Tropicalization The Cartesian product ( K ∗ ) n is called an algebraic torus. (In complex case, ( C ∗ ) n is the natural analog of ( S 1 ) n .) An algebraic variety in ( K ∗ ) n is the common zero locus of a system of Laurent polynomials in n variables with coefficients in K . Tropicalization is a procedure that takes subvarieties of an algebraic torus to polyhedral complexes. The tropicalization of a variety X ⊂ ( K ∗ ) n is defined to be Trop( X ) = v ( X ) ⊂ R n where the closure is topological. Question: Why is this even reasonable? Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 8 / 34
Tropicalization of a line Let f ( x , y ) = x + y + 1. Let X = V ( f ), the classical zero-locus of f . What is the tropicalization of X ? Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 9 / 34
Tropicalization of a line Let f ( x , y ) = x + y + 1. Let X = V ( f ), the classical zero-locus of f . What is the tropicalization of X ? For x + y + 1 = 0, the coefficient of the lowest power of t must be 0. Say that power is t r . Now, where can that lowest power come from? Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 9 / 34
Tropicalization of a line Let f ( x , y ) = x + y + 1. Let X = V ( f ), the classical zero-locus of f . What is the tropicalization of X ? For x + y + 1 = 0, the coefficient of the lowest power of t must be 0. Say that power is t r . Now, where can that lowest power come from? If it comes from x = at r + . . . then the coefficient of t r in x must be cancelled by the coefficient of lowest power in y or in 1. So, if it comes only from y then y = ( − a ) t r + . . . and we have v ( x ) = v ( y ) < v (1) Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 9 / 34
Tropicalization of a line Let f ( x , y ) = x + y + 1. Let X = V ( f ), the classical zero-locus of f . What is the tropicalization of X ? For x + y + 1 = 0, the coefficient of the lowest power of t must be 0. Say that power is t r . Now, where can that lowest power come from? If it comes from x = at r + . . . then the coefficient of t r in x must be cancelled by the coefficient of lowest power in y or in 1. So, if it comes only from y then y = ( − a ) t r + . . . and we have v ( x ) = v ( y ) < v (1) In general, must have the minimum of { v ( x ) , v ( y ) , v (1) = 0 } be achieved at least twice. So tropicalization must be contained in Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 9 / 34
Tropicalization of a line Let f ( x , y ) = x + y + 1. Let X = V ( f ), the classical zero-locus of f . What is the tropicalization of X ? For x + y + 1 = 0, the coefficient of the lowest power of t must be 0. Say that power is t r . Now, where can that lowest power come from? If it comes from x = at r + . . . then the coefficient of t r in x must be cancelled by the coefficient of lowest power in y or in 1. So, if it comes only from y then y = ( − a ) t r + . . . and we have v ( x ) = v ( y ) < v (1) In general, must have the minimum of { v ( x ) , v ( y ) , v (1) = 0 } be achieved at least twice. So tropicalization must be contained in and, in fact, is equal by a theorem due to Kapranov. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 9 / 34
Kapranov’s theorem Theorem (Kapranov) If f is a Laurent polynomial in x 1 , . . . , x n with support set A ⊂ Z n , � a ω x ω f = ω ∈A � v ( a ω ) ⊙ x ⊙ ω . trop( f ) = ω ∈A Let Z ( f ) ⊂ ( K ∗ ) n be the zero-locus of f . Then Trop( Z ( f )) is equal to the tropical zero-locus of trop( f ). Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 10 / 34
Kapranov’s theorem Theorem (Kapranov) If f is a Laurent polynomial in x 1 , . . . , x n with support set A ⊂ Z n , � a ω x ω f = ω ∈A � v ( a ω ) ⊙ x ⊙ ω . trop( f ) = ω ∈A Let Z ( f ) ⊂ ( K ∗ ) n be the zero-locus of f . Then Trop( Z ( f )) is equal to the tropical zero-locus of trop( f ). So the valuation definition generalizes the min-plus definition in the case of hypersurfaces. This lets you talk about the tropicalization of higher codimensional subvarieties. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 10 / 34
Tropicalization of curves Tropicalization map: Trop : { curves C ⊂ ( K ∗ ) n } → { tropical graphs Σ = Trop( C ) ⊂ R n } Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 11 / 34
Tropicalization of curves Tropicalization map: Trop : { curves C ⊂ ( K ∗ ) n } → { tropical graphs Σ = Trop( C ) ⊂ R n } Tropical graphs are balanced, weighted, integral graphs Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 11 / 34
Tropicalization of curves Tropicalization map: Trop : { curves C ⊂ ( K ∗ ) n } → { tropical graphs Σ = Trop( C ) ⊂ R n } Tropical graphs are balanced, weighted, integral graphs u ∈ Z n . Integral: Each edge is a line-segment or a ray parallel to � Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 11 / 34
Tropicalization of curves Tropicalization map: Trop : { curves C ⊂ ( K ∗ ) n } → { tropical graphs Σ = Trop( C ) ⊂ R n } Tropical graphs are balanced, weighted, integral graphs u ∈ Z n . Integral: Each edge is a line-segment or a ray parallel to � Weighted: Each edge has a weight (multiplicity) m ( E ) ∈ N . Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 11 / 34
Tropicalization Balanced: For v , a vertex of Σ and adjacent edges E 1 , . . . , E k in primitive Z n directions, � u k then u 1 , . . . ,� � u i = � m ( E i ) � 0 . Example: m = 1 m = 2 m = 1 Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 12 / 34
An elliptic curve in the plane All multiplicities are 1. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 13 / 34
An elliptic curve in space All multiplicities are 1. Note that the cycle in the graph is contained in the plane of the screen. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 14 / 34
More generally... Tropicalizations of general subvarieties are balanced, weighted, integral polyhedral complexes (by results of Bieri-Groves and Speyer). Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 15 / 34
More generally... Tropicalizations of general subvarieties are balanced, weighted, integral polyhedral complexes (by results of Bieri-Groves and Speyer). Can think of varieties in ( K ∗ ) n as families. Their coefficients are formal 1 N )). Set Puiseux series and so are formal Laurent series in some C (( t 1 N . u = t Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 15 / 34
More generally... Tropicalizations of general subvarieties are balanced, weighted, integral polyhedral complexes (by results of Bieri-Groves and Speyer). Can think of varieties in ( K ∗ ) n as families. Their coefficients are formal 1 N )). Set Puiseux series and so are formal Laurent series in some C (( t 1 N . u = t Ignoring issues of convergence, if we fix a particular value of u , we get a variety in ( C ∗ ) n . So by including all values of u in a punctured neighborhood of u = 0, we get a family of varieties in ( C ∗ ) n over a punctured disc. So in a certain sense we are tropicalizing a family of varieties. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 15 / 34
Natural questions Q: What does Trop( X ) know about X ? Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 16 / 34
Natural questions Q: What does Trop( X ) know about X ? A: Some intersection theory, some topology of X , some of the Hodge theory of X by K., Sturmfels-Tevelev, Hacking, Helm-K., K.-Stapledon, Osserman-Payne. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 16 / 34
Natural questions Q: What does Trop( X ) know about X ? A: Some intersection theory, some topology of X , some of the Hodge theory of X by K., Sturmfels-Tevelev, Hacking, Helm-K., K.-Stapledon, Osserman-Payne. Q: How are tropicalizations special among balanced weighted integral polyhedral complexes? Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 16 / 34
Natural questions Q: What does Trop( X ) know about X ? A: Some intersection theory, some topology of X , some of the Hodge theory of X by K., Sturmfels-Tevelev, Hacking, Helm-K., K.-Stapledon, Osserman-Payne. Q: How are tropicalizations special among balanced weighted integral polyhedral complexes? A: Today’s talk. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 16 / 34
Statement of lifting problem for curves Lifting Problem: Which tropical (that is, balanced, weighted, integral) graphs are tropicalizations of curves? Today: necessary conditions. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 17 / 34
Statement of lifting problem for curves Lifting Problem: Which tropical (that is, balanced, weighted, integral) graphs are tropicalizations of curves? Today: necessary conditions. Speyer: Elliptic Curves, necessary and sufficient conditions in genus 1. Nishinou and Brugall´ e-Mikhalkin: Generalization of Speyer’s result in one-bouquet case. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 17 / 34
Statement of lifting problem for curves Lifting Problem: Which tropical (that is, balanced, weighted, integral) graphs are tropicalizations of curves? Today: necessary conditions. Speyer: Elliptic Curves, necessary and sufficient conditions in genus 1. Nishinou and Brugall´ e-Mikhalkin: Generalization of Speyer’s result in one-bouquet case. The condition we’ll talk about today implies the necessity of these previously known conditions. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 17 / 34
Why? There are tropical curves that are not tropicalizations, telling the difference is subtle. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 18 / 34
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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 18 / 34
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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 18 / 34
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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 19 / 34
Example of non-liftable curve (cont’d) This is not liftable to a curve over K because Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 20 / 34
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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 20 / 34
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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 20 / 34
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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 20 / 34
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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 20 / 34
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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 21 / 34
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 ˜ Σ is a tropical graph (balanced where each edge is given the direction 1 of its image), Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 21 / 34
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 ˜ Σ is a tropical graph (balanced where each edge is given the direction 1 of its image), 2 � m (˜ ˜ E ) = m ( E ) . ˜ E ∈ p − 1 ( E ) Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 21 / 34
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 ˜ Σ is a tropical graph (balanced where each edge is given the direction 1 of its image), 2 � m (˜ ˜ E ) = m ( E ) . ˜ E ∈ p − 1 ( E ) Note: If all the multiplicities of Σ are 1 and all vertices are trivalent, then the only parameterization of Σ is the identity. In fact, the only parameterization used in explicit examples will be the identity. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 21 / 34
Linear Systems on Graphs (following Baker-Norine) If ̟ is a piecewise-linear function on ˜ Σ (linear on all edges), Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 22 / 34
Linear Systems on Graphs (following Baker-Norine) If ̟ is a piecewise-linear function on ˜ Σ (linear on all edges), if v ∈ ˜ Σ, E ∋ v , write s ( v , E ) for the slope of ̟ on E coming from v . Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 22 / 34
Linear Systems on Graphs (following Baker-Norine) If ̟ is a piecewise-linear function on ˜ Σ (linear on all edges), if v ∈ ˜ Σ, E ∋ v , write s ( v , E ) for the slope of ̟ on E coming from v . Define the Laplacian of ̟ by � ∆( ̟ )( v ) = − s ( v , E ) E ∋ v Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 22 / 34
Linear Systems on Graphs (following Baker-Norine) If ̟ is a piecewise-linear function on ˜ Σ (linear on all edges), if v ∈ ˜ Σ, E ∋ v , write s ( v , E ) for the slope of ̟ on E coming from v . Define the Laplacian of ̟ by � ∆( ̟ )( v ) = − s ( v , E ) E ∋ v A divisor Λ on ˜ Σ is a Z -combination of vertices of ˜ Σ. We write ̟ ∈ L (Λ) ( ̟ is the linear system associated to Λ) if 0 ≤ Λ( w ) + ∆ ̟ ( w ) . Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 22 / 34
Linear Systems on Graphs (following Baker-Norine) If ̟ is a piecewise-linear function on ˜ Σ (linear on all edges), if v ∈ ˜ Σ, E ∋ v , write s ( v , E ) for the slope of ̟ on E coming from v . Define the Laplacian of ̟ by � ∆( ̟ )( v ) = − s ( v , E ) E ∋ v A divisor Λ on ˜ Σ is a Z -combination of vertices of ˜ Σ. We write ̟ ∈ L (Λ) ( ̟ is the linear system associated to Λ) if 0 ≤ Λ( w ) + ∆ ̟ ( w ) . ˜ Σ has canonical divisor: � K ˜ Σ = (deg( v ) − 2)( v ) v Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 22 / 34
Main theorem Theorem: If Σ ⊂ R n is a tropicalization of a curve then there exists Σ → Σ and for all m ∈ Z n (which will be the normal vector to a p : ˜ plane), 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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 23 / 34
Main theorem Theorem: If Σ ⊂ R n is a tropicalization of a curve then there exists Σ → Σ and for all m ∈ Z n (which will be the normal vector to a p : ˜ plane), 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 ˜ Σ l ), Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 23 / 34
Main theorem Theorem: If Σ ⊂ R n is a tropicalization of a curve then there exists Σ → Σ and for all m ∈ Z n (which will be the normal vector to a p : ˜ plane), 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 ˜ Σ l ), 2 ϕ m = 0 on E with m · E � = 0, Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 23 / 34
Main theorem Theorem: If Σ ⊂ R n is a tropicalization of a curve then there exists Σ → Σ and for all m ∈ Z n (which will be the normal vector to a p : ˜ plane), 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 ˜ Σ l ), 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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 23 / 34
Main theorem Theorem: If Σ ⊂ R n is a tropicalization of a curve then there exists Σ → Σ and for all m ∈ Z n (which will be the normal vector to a p : ˜ plane), 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 ˜ Σ l ), 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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 23 / 34
Cycle-ampleness condition Let H be a hyperplane given by H = { x | x · m = c } . Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 24 / 34
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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 24 / 34
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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 24 / 34
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, � � ≥ 2 . D ϕ m ≡ ( − s ( v , E )) v ∈ Γ | ϕ m ( v )= h E �∈ Γ | s ( v , E ) < 0 “sum of positive slopes coming into the cycle at min’s of ϕ m must be at least 2.” Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 24 / 34
Sections of canonical bundle Before we use these conditions, we need the following observation: ϕ m ∈ L ( K Σ l ) translates into � ∆( ϕ m )( v ) = − s ( v , E ) ≥ 2 − deg( v ) . E ∋ v Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 25 / 34
Sections of canonical bundle Before we use these conditions, we need the following observation: ϕ m ∈ L ( K Σ l ) translates into � ∆( ϕ m )( v ) = − s ( v , E ) ≥ 2 − deg( v ) . E ∋ 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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 25 / 34
Sections of canonical bundle Before we use these conditions, we need the following observation: ϕ m ∈ L ( K Σ l ) translates into � ∆( ϕ m )( v ) = − s ( v , E ) ≥ 2 − deg( v ) . E ∋ 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 , F j ) ≤ − s ( v , E i ) + (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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 25 / 34
Sections of canonical bundle Before we use these conditions, we need the following observation: ϕ m ∈ L ( K Σ l ) translates into � ∆( ϕ m )( v ) = − s ( v , E ) ≥ 2 − deg( v ) . E ∋ 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 , F j ) ≤ − s ( v , E i ) + (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).” If deg( v ) = 2, then the slope is non-increasing through v ( ϕ m is concave at v ). Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 25 / 34
Elliptic curve example Note: This is p − 1 ( H ) where H is the plane of the screen. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 26 / 34
Elliptic curve example (cont’d) Need to pay attention to positive incoming slope coming into the cycle. 1 Direct edges towards cycle. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 27 / 34
Elliptic curve example (cont’d) Need to pay attention to positive incoming slope coming into the cycle. 1 Direct edges towards cycle. 2 ϕ m must be decreasing on unbounded edges. ( ϕ m ≥ 0) Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 27 / 34
Elliptic curve example (cont’d) Need to pay attention to positive incoming slope coming into the cycle. 1 Direct edges towards cycle. 2 ϕ m must be decreasing on unbounded edges. ( ϕ m ≥ 0) 3 ϕ m is equal to 0 on ∂ ( p − 1 ( H )) and has slope at most 1 there. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 27 / 34
Elliptic curve example (cont’d) Need to pay attention to positive incoming slope coming into the cycle. 1 Direct edges towards cycle. 2 ϕ m must be decreasing on unbounded edges. ( ϕ m ≥ 0) 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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 27 / 34
Elliptic curve example (cont’d) Need to pay attention to positive incoming slope coming into the cycle. 1 Direct edges towards cycle. 2 ϕ m must be decreasing on unbounded edges. ( ϕ m ≥ 0) 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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 27 / 34
Elliptic curve example (cont’d) Need to pay attention to positive incoming slope coming into the cycle. 1 Direct edges towards cycle. 2 ϕ m must be decreasing on unbounded edges. ( ϕ m ≥ 0) 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 (University of Waterloo) Lifting Tropical Curves September 4, 2012 27 / 34
Elliptic curve example (cont’d) Need to pay attention to positive incoming slope coming into the cycle. 1 Direct edges towards cycle. 2 ϕ m must be decreasing on unbounded edges. ( ϕ m ≥ 0) 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 ϕ m ) ≥ 2, the minimum distance must be achieved at least twice. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 27 / 34
Elliptic curve example (concluded) In summary, minimum distance from Γ to ˜ Σ \ p − 1 ( H ) must be achieved by at least two paths. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 28 / 34
Elliptic curve example (concluded) In summary, minimum distance from Γ to ˜ Σ \ p − 1 ( H ) must be achieved by at least two paths. This is Speyer’s well-spacedness condition! Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 28 / 34
Elliptic curve example (concluded) In summary, minimum distance from Γ to ˜ Σ \ p − 1 ( H ) must be achieved by at least two paths. This is Speyer’s well-spacedness condition! Also get generalization to higher genus as given by Nishinou and Brugall´ e-Mikhalkin. This requires strong conditions on combinatorics of Σ. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 28 / 34
Weak well-spacedness condition There’s a new generalized version of a weak form of Speyer’s condition in higher genera that holds for curves of complicated combinatorial type. It’s a consequence of the main theorem. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 29 / 34
Weak well-spacedness condition There’s a new generalized version of a weak form of Speyer’s condition in higher genera that holds for curves of complicated combinatorial type. It’s a consequence of the main theorem. Theorem: Let Σ ⊂ R n be a tropicalization. Then there exists p : ˜ Σ → Σ that satisfies the following property: Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 29 / 34
Weak well-spacedness condition There’s a new generalized version of a weak form of Speyer’s condition in higher genera that holds for curves of complicated combinatorial type. It’s a consequence of the main theorem. Theorem: Let Σ ⊂ R n be a tropicalization. Then there exists p : ˜ Σ → Σ that satisfies the following property: if H ⊂ R n is a hyperplane and Γ ′ is any component of p − 1 ( H ) ⊂ ˜ Σ with h 1 (Γ ′ ) > 0 Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 29 / 34
Weak well-spacedness condition There’s a new generalized version of a weak form of Speyer’s condition in higher genera that holds for curves of complicated combinatorial type. It’s a consequence of the main theorem. Theorem: Let Σ ⊂ R n be a tropicalization. Then there exists p : ˜ Σ → Σ that satisfies the following property: if H ⊂ R n is a hyperplane and Γ ′ is any component of p − 1 ( H ) ⊂ ˜ Σ with h 1 (Γ ′ ) > 0 then ∂ Γ ′ is not a single trivalent vertex of ˜ Σ. Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 29 / 34
A new example b a c d Eric Katz (University of Waterloo) Lifting Tropical Curves September 4, 2012 30 / 34
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