Tropical Secant Graphs of Monomial Curves M. Angelica Cueto UC Berkeley Joint work with Shaowei Lin arXiv:1005.3364v1 2nd PhD Students Conference on Tropical Geometry July 16-17th, 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, weighted balanced rational polyhedral fan of dim. 4 in R n +1 , with a 2-dimensional lineality space R � 1 , (0 , 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, weighted balanced rational polyhedral fan of dim. 4 in R n +1 , with a 2-dimensional lineality space R � 1 , (0 , 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 useful information about X . E.g.: its Chow polytope (hence, its degree , . . . ). 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, weighted balanced rational polyhedral fan of dim. 4 in R n +1 , with a 2-dimensional lineality space R � 1 , (0 , 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 useful information about X . E.g.: its Chow polytope (hence, its degree , . . . ). 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 } ⊂ ( C ∗ ) 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 } ⊂ ( C ∗ ) 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 ∈ surface Z 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 } ⊂ ( C ∗ ) 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 ∈ surface Z Definition Let X, Y ⊂ ( C ∗ ) 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 } ⊂ ( C ∗ ) N . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 3 / 10
Theorem ([C. - Tobis - Yu], [Allermann-Rau], . . . ) Let X, Y ⊂ ( C ∗ ) N be closed subvarieties and consider their Hadamard product X � Y ⊂ ( C ∗ ) N . Then as 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 ) ⊂ ( C ∗ ) n +1 . Then: T Sec 1 ( C ) = T Z + R ⊗ Z Λ where Λ = Z � 1 , (0 , i 1 , . . . , i n ) � generates the lineality space of T Sec 1 ( C ) . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 4 / 10
Theorem ([C. - Tobis - Yu], [Allermann-Rau], . . . ) Let X, Y ⊂ ( C ∗ ) N be closed subvarieties and consider their Hadamard product X � Y ⊂ ( C ∗ ) N . Then as 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 ) ⊂ ( C ∗ ) n +1 . Then: T Sec 1 ( C ) = T Z + R ⊗ Z Λ where Λ = Z � 1 , (0 , i 1 , . . . , i n ) � generates the lineality space of T Sec 1 ( C ) . Strategy Construct the weighted graph T Z . Modify T Z to get a weighted graph 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 ( C ∗ ) N with coordinate functions t 1 , . . . , t N , and let Z ⊂ ( C ∗ ) 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 � = ∅ . We realize ∆ in R N via { k } �→ [ D k ]:=( val D k ( t 1 ) , . . . , val D k ( t N )) ∈ Z N . Then, T Z is the cone over this graph in R N . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 5 / 10
Construction of T Z Theorem (Geometric Tropicalization [Hacking - Keel - Tevelev]) Consider ( C ∗ ) N with coordinate functions t 1 , . . . , t N , and let Z ⊂ ( C ∗ ) 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 � = ∅ . We realize ∆ in R N via { k } �→ [ D k ]:=( val D k ( t 1 ) , . . . , val D k ( t N )) ∈ Z N . Then, T Z is the cone over this graph in R N . Theorem ([C.]) Combinatorial formula for computing the weights of the edges of ∆ . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 5 / 10
• How to proceed if Z doesn’t satisfy the C.N.C. hypothesis? � Find nice compactification by resolving singularities! M.A. Cueto (UC Berkeley) Tropical Secant Graphs 6 / 10
• How to proceed if Z doesn’t satisfy the C.N.C. hypothesis? � Find nice compactification by resolving singularities! → Z ⊂ ( C ∗ ) n +1 , ( λ, w ) �→ (1 − λ, w i 1 − λ, . . . , w i n − λ ) • Recall: β : X ֒ and n � X = ( C ∗ ) 2 � ( w i j − λ = 0) . j =0 • 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
• How to proceed if Z doesn’t satisfy the C.N.C. hypothesis? � Find nice compactification by resolving singularities! → Z ⊂ ( C ∗ ) n +1 , ( λ, w ) �→ (1 − λ, w i 1 − λ, . . . , w i n − λ ) • Recall: β : X ֒ and n � X = ( C ∗ ) 2 � ( w i j − λ = 0) . j =0 • Idea: work with X instead of Z and use β to translate back to Z . • Compactify X inside P 2 and extend β to β : P 2 ⊃ X ֒ → ( C ∗ ) n +1 . Our boundary divisors in X ⊂ P 2 are D i j = ( w i j − λ = 0) ( j =0 ,...,n ) , D ∞ . M.A. Cueto (UC Berkeley) Tropical Secant Graphs 6 / 10
• How to proceed if Z doesn’t satisfy the C.N.C. hypothesis? � Find nice compactification by resolving singularities! → Z ⊂ ( C ∗ ) n +1 , ( λ, w ) �→ (1 − λ, w i 1 − λ, . . . , w i n − λ ) • Recall: β : X ֒ and n � X = ( C ∗ ) 2 � ( w i j − λ = 0) . j =0 • Idea: work with X instead of Z and use β to translate back to Z . • Compactify X inside P 2 and extend β to β : P 2 ⊃ X ֒ → ( C ∗ ) n +1 . Our boundary divisors in X ⊂ P 2 are D i j = ( w i j − λ = 0) ( j =0 ,...,n ) , D ∞ . • Triple intersections at: the origin, a point at infinity and at points in ( C ∗ ) 2 . � Three types of points to blow-up ! M.A. Cueto (UC Berkeley) Tropical Secant Graphs 6 / 10
• How to proceed if Z doesn’t satisfy the C.N.C. hypothesis? � Find nice compactification by resolving singularities! → Z ⊂ ( C ∗ ) n +1 , ( λ, w ) �→ (1 − λ, w i 1 − λ, . . . , w i n − λ ) • Recall: β : X ֒ and n � X = ( C ∗ ) 2 � ( w i j − λ = 0) . j =0 • Idea: work with X instead of Z and use β to translate back to Z . • Compactify X inside P 2 and extend β to β : P 2 ⊃ X ֒ → ( C ∗ ) n +1 . Our boundary divisors in X ⊂ P 2 are D i j = ( w i j − λ = 0) ( j =0 ,...,n ) , D ∞ . • Triple intersections at: the origin, a point at infinity and at points in ( C ∗ ) 2 . � Three types of points to blow-up ! • The resolution diagrams come in three flavors: two caterpillar trees and families of star trees. We glue together these graphs along common nodes to obtain the intersection complex ∆ from the theorem. M.A. Cueto (UC Berkeley) Tropical Secant Graphs 6 / 10
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