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Weierstrass Points on a Tropical Curve Weierstrass points on a tropical curve Harry Richman University of Michigan [email protected] University of Kentucky Algebra Seminar November 20, 2019 Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 1 / 20 What is a Weierstrass point? Definition: X a smooth algebraic curve, DN a divisor of degree N r projective embedding φ : X ! P . Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 2 / 20 What is a Weierstrass point? Definition: X a smooth algebraic curve, DN a divisor of degree N r projective embedding φ : X ! P . r W (DN )= fx 2 X : 9H ⊂ P s.t. mx (H \ X ) ≥ r + 1g \higher-than-expected" tangency with = x 2 X : some hyperplane H at x Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 2 / 20 ,#W (D)=9 What is a Weierstrass point? Definition: \higher-than-expected" tangency with W (D ) = x 2 X : N some hyperplane H at x Example: X = fxyz + x3 + y 3 + z3 = 0g ⊂ 2 PC N = 3 Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 2 / 20 ,#W (D)=9 What is a Weierstrass point? Definition: \higher-than-expected" tangency with W (D ) = x 2 X : N some hyperplane H at x Example: X = fxyz + x3 + y 3 + z3 = 0g ⊂ 2 PC N = 3 Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 2 / 20 ,#W (D)=9 What is a Weierstrass point? Definition: \higher-than-expected" tangency with W (D ) = x 2 X : N some hyperplane H at x Example: X = fxyz + x3 + y 3 + z3 = 0g ⊂ 2 PC N = 3 Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 2 / 20 What is a Weierstrass point? Definition: \higher-than-expected" tangency with W (D ) = x 2 X : N some hyperplane H at x Example: X = fxyz + x3 + y 3 + z3 = 0g ⊂ 2 PC N = 3 ,#W (D)=9 Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 2 / 20 Numerical \evidence": as N grows, 2 #(Weierstrass points of DN ) = gN + O(N) Problem How are Weierstrass points distributed on an algebraic curve? What is a Weierstrass point? Intution (Mumford): N-torsion points $ Weierstrass points of DN on an elliptic curve on a higher-genus curve Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 3 / 20 Problem How are Weierstrass points distributed on an algebraic curve? What is a Weierstrass point? Intution (Mumford): N-torsion points $ Weierstrass points of DN on an elliptic curve on a higher-genus curve Numerical \evidence": as N grows, 2 #(Weierstrass points of DN ) = gN + O(N) Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 3 / 20 What is a Weierstrass point? Intution (Mumford): N-torsion points $ Weierstrass points of DN on an elliptic curve on a higher-genus curve Numerical \evidence": as N grows, 2 #(Weierstrass points of DN ) = gN + O(N) Problem How are Weierstrass points distributed on an algebraic curve? Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 3 / 20 Weierstrass points: genus 1, complex case Problem How are Weierstrass points distributed on genus 1 curve X =C? Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 4 / 20 Weierstrass points: genus 1, complex case Problem How are Weierstrass points distributed on genus 1 curve X =C? Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 4 / 20 Weierstrass points: genus 1, complex case Problem How are Weierstrass points distributed on genus 1 curve X =C? Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 4 / 20 Weierstrass points: genus 1, complex case Problem How are Weierstrass points distributed on genus 1 curve X =C? Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 4 / 20 Weierstrass points: genus 1, complex case Problem How are Weierstrass points distributed on genus 1 curve X =C? Weierstrass points distribute uniformly, w.r.t. C ! C=Λ Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 4 / 20 Theorem (Neeman, 1984) Suppose X is a complex algebraic curve of genus g ≥ 2. Then W (DN ) distributes according to the Bergman measure as N ! 1. Weierstrass points: genus ≥ 2, complex case Problem How are Weierstrass points distributed on higher genus curve X =C? Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 5 / 20 Theorem (Neeman, 1984) Suppose X is a complex algebraic curve of genus g ≥ 2. Then W (DN ) distributes according to the Bergman measure as N ! 1. Weierstrass points: genus ≥ 2, complex case Problem How are Weierstrass points distributed on higher genus curve X =C? Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 5 / 20 Theorem (Neeman, 1984) Suppose X is a complex algebraic curve of genus g ≥ 2. Then W (DN ) distributes according to the Bergman measure as N ! 1. Weierstrass points: genus ≥ 2, complex case Problem How are Weierstrass points distributed on higher genus curve X =C? Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 5 / 20 Theorem (Neeman, 1984) Suppose X is a complex algebraic curve of genus g ≥ 2. Then W (DN ) distributes according to the Bergman measure as N ! 1. Weierstrass points: genus ≥ 2, complex case Problem How are Weierstrass points distributed on higher genus curve X =C? Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 5 / 20 Weierstrass points: genus ≥ 2, complex case Problem How are Weierstrass points distributed on higher genus curve X =C? Theorem (Neeman, 1984) Suppose X is a complex algebraic curve of genus g ≥ 2. Then W (DN ) distributes according to the Bergman measure as N ! 1. Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 5 / 20 Theorem (Amini, 2014) an Suppose X is a Berkovich curve of genus g ≥ 2. Then W (DN ) distributes according to the Zhang measure as N ! 1. Weierstrass points: non-Archimedean case Problem × How are Weierstrass points distributed on X =K, val : K ! R? Source: Matt Baker's math blog Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 6 / 20 Theorem (Amini, 2014) an Suppose X is a Berkovich curve of genus g ≥ 2. Then W (DN ) distributes according to the Zhang measure as N ! 1. Weierstrass points: non-Archimedean case Problem an How are Weierstrass points distributed on X =K X ? Source: Matt Baker's math blog Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 6 / 20 Weierstrass points: non-Archimedean case Problem an How are Weierstrass points distributed on X =K X ? Source: Matt Baker's math blog Theorem (Amini, 2014) an Suppose X is a Berkovich curve of genus g ≥ 2. Then W (DN ) distributes according to the Zhang measure as N ! 1. Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 6 / 20 Weierstrass points: non-Archimedean case Problem an How are Weierstrass points distributed on X =K X ? Source: Matt Baker's math blog Problem (Amini, 2014) Does the distribution follow from considering only the skeleton Γ ⊂ X an? Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 6 / 20 Weierstrass points: non-Archimedean case Problem an How are Weierstrass points distributed on X =K X ? Source: Matt Baker's math blog Problem (Amini, 2014) Does the distribution follow from considering only the skeleton Γ ⊂ X an? Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 6 / 20 dual graph of X0 (combinatorics)= (alg. geometry) = model for a degenerating algebraic curve What is a tropical curve? Tropical curve ( = a skeleton of X an) Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 7 / 20 dual graph of X0 (alg. geometry) = model for a degenerating algebraic curve What is a tropical curve? Tropical curve ( = a skeleton of X an) (combinatorics) = finite graph with edge lengths Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 7 / 20 dual graph of X0 What is a tropical curve? Tropical curve ( = a skeleton of X an) (combinatorics) = finite graph with edge lengths (alg. geometry) = model for a degenerating algebraic curve Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 7 / 20 dual graph of X0 What is a tropical curve? Tropical curve ( = a skeleton of X an) (combinatorics) = finite graph with edge lengths (alg. geometry) = model for a degenerating algebraic curve Example: X = fxyz − tx3 + t2y 3 + t5z3 = 0g ⊂ 2 t PC Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 7 / 20 What is a tropical curve? Tropical curve ( = a skeleton of X an) (combinatorics) = finite graph with edge lengths (alg. geometry) = model for a degenerating algebraic curve Example: X = fxyz − t1x3 + t2y 3 + t5z3 = 0g ⊂ 2 t PC dual graph of X0 Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 7 / 20 What is a tropical curve? Tropical curve ( = a skeleton of X an) (combinatorics)= finite graph with edge lengths metric graph (alg. geometry) = model for a degenerating algebraic curve Example: X = fxyz − t1x3 + t2y 3 + t5z3 = 0g ⊂ 2 t PC dual graph of X0 Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 7 / 20 Intuition: linear equivalence on Γ = \discrete current flow" jDj = fE lin. equiv. to D; E ≥ 0g q-reduced divisor redq[D] = \energy-minimizing" divisor in jDj Tropical curves: divisor theory Tropical curve = metric graph alg. curve X tropical curve Γ divisors Div(X ) divisors Div(Γ) meromorphic functions piecewise Z-linear functions linear system jDj linear system jDj r = P = polyhedral complex of dim ≥ r rank r = dim jDj rank r = Baker-Norine rank Harry Richman (U. Michigan) Tropical Weierstrass points 20 November 2019 8 / 20 q-reduced divisor redq[D] = \energy-minimizing" divisor in jDj Tropical curves: divisor theory Tropical curve = metric graph alg.
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