Lifting Tropical Curves and Linear Systems on Graphs

Transcription

1 Lifting Tropical Curves and Linear Systems on Graphs Eric Katz (Texas/MSRI) November 7, 2010 Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

2 Tropicalization Let K = C{{t}} = C((t)), the field of Puiseux series. K has non-archimedean valuation v : K Q R. Tropicalization map: Trop : {curves C (K ) n } {tropical graphs Σ = Trop(C) R n } Trop(C) is defined as v(c) R n where v : (K ) n R n is the product of valuation maps and closure is topological. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

3 Tropicalization Let K = C{{t}} = C((t)), the field of Puiseux series. K has non-archimedean valuation v : K Q R. Tropicalization map: Trop : {curves C (K ) n } {tropical graphs Σ = Trop(C) R n } Trop(C) is defined as v(c) R n where v : (K ) n R n is the product of valuation maps and closure is topological. Tropical graphs are balanced, weighted, integral graphs Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

4 Tropicalization Let K = C{{t}} = C((t)), the field of Puiseux series. K has non-archimedean valuation v : K Q R. Tropicalization map: Trop : {curves C (K ) n } {tropical graphs Σ = Trop(C) R n } Trop(C) is defined as v(c) R n where v : (K ) n R n is the product of valuation maps and closure is topological. Tropical graphs are balanced, weighted, integral graphs Integral: Each edge is a line-segment or a ray parallel to u Z n. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

5 Tropicalization Let K = C{{t}} = C((t)), the field of Puiseux series. K has non-archimedean valuation v : K Q R. Tropicalization map: Trop : {curves C (K ) n } {tropical graphs Σ = Trop(C) R n } Trop(C) is defined as v(c) R n where v : (K ) n R n is the product of valuation maps and closure is topological. Tropical graphs are balanced, weighted, integral graphs Integral: Each edge is a line-segment or a ray parallel to u Z n. Weighted: Each edge has a weight (multiplicity) m(σ) N. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

6 Tropicalization Balanced: For v, a vertex of Σ and adjacent edges E 1,...,E k in primitive Z n directions, u 1,..., u k then m(ei ) u i = 0. Example: m = 1 m = 2 m = 1 Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

7 An Elliptic Curve in the Plane All multiplicities are 1. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

8 An Elliptic Curve in Space All multiplicities are 1. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

9 Statement of Lifting Problem Lifting Problem: Which tropical graphs are tropicalizations of curves? Today: necessary conditions. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

10 Statement of Lifting Problem Lifting Problem: Which tropical graphs are tropicalizations of curves? Today: necessary conditions. Speyer: Elliptic Curves, necessary and sufficient conditions in genus 1. Nishinou and Brugallé-Mikhalkin: Generalization of Speyer s result in one-bouquet case. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

11 Why? There are tropical curves that are not tropicalizations, telling the difference is subtle. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

12 Why? There are tropical curves that are not tropicalizations, telling the difference is subtle. The problem is combinatorial, but what kind of combinatorics even encodes this? Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

13 Why? There are tropical curves that are not tropicalizations, telling the difference is subtle. The problem is combinatorial, but what kind of combinatorics even encodes this? Closely tied to deformation theory which is often grungy, maybe there s a combinatorial approach. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

14 Example of non-liftable curve Change the length of a bounded edge in the spatial elliptic curve so that it does not lie on the tropicalization of any plane (possible by dimension counting). Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

15 Example of non-liftable curve (cont d) This is not liftable to a curve over K because Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

16 Example of non-liftable curve (cont d) This is not liftable to a curve over K because 1 three unbounded edges in each direction in the curve shows that it must be a cubic, Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

17 Example of non-liftable curve (cont d) This is not liftable to a curve over K because 1 three unbounded edges in each direction in the curve shows that it must be a cubic, 2 the loop in the curve shows that any lift must have genus at least 1, Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

18 Example of non-liftable curve (cont d) This is not liftable to a curve over K because 1 three unbounded edges in each direction in the curve shows that it must be a cubic, 2 the loop in the curve shows that any lift must have genus at least 1, 3 any classical cubic is either genus 0 and spatial or genus 1 and planar, Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

19 Example of non-liftable curve (cont d) This is not liftable to a curve over K because 1 three unbounded edges in each direction in the curve shows that it must be a cubic, 2 the loop in the curve shows that any lift must have genus at least 1, 3 any classical cubic is either genus 0 and spatial or genus 1 and planar, no lift of the curve can be planar or genus 0, so the curve does not lift. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

20 Aside for number theorists Instead of considering C (K ) n, we may consider C A where A is an Abelian variety. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

21 Aside for number theorists Instead of considering C (K ) n, we may consider C A where A is an Abelian variety. If C is a curve over K with maximally degenerate reduction, way consider the Abel-Jacobi map, i : C J(C). By uniformization, J(C) = (K ) n /Λ. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

22 Aside for number theorists Instead of considering C (K ) n, we may consider C A where A is an Abelian variety. If C is a curve over K with maximally degenerate reduction, way consider the Abel-Jacobi map, i : C J(C). By uniformization, J(C) = (K ) n /Λ. We can tropicalize this to get Trop(i) : Σ R n /v(λ). Here R n /v(λ) is a real torus. Our lifting method involves the combinatorics of pulling back 1-forms from ambient space to C and restricting to closed fiber of semistable model. More later! Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

23 Parameterized Tropical Graphs A tropical parameterization of a tropical graph Σ is a map p : Σ Σ (maps vertices to vertices but may contract edges) such that Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

24 Parameterized Tropical Graphs A tropical parameterization of a tropical graph Σ is a map p : Σ Σ (maps vertices to vertices but may contract edges) such that 1 Σ is a tropical graph (balanced where each edge is given the direction of its image), Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

25 Parameterized Tropical Graphs A tropical parameterization of a tropical graph Σ is a map p : Σ Σ (maps vertices to vertices but may contract edges) such that 1 Σ is a tropical graph (balanced where each edge is given the direction of its image), 2 Ẽ p 1 (E) m(ẽ) = m(e). Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

26 Parameterized Tropical Graphs A tropical parameterization of a tropical graph Σ is a map p : Σ Σ (maps vertices to vertices but may contract edges) such that 1 Σ is a tropical graph (balanced where each edge is given the direction of its image), 2 Ẽ p 1 (E) m(ẽ) = m(e). Note: If all the multiplicities of Σ are 1 and all vertices are trivalent, then the only parameterization of Σ is the identity. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

27 Specialization of line bundles Quick Review of Baker s Specialization: Let C be a curve over K. C is defined over some field C((t)) after possibly replacing t 1 N by t. Pick a regular semistable model C over O = C[[t]]. Let Σ be the dual graph of C 0. For v V (Σ), let C v be the corresponding component. For E, a bounded edge, let p E be the corresponding node. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

28 Specialization of line bundles Quick Review of Baker s Specialization: Let C be a curve over K. C is defined over some field C((t)) after possibly replacing t 1 N by t. Pick a regular semistable model C over O = C[[t]]. Let Σ be the dual graph of C 0. For v V (Σ), let C v be the corresponding component. For E, a bounded edge, let p E be the corresponding node. A divisor on Σ is a Z-combination of points of Σ. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

29 Specialization of line bundles Quick Review of Baker s Specialization: Let C be a curve over K. C is defined over some field C((t)) after possibly replacing t 1 N by t. Pick a regular semistable model C over O = C[[t]]. Let Σ be the dual graph of C 0. For v V (Σ), let C v be the corresponding component. For E, a bounded edge, let p E be the corresponding node. A divisor on Σ is a Z-combination of points of Σ. Let L be a line bundle on C. The divisor associated to L is Λ = v deg(l Cv )(v). Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

30 Specialization of sections If s Γ(C, L), write (s) = D + v (v)c v where D is a horizontal divisor. is called the vanishing functions (order of vanishing of s on generic points of components of central fiber). Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

31 Specialization of sections If s Γ(C, L), write (s) = D + v (v)c v where D is a horizontal divisor. is called the vanishing functions (order of vanishing of s on generic points of components of central fiber). D = (s) v (v)c v so for w V (Σ), (since D intersects C w properly) 0 D C w = (s) C w ( (w) (v)) E=vw Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

32 Specialization of sections If s Γ(C, L), write (s) = D + v (v)c v where D is a horizontal divisor. is called the vanishing functions (order of vanishing of s on generic points of components of central fiber). D = (s) v (v)c v so for w V (Σ), (since D intersects C w properly) 0 D C w = (s) C w ( (w) (v)) E=vw which can be expressed as 0 Λ(w) + (w) Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

33 Specialization of sections If s Γ(C, L), write (s) = D + v (v)c v where D is a horizontal divisor. is called the vanishing functions (order of vanishing of s on generic points of components of central fiber). D = (s) v (v)c v so for w V (Σ), (since D intersects C w properly) 0 D C w = (s) C w ( (w) (v)) E=vw which can be expressed as 0 Λ(w) + (w) or as L(Λ). Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

34 Canonical divisor Σ has canonical divisor: K Σ = v (2g(v) 2 + deg(v)) Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

35 Canonical divisor Σ has canonical divisor: K Σ = v (2g(v) 2 + deg(v)) We will need to make sense of the condition that L(K Σ ). Interpolate as a piecewise-linear function on Σ. Assumption:g(v) = 0. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

36 Canonical divisor Σ has canonical divisor: K Σ = v (2g(v) 2 + deg(v)) We will need to make sense of the condition that L(K Σ ). Interpolate as a piecewise-linear function on Σ. Assumption:g(v) = 0. if v Σ, E v, write s(v,e) for the slope of on E coming from v. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

37 Canonical divisor Σ has canonical divisor: K Σ = v (2g(v) 2 + deg(v)) We will need to make sense of the condition that L(K Σ ). Interpolate as a piecewise-linear function on Σ. Assumption:g(v) = 0. if v Σ, E v, write s(v,e) for the slope of on E coming from v. The Laplacian of ϕ is ( )(v) = E v s(v,e) Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

38 Canonical divisor Σ has canonical divisor: K Σ = v (2g(v) 2 + deg(v)) We will need to make sense of the condition that L(K Σ ). Interpolate as a piecewise-linear function on Σ. Assumption:g(v) = 0. if v Σ, E v, write s(v,e) for the slope of on E coming from v. The Laplacian of ϕ is ( )(v) = E v s(v,e) Note: For L(K Σ ), at bivalent points v, K Σ (v) = 0 and the condition ( )(v) = ( )(v) + K Σ (v) 0 means that slopes only decrease. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

39 Main Theorem Theorem: If Σ R n is a tropicalization then there exists p : Σ Σ and for all m Z n, there is a piecewise-linear function ϕ m : Σ l R 0 ( Σ l is the l-fold subdivision of Σ) with Z-slopes such that Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

40 Main Theorem Theorem: If Σ R n is a tropicalization then there exists p : Σ Σ and for all m Z n, there is a piecewise-linear function ϕ m : Σ l R 0 ( Σ l is the l-fold subdivision of Σ) with Z-slopes such that 1 ϕ m L(K Σl ), Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

41 Main Theorem Theorem: If Σ R n is a tropicalization then there exists p : Σ Σ and for all m Z n, there is a piecewise-linear function ϕ m : Σ l R 0 ( Σ l is the l-fold subdivision of Σ) with Z-slopes such that 1 ϕ m L(K Σl ), 2 ϕ m = 0 on E with m E 0, Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

42 Main Theorem Theorem: If Σ R n is a tropicalization then there exists p : Σ Σ and for all m Z n, there is a piecewise-linear function ϕ m : Σ l R 0 ( Σ l is the l-fold subdivision of Σ) with Z-slopes such that 1 ϕ m L(K Σl ), 2 ϕ m = 0 on E with m E 0, 3 ϕ m never has slope 0 on edges E with m E = 0, Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

43 Main Theorem Theorem: If Σ R n is a tropicalization then there exists p : Σ Σ and for all m Z n, there is a piecewise-linear function ϕ m : Σ l R 0 ( Σ l is the l-fold subdivision of Σ) with Z-slopes such that 1 ϕ m L(K Σl ), 2 ϕ m = 0 on E with m E 0, 3 ϕ m never has slope 0 on edges E with m E = 0, 4 ϕ m obeys the cycle-ampleness condition. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

44 Cycle-Ampleness Condition Let H be a hyperplane given by H = {x x m = c}. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

45 Cycle-Ampleness Condition Let H be a hyperplane given by H = {x x m = c}. Let Γ be a cycle in the interior of p 1 (H) Σ. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

46 Cycle-Ampleness Condition Let H be a hyperplane given by H = {x x m = c}. Let Γ be a cycle in the interior of p 1 (H) Σ. Set h = min v Γ (ϕ m (v)) then, Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

47 Cycle-Ampleness Condition Let H be a hyperplane given by H = {x x m = c}. Let Γ be a cycle in the interior of p 1 (H) Σ. Set h = min v Γ (ϕ m (v)) then, D ϕm v Γ ϕ m(v)=h E Γ s(v,e)<0 ( s(v,e)) 2. sum of positive slopes coming into the cycle at min s of ϕ m must be at least 2. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

48 Sections of Canonical Bundle Before we use these conditions, we need the following observation: ϕ m L(K Σl ) translates into (ϕ m )(v) = E v s(v,e) 2 deg(v). Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

49 Sections of Canonical Bundle Before we use these conditions, we need the following observation: ϕ m L(K Σl ) translates into (ϕ m )(v) = E v s(v,e) 2 deg(v). If v Γ is a vertex with edges E 1,...,E k,f 1,...,F l (partitioned in any way). By hypothesis s(v,e i ),s(v,f j ) 0. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

50 Sections of Canonical Bundle Before we use these conditions, we need the following observation: ϕ m L(K Σl ) translates into (ϕ m )(v) = E v s(v,e) 2 deg(v). If v Γ is a vertex with edges E 1,...,E k,f 1,...,F l (partitioned in any way). By hypothesis s(v,e i ),s(v,f j ) 0. ( s(v,fj ) s(v,ei )) + (deg(v) 2)) At v, sum of outgoing slope along edges F j is less than sum of incoming slopes along edges E i plus (deg(v) 2). Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

51 Elliptic Curve Example Note: This is p 1 (H) where H is the plane of the screen. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

52 Elliptic Curve Example (cont d) 1 Direct edges towards cycle. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

53 Elliptic Curve Example (cont d) 1 Direct edges towards cycle. 2 ϕ m must be decreasing on unbounded edges. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

54 Elliptic Curve Example (cont d) 1 Direct edges towards cycle. 2 ϕ m must be decreasing on unbounded edges. 3 ϕ m is equal to 0 on (p 1 (H)) and has slope at most 1 there. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

55 Elliptic Curve Example (cont d) 1 Direct edges towards cycle. 2 ϕ m must be decreasing on unbounded edges. 3 ϕ m is equal to 0 on (p 1 (H)) and has slope at most 1 there. 4 Slopes of ϕ m only decrease along edge as we move towards cycle. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

56 Elliptic Curve Example (cont d) 1 Direct edges towards cycle. 2 ϕ m must be decreasing on unbounded edges. 3 ϕ m is equal to 0 on (p 1 (H)) and has slope at most 1 there. 4 Slopes of ϕ m only decrease along edge as we move towards cycle. 5 Slope of ϕ m is at most 1 as it turns the corner and heads to cycle. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

57 Elliptic Curve Example (cont d) 1 Direct edges towards cycle. 2 ϕ m must be decreasing on unbounded edges. 3 ϕ m is equal to 0 on (p 1 (H)) and has slope at most 1 there. 4 Slopes of ϕ m only decrease along edge as we move towards cycle. 5 Slope of ϕ m is at most 1 as it turns the corner and heads to cycle. 6 There is positive incoming slope at 3 points on the cycle. At those points, ϕ m is equal to distance to (p 1 (H)) Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

58 Elliptic Curve Example (cont d) 1 Direct edges towards cycle. 2 ϕ m must be decreasing on unbounded edges. 3 ϕ m is equal to 0 on (p 1 (H)) and has slope at most 1 there. 4 Slopes of ϕ m only decrease along edge as we move towards cycle. 5 Slope of ϕ m is at most 1 as it turns the corner and heads to cycle. 6 There is positive incoming slope at 3 points on the cycle. At those points, ϕ m is equal to distance to (p 1 (H)) 7 For deg(d ϕm ) 2, the minimum distance must be achieved at least twice. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

59 Elliptic Curve Example (concluded) In summary, minimum distance from Γ to Σ \ p 1 (H) must be achieved by two paths. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

60 Elliptic Curve Example (concluded) In summary, minimum distance from Γ to Σ \ p 1 (H) must be achieved by two paths. This is Speyer s well-spacedness condition! Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

61 Elliptic Curve Example (concluded) In summary, minimum distance from Γ to Σ \ p 1 (H) must be achieved by two paths. This is Speyer s well-spacedness condition! Also get generalization to higher genus as given by Nishinou and Brugallé-Mikhalkin. This requires strong conditions on combinatorics of Σ. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

62 A New Example a b c d Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

63 A New Example a b c d Embed in the plane so that it is balanced in the plane. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

64 A New Example a b c d Embed in the plane so that it is balanced in the plane. Add unbounded edges pointing out of the plane to ensure that is globally balanced. Give every edge multiplicity 1. Can ensure that only parameterization is the identity. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

65 A New Example a b c d Embed in the plane so that it is balanced in the plane. Add unbounded edges pointing out of the plane to ensure that is globally balanced. Give every edge multiplicity 1. Can ensure that only parameterization is the identity. There does not exist the desired ϕ m, so it does not lift. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

66 A New Example (cont d) 1 Direct edges towards cycle. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

67 A New Example (cont d) 1 Direct edges towards cycle. 2 ϕ m is equal to 0 on (p 1 (H)) and has slope at most 1 there. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

68 A New Example (cont d) 1 Direct edges towards cycle. 2 ϕ m is equal to 0 on (p 1 (H)) and has slope at most 1 there. 3 Slopes of ϕ m only decrease along edge as we move towards cycle. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

69 A New Example (cont d) 1 Direct edges towards cycle. 2 ϕ m is equal to 0 on (p 1 (H)) and has slope at most 1 there. 3 Slopes of ϕ m only decrease along edge as we move towards cycle. 4 Slope on edge a is at most 3. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

70 A New Example (cont d) 1 Direct edges towards cycle. 2 ϕ m is equal to 0 on (p 1 (H)) and has slope at most 1 there. 3 Slopes of ϕ m only decrease along edge as we move towards cycle. 4 Slope on edge a is at most 3. 5 Slopes on edges b,c,d sum to at most 5, so they contribute at most one point to D ϕm on one of the cycles. 6 Long edges are too long for ϕ m to have positive slope and to also intersect a cycle in a minimum of ϕ m. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

71 A New Example (cont d) 1 Direct edges towards cycle. 2 ϕ m is equal to 0 on (p 1 (H)) and has slope at most 1 there. 3 Slopes of ϕ m only decrease along edge as we move towards cycle. 4 Slope on edge a is at most 3. 5 Slopes on edges b,c,d sum to at most 5, so they contribute at most one point to D ϕm on one of the cycles. 6 Long edges are too long for ϕ m to have positive slope and to also intersect a cycle in a minimum of ϕ m. 7 deg(d ϕm ) 1 on one cycle. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

72 Weak well-spacedness condition There s a weak version of Speyer s condition in higher genera that holds for curves of complicated combinatorial type. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

73 Weak well-spacedness condition There s a weak version of Speyer s condition in higher genera that holds for curves of complicated combinatorial type. Theorem: Let H be a hyperplane with normal vector m Z n. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

74 Weak well-spacedness condition There s a weak version of Speyer s condition in higher genera that holds for curves of complicated combinatorial type. Theorem: Let H be a hyperplane with normal vector m Z n. Let Γ be a component of p 1 (H) Σ with h 1 (Γ ) > 0. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

75 Weak well-spacedness condition There s a weak version of Speyer s condition in higher genera that holds for curves of complicated combinatorial type. Theorem: Let H be a hyperplane with normal vector m Z n. Let Γ be a component of p 1 (H) Σ with h 1 (Γ ) > 0. Then Γ does not consist of a single trivalent vertex of Σ. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

76 Observation: controls poles of sections! Let O be the valuation ring of K = C((t)), suppose everything over K. C is a family of curves over O extending C, Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

77 Observation: controls poles of sections! Let O be the valuation ring of K = C((t)), suppose everything over K. C is a family of curves over O extending C, let Σ be its log dual graph (infinite ends correspond to marked points), Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

78 Observation: controls poles of sections! Let O be the valuation ring of K = C((t)), suppose everything over K. C is a family of curves over O extending C, let Σ be its log dual graph (infinite ends correspond to marked points), let L be a line bundle on C and s, a section, Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

79 Observation: controls poles of sections! Let O be the valuation ring of K = C((t)), suppose everything over K. C is a family of curves over O extending C, let Σ be its log dual graph (infinite ends correspond to marked points), let L be a line bundle on C and s, a section, after base-extension, the zeroes of s K on C K consist of K-rational points and therefore, specialize to smooth points on the special fiber, Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

80 Observation: controls poles of sections! Let O be the valuation ring of K = C((t)), suppose everything over K. C is a family of curves over O extending C, let Σ be its log dual graph (infinite ends correspond to marked points), let L be a line bundle on C and s, a section, after base-extension, the zeroes of s K on C K consist of K-rational points and therefore, specialize to smooth points on the special fiber, let be the vanishing function of s, Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

81 Observation: controls poles of sections! Let O be the valuation ring of K = C((t)), suppose everything over K. C is a family of curves over O extending C, let Σ be its log dual graph (infinite ends correspond to marked points), let L be a line bundle on C and s, a section, after base-extension, the zeroes of s K on C K consist of K-rational points and therefore, specialize to smooth points on the special fiber, let be the vanishing function of s, for v V (Σ), let s v = s t (v) Cv, a nonvanishing rational function on C v Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

82 Observation: controls poles of sections! Let O be the valuation ring of K = C((t)), suppose everything over K. C is a family of curves over O extending C, let Σ be its log dual graph (infinite ends correspond to marked points), let L be a line bundle on C and s, a section, after base-extension, the zeroes of s K on C K consist of K-rational points and therefore, specialize to smooth points on the special fiber, let be the vanishing function of s, for v V (Σ), let s v = s t (v) Cv, a nonvanishing rational function on C v If E = vw is an edge, then s vanishes generically to order (v) on C v and to order (w) on C v. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

83 Observation: controls poles of sections! Let O be the valuation ring of K = C((t)), suppose everything over K. C is a family of curves over O extending C, let Σ be its log dual graph (infinite ends correspond to marked points), let L be a line bundle on C and s, a section, after base-extension, the zeroes of s K on C K consist of K-rational points and therefore, specialize to smooth points on the special fiber, let be the vanishing function of s, for v V (Σ), let s v = s t (v) Cv, a nonvanishing rational function on C v If E = vw is an edge, then s vanishes generically to order (v) on C v and to order (w) on C v. Consequently if p e is the node between C v and C w then ord pe (s v ) = (w) (v). Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

84 Outline of Proof Try to extend relevant objects over O. (Nishinou-Siebert stable reduction theorem) Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

85 Outline of Proof Try to extend relevant objects over O. (Nishinou-Siebert stable reduction theorem) 1 C extends to a regular semistable model C. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

86 Outline of Proof Try to extend relevant objects over O. (Nishinou-Siebert stable reduction theorem) 1 C extends to a regular semistable model C. 2 (K ) n extends to a toric scheme P. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

87 Outline of Proof Try to extend relevant objects over O. (Nishinou-Siebert stable reduction theorem) 1 C extends to a regular semistable model C. 2 (K ) n extends to a toric scheme P. 3 C (K ) n extends to a stable map f : C P. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

88 Outline of Proof Try to extend relevant objects over O. (Nishinou-Siebert stable reduction theorem) 1 C extends to a regular semistable model C. 2 (K ) n extends to a toric scheme P. 3 C (K ) n extends to a stable map f : C P. 4 C and P have log structures (magic dust) giving them locally free log cotangent sheaves Ω 1 C /O,Ω 1 P /O. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

89 Outline of Proof (cont d) Log structures are particularly nice: 1 Ω 1 C /O is relative dualizing sheaf twisted by points mapped to toric boundary. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

90 Outline of Proof (cont d) Log structures are particularly nice: 1 Ω 1 C /O is relative dualizing sheaf twisted by points mapped to toric boundary. 2 Ω 1 is trivial bundle. Sections of the form dzm P /O z m character of (K ) n. where z m is a Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

91 Outline of Proof (cont d) Log structures are particularly nice: 1 Ω 1 C /O is relative dualizing sheaf twisted by points mapped to toric boundary. 2 Ω 1 is trivial bundle. Sections of the form dzm P /O z m character of (K ) n. 3 f is a log morphism. ω m = f dzm Ω 1 C /O. z m where z m is a is a section of log cotangent bundle Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

92 Outline of Proof (cont d) Let Σ be the dual graph of C 0. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

93 Outline of Proof (cont d) Let Σ be the dual graph of C 0. The cotangent sheaf Ω 1 C /O specializes to K Σ. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

94 Outline of Proof (cont d) Let Σ be the dual graph of C 0. The cotangent sheaf Ω 1 C /O specializes to K Σ. Specializing ω m (almost) gives ϕ m L(K Σ ). Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

95 Outline of Proof (cont d) Cycle-ampleness condition is the only hard part. Look at ω m in a formal neighborhood U Σ of components of C 0 corresponding to cycle Σ. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

96 Outline of Proof (cont d) Cycle-ampleness condition is the only hard part. Look at ω m in a formal neighborhood U Σ of components of C 0 corresponding to cycle Σ. Since Γ is mapped into a hyperplane orthogonal to m, t c f z m is a regular function on U Σ. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

97 Outline of Proof (cont d) Cycle-ampleness condition is the only hard part. Look at ω m in a formal neighborhood U Σ of components of C 0 corresponding to cycle Σ. Since Γ is mapped into a hyperplane orthogonal to m, t c f z m is a regular function on U Σ. If h = min v Σ (ϕ m (v)), ω m = t c d(f z m ) vanishes to order h. t c f z m Consequently, t c f z m is equal to a constant L up to order h. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

98 Outline of Proof (cont d) Cycle-ampleness condition is the only hard part. Look at ω m in a formal neighborhood U Σ of components of C 0 corresponding to cycle Σ. Since Γ is mapped into a hyperplane orthogonal to m, t c f z m is a regular function on U Σ. If h = min v Σ (ϕ m (v)), ω m = t c d(f z m ) vanishes to order h. t c f z m Consequently, t c f z m is equal to a constant L up to order h. ω m is equal to an exact 1-form up to low orders: let s = f (t c z m ) L t h. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

99 Outline of Proof (cont d) Cycle-ampleness condition is the only hard part. Look at ω m in a formal neighborhood U Σ of components of C 0 corresponding to cycle Σ. Since Γ is mapped into a hyperplane orthogonal to m, t c f z m is a regular function on U Σ. If h = min v Σ (ϕ m (v)), ω m = t c d(f z m ) vanishes to order h. t c f z m Consequently, t c f z m is equal to a constant L up to order h. ω m is equal to an exact 1-form up to low orders: let s = f (t c z m ) L t h. ω m t h = ds t c which is close to being an exact form. zm Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

100 Outline of Proof (cont d) Cycle-ampleness condition is the only hard part. Look at ω m in a formal neighborhood U Σ of components of C 0 corresponding to cycle Σ. Since Γ is mapped into a hyperplane orthogonal to m, t c f z m is a regular function on U Σ. If h = min v Σ (ϕ m (v)), ω m = t c d(f z m ) vanishes to order h. t c f z m Consequently, t c f z m is equal to a constant L up to order h. ω m is equal to an exact 1-form up to low orders: let s = f (t c z m ) L t h. ω m t h = ds t c which is close to being an exact form. zm Upshot: at components of the central fiber C v with ϕ m (v) = h, s Cv and ω m t h have the same poles (where ωm is considered as a log 1-form)! t h Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

101 Outline of Proof (concluded) The components of C 0 corresponding to Σ are a cycle, (C 0 ) Σ of rational curves. This is a degenerate elliptic curve. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

102 Outline of Proof (concluded) The components of C 0 corresponding to Σ are a cycle, (C 0 ) Σ of rational curves. This is a degenerate elliptic curve. Punchline: A rational function on an elliptic curve needs at least two poles. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

103 Outline of Proof (concluded) The components of C 0 corresponding to Σ are a cycle, (C 0 ) Σ of rational curves. This is a degenerate elliptic curve. Punchline: A rational function on an elliptic curve needs at least two poles. D ϕm picks up the poles of s (C0 ) Σ. This immediately gives cycle-ampleness condition. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

104 Asides and Future Directions 1 This method is a combinatorial approach to deformation theory. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

105 Asides and Future Directions 1 This method is a combinatorial approach to deformation theory. 2 Gives an additional combinatorial structure on tropicalizations of curves. Higher dimensions? Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

106 Asides and Future Directions 1 This method is a combinatorial approach to deformation theory. 2 Gives an additional combinatorial structure on tropicalizations of curves. Higher dimensions? 3 Method works in finite residue characteristic as long as you exclude wild phenomena. Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

107 Asides and Future Directions 1 This method is a combinatorial approach to deformation theory. 2 Gives an additional combinatorial structure on tropicalizations of curves. Higher dimensions? 3 Method works in finite residue characteristic as long as you exclude wild phenomena. 4 Abelian varieties also have lots of 1-forms. What does 1-form in effective Chabauty give? Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31

108 Thanks! Lifting Tropical Curves in Space and Linear Systems on Graphs, arxiv: Speyer, David. Uniformizing Tropical Curves I: Genus Zero and One, arxiv: Nishinou, Takeo. Correspondence Theorems for Tropical Curves, arxiv: Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, / 31