n y n ) | x ∈ X, y ∈ Y } ⊂ T N . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 3 / 10

Theorem ([C. - Tobis - Yu]) Let X, Y ⊂ T N be closed subvarieties and consider their Hadamard product X � Y ⊂ T N . Then as weighted sets : T ( X � Y ) = T X + T Y. Corollary ([C. - Lin]) Given a monomial curve C : t �→ (1 : t i 1 : . . . : t i n ) , and the surface Z : ( λ, ω ) �→ (1 − λ, ω i 1 − λ, . . . , ω i n − λ ) ⊂ T n +1 . Then: T Sec 1 ( C ) = T Z + R ⊗ Z Λ where Λ = Z � 1 , (0 , i 1 , . . . , i n ) � is the intrinsic lin. lattice of T Sec 1 ( C ) . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 4 / 10

Theorem ([C. - Tobis - Yu]) Let X, Y ⊂ T N be closed subvarieties and consider their Hadamard product X � Y ⊂ T N . Then as weighted sets : T ( X � Y ) = T X + T Y. Corollary ([C. - Lin]) Given a monomial curve C : t �→ (1 : t i 1 : . . . : t i n ) , and the surface Z : ( λ, ω ) �→ (1 − λ, ω i 1 − λ, . . . , ω i n − λ ) ⊂ T n +1 . Then: T Sec 1 ( C ) = T Z + R ⊗ Z Λ where Λ = Z � 1 , (0 , i 1 , . . . , i n ) � is the intrinsic lin. lattice of T Sec 1 ( C ) . Strategy Construct the graph T Z . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 4 / 10

Theorem ([C. - Tobis - Yu]) Let X, Y ⊂ T N be closed subvarieties and consider their Hadamard product X � Y ⊂ T N . Then as weighted sets : T ( X � Y ) = T X + T Y. Corollary ([C. - Lin]) Given a monomial curve C : t �→ (1 : t i 1 : . . . : t i n ) , and the surface Z : ( λ, ω ) �→ (1 − λ, ω i 1 − λ, . . . , ω i n − λ ) ⊂ T n +1 . Then: T Sec 1 ( C ) = T Z + R ⊗ Z Λ where Λ = Z � 1 , (0 , i 1 , . . . , i n ) � is the intrinsic lin. lattice of T Sec 1 ( C ) . Strategy Construct the graph T Z . � “Geometric tropicalization” Modify T Z to get a weighted graph (the tropical secant graph (TSG)) representing T Sec 1 ( C ) as a weighted set . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 4 / 10

Construction of T Z Theorem (Geometric Tropicalization [Hacking - Keel - Tevelev]) Consider T N with coordinate functions t 1 , . . . , t N , and let Z ⊂ T N be a closed smooth surface. Suppose Z ⊃ Z is any compactification whose boundary D is a smooth divisor with C.N.C. Let D 1 , . . . , D m be the irred. comp. of D , and write ∆ for the graph on { 1 , . . . , m } defined by { k i , k j } ∈ ∆ ⇐ ⇒ D k i ∩ D k j � = ∅ . Let [ D k ]:=( val D k ( t 1 ) , . . . , val D k ( t N )) ∈ Z N , and [ σ ] := Z ≥ 0 � [ D k ] : k ∈ σ � , for σ ∈ ∆ . Then, � T Z = R ≥ 0 [ σ ] . σ ∈ ∆ M.A. Cueto (UC Berkeley) Tropical Secant Graphs 5 / 10

Construction of T Z Theorem (Geometric Tropicalization [Hacking - Keel - Tevelev]) Consider T N with coordinate functions t 1 , . . . , t N , and let Z ⊂ T N be a closed smooth surface. Suppose Z ⊃ Z is any compactification whose boundary D is a smooth divisor with C.N.C. Let D 1 , . . . , D m be the irred. comp. of D , and write ∆ for the graph on { 1 , . . . , m } defined by { k i , k j } ∈ ∆ ⇐ ⇒ D k i ∩ D k j � = ∅ . Let [ D k ]:=( val D k ( t 1 ) , . . . , val D k ( t N )) ∈ Z N , and [ σ ] := Z ≥ 0 � [ D k ] : k ∈ σ � , for σ ∈ ∆ . Then, � T Z = R ≥ 0 [ σ ] . σ ∈ ∆ Theorem ([C.]) � � ( R ⊗ Z [ σ ]) ∩ Z N : Z [ σ ] � m w = ( D k 1 · D k 2 ) index σ ∈ ∆ s.t. w ∈ R ≥ 0 [ σ ] M.A. Cueto (UC Berkeley) Tropical Secant Graphs 5 / 10

• Today, β = ( f 0 , f i 1 , . . . , f i n ): X → Z ⊂ T n +1 , f i j ( w, λ ) := w i j − λ , and n X = T 2 � � ( f i j ( w, λ ) = 0) j =0 . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 6 / 10

• Today, β = ( f 0 , f i 1 , . . . , f i n ): X → Z ⊂ T n +1 , f i j ( w, λ ) := w i j − λ , and n X = T 2 � � ( f i j ( w, λ ) = 0) j =0 . • How to proceed if Z doesn’t satisfy the C.N.C. hypothesis? � Find nice (tropical) compactification by resolving singularities! M.A. Cueto (UC Berkeley) Tropical Secant Graphs 6 / 10

• Today, β = ( f 0 , f i 1 , . . . , f i n ): X → Z ⊂ T n +1 , f i j ( w, λ ) := w i j − λ , and n X = T 2 � � ( f i j ( w, λ ) = 0) j =0 . • How to proceed if Z doesn’t satisfy the C.N.C. hypothesis? � Find nice (tropical) compactification by resolving singularities! • Idea: work with X instead of Z and use β to translate back to Z . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 6 / 10