Tropical Secant Graphs of Monomial Curves Mar´ ıa Ang´ elica Cueto Shaowei Lin Department of Mathematics University of California, Berkeley Combinatorics Seminar - UC Berkeley M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 1 / 16
Outline 1 Introducing our favorite graphs: the abstract tropical secant surface graph and the master graph . 2 What is behind these graphs? � A surface in P n parameterized by binomials, and its tropicalization. 3 Geometric tropicalization by example (with lots of blow-ups!) 4 Towards the first secant of monomial curves in P n � Our other two favorite graphs: the tropical secant graph and its planar buddy, the Gr¨ obner tropical secant graph. 5 The hypersurface case: from the tropical secant graph to the Newton polytope. M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 2 / 16
The abstract tropical secant surface graph • Fix n ≥ 4 and n coprime distinct integers I := { 0 = i 0 < i 1 < . . . < i n } . • Consider all sequences a ⊂ I arising from arith. prog. in Z , with | a | ≥ 2 . M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 3 / 16
The abstract tropical secant surface graph • Fix n ≥ 4 and n coprime distinct integers I := { 0 = i 0 < i 1 < . . . < i n } . • Consider all sequences a ⊂ I arising from arith. prog. in Z , with | a | ≥ 2 . • Build two caterpillar trees G E,D , G h,D and a family of star trees G F a ,D : a = { i j 1 , . . . , i j k } M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 3 / 16
The abstract tropical secant surface graph • Fix n ≥ 4 and n coprime distinct integers I := { 0 = i 0 < i 1 < . . . < i n } . • Consider all sequences a ⊂ I arising from arith. prog. in Z , with | a | ≥ 2 . • Build two caterpillar trees G E,D , G h,D and a family of star trees G F a ,D : a = { i j 1 , . . . , i j k } • Glue the graphs G E,D , G h,D and G F a ,D along common nodes to form the abstract tropical secant surface graph. M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 3 / 16
The master graph (a.k.a. the tropical secant surface graph) It is defined by a weighted embedding of the abstract graph in R n +1 . Definition ( master graph ) 1 D i j = e j := (0 , . . . , 0 , 1 , 0 , . . . , 0) ( 0 ≤ j ≤ n ), 2 E i j = (0 , i 1 , . . . , i j − 1 , i j , . . . , i j ) ( 1 ≤ j ≤ n − 1 ), 3 h i j = ( − i j , − i j , . . . , − i j , − i j +1 , . . . , − i n ) ( 1 ≤ j ≤ n − 1 ), 4 F a = � i j ∈ a e j for a ⊆ { 0 , i 1 , . . . , i n } arith. progr., | a | ≥ 2 . The edges have weights: 1 m D ij ,E ij = gcd( i 1 , . . . , i j ) , m D ij ,h ij = gcd( i j , . . . , i n ) , 2 m D i 0 ,h i 1 = 1 , m D in ,E in − 1 = gcd( i 1 , . . . , i n − 1 ) , m D in ,h in − 1 = i n , 3 m E ij ,E ij +1 = gcd( i 1 , . . . , i j ) , m h ij ,h ij +1 = gcd( i j +1 , . . . , i n ) , 4 m F a ,D ij = � r ϕ ( r ) , where we sum over all common diff. r of all possible arith. prog. containing i j and giving a . Here, ϕ is Euler’s phi . M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 4 / 16
Our favorite example: I = { 0 , 30 , 45 , 55 , 78 } (K. Ranestad) • 9 bivalent nodes F i j ,i k are eliminated from the picture and replace its two adjacent edges by edge D i j D i k . • 16 vertices (incl. bivalent node E 30 ), and 36 edges. • Five red non-bivalent (unla- beled) nodes F a : F 0 , 30 , 45 , 55 , 78 = (1 , 1 , 1 , 1 , 1) , F 0 , 30 , 45 , 78 = (1 , 1 , 1 , 0 , 1) , F 0 , 30 , 45 , 55 = (1 , 1 , 1 , 1 , 0) , F 0 , 30 , 45 = (1 , 1 , 1 , 0 , 0) , F 0 , 30 , 78 = (1 , 1 , 0 , 0 , 1) . M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 5 / 16
Remark (Disclaimer) 1 If a = { i j , i k } , we eliminate the bivalent node F a , replacing its two adj. edges by a single edge D i j D i k , with the inherited weight. 2 E i 1 is bivalent node, but we keep this one to simplify notation. 3 F i j 1 ,...,i jk is a node ⇐ ⇒ gcd( i j k − i j 1 , . . . , i j 2 − i j 1 ) � = 1 , k maximal with the same gcd . M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 6 / 16
Remark (Disclaimer) 1 If a = { i j , i k } , we eliminate the bivalent node F a , replacing its two adj. edges by a single edge D i j D i k , with the inherited weight. 2 E i 1 is bivalent node, but we keep this one to simplify notation. 3 F i j 1 ,...,i jk is a node ⇐ ⇒ gcd( i j k − i j 1 , . . . , i j 2 − i j 1 ) � = 1 , k maximal with the same gcd . Theorem ( — - Lin) The master graph satisfies the balancing condition. M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 6 / 16
The master graph is a tropical surface Definition For an irreducible algebraic variety X ⊂ T N = ( C ∗ ) N with defining ideal I = I ( X ) ⊂ C [ x ± 1 1 , . . . , x ± 1 N ] , the tropicalization of X or I is defined as: T ( X ) = T ( I ) = { w ∈ Q N | 1 / ∈ in w ( I ) } where in w ( I ) = � in w ( f ) : f ∈ I � , and in w ( f ) is the sum of all nonzero α c α x α such that α · w is minimum . terms of f = � Remark 1 T ( X ) is a pure dim X -dim’l poly. subfan of the Gr¨ obner fan of I ( X ) . 2 The lineality space of the fan T ( X ) is the set L = { w ∈ T X : in w ( I ) = I } . It describes action of the maximal torus acting on X (by the lattice Λ := L ∩ Z n +1 ). M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 7 / 16
The master graph is a tropical surface Definition For an irreducible algebraic variety X ⊂ T N = ( C ∗ ) N with defining ideal I = I ( X ) ⊂ C [ x ± 1 1 , . . . , x ± 1 N ] , the tropicalization of X or I is defined as: T ( X ) = T ( I ) = { w ∈ Q N | 1 / ∈ in w ( I ) } where in w ( I ) = � in w ( f ) : f ∈ I � , and in w ( f ) is the sum of all nonzero α c α x α such that α · w is minimum . terms of f = � Remark 1 T ( X ) is a pure dim X -dim’l poly. subfan of the Gr¨ obner fan of I ( X ) . 2 The lineality space of the fan T ( X ) is the set L = { w ∈ T X : in w ( I ) = I } . It describes action of the maximal torus acting on X (by the lattice Λ := L ∩ Z n +1 ). � View T X in the ( N − rk Λ − 1) -sphere of R N /L . M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 7 / 16
• A point w ∈ T X is regular if T X is a linear space locally near w . • We can assing a positive multiplicity to every maximal cones in T X , and give regular points the multiplicity of the corresp. mxl. cone. • Tropical varieties satisfy the balancing condition. M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 8 / 16
• A point w ∈ T X is regular if T X is a linear space locally near w . • We can assing a positive multiplicity to every maximal cones in T X , and give regular points the multiplicity of the corresp. mxl. cone. • Tropical varieties satisfy the balancing condition. Theorem (— - Lin) Let Z ⊂ T n +1 be the surface parameterized by ( λ, ω ) �→ (1 − λ, ω i 1 − λ, . . . , ω i n − λ ) . Then, the cone over the master graph is the tropical surface T Z . M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 8 / 16
• A point w ∈ T X is regular if T X is a linear space locally near w . • We can assing a positive multiplicity to every maximal cones in T X , and give regular points the multiplicity of the corresp. mxl. cone. • Tropical varieties satisfy the balancing condition. Theorem (— - Lin) Let Z ⊂ T n +1 be the surface parameterized by ( λ, ω ) �→ (1 − λ, ω i 1 − λ, . . . , ω i n − λ ) . Then, the cone over the master graph is the tropical surface T Z . • Main tool: “Geometric Tropicalization” (Hacking-Keel-Tevelev) M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 8 / 16
Geometric tropicalization for SURFACES: an overview • IDEA: Given β : T 2 ⊃ X ֒ → T N , compute T β ( X ) from the geometry of X . Theorem (Geometric Tropicalization [Hacking-Keel-Tevelev]) Consider T N with coordinate functions t 1 , . . . , t N , and let Y ⊂ T N be a closed smooth surface. Suppose ¯ Y ⊃ Y 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 ] := N 0 � [ D k ] : k ∈ σ � , for σ ∈ ∆ . Then, � T Y = Q ≥ 0 [ σ ] . σ ∈ ∆ M.A. Cueto et al. (UC Berkeley) Tropical Secant Graphs January 25th 2010 9 / 16
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