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Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Specialization of Divisors from Curves to Graphs
Matthew Baker
Georgia Institute of Technology
AMS Southeastern Section MeetingUniversity of Richmond
November 2010
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Outline
1 The secret life of graphs
2 Tropical curves and their Jacobians
3 Graphs and arithmetic surfaces
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Outline
1 The secret life of graphs
2 Tropical curves and their Jacobians
3 Graphs and arithmetic surfaces
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Outline
1 The secret life of graphs
2 Tropical curves and their Jacobians
3 Graphs and arithmetic surfaces
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Outline
1 The secret life of graphs
2 Tropical curves and their Jacobians
3 Graphs and arithmetic surfaces
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Graphs
By a graph G , we mean a connected, finite, undirectedmultigraph without loop edges.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Divisors
The group Div(G ) of divisors on G is the free abeliangroup on V (G ).
We write elements of Div(G ) as formal sums
D =∑
v∈V (G)
av (v)
with av ∈ Z.
A divisor D is effective if av ≥ 0 for all v .
The degree of D =∑
av (v) is deg(D) =∑
av .
We set
Div0(G ) = {D ∈ Div(G ) : deg(D) = 0}.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Divisors
The group Div(G ) of divisors on G is the free abeliangroup on V (G ).
We write elements of Div(G ) as formal sums
D =∑
v∈V (G)
av (v)
with av ∈ Z.
A divisor D is effective if av ≥ 0 for all v .
The degree of D =∑
av (v) is deg(D) =∑
av .
We set
Div0(G ) = {D ∈ Div(G ) : deg(D) = 0}.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Divisors
The group Div(G ) of divisors on G is the free abeliangroup on V (G ).
We write elements of Div(G ) as formal sums
D =∑
v∈V (G)
av (v)
with av ∈ Z.
A divisor D is effective if av ≥ 0 for all v .
The degree of D =∑
av (v) is deg(D) =∑
av .
We set
Div0(G ) = {D ∈ Div(G ) : deg(D) = 0}.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Divisors
The group Div(G ) of divisors on G is the free abeliangroup on V (G ).
We write elements of Div(G ) as formal sums
D =∑
v∈V (G)
av (v)
with av ∈ Z.
A divisor D is effective if av ≥ 0 for all v .
The degree of D =∑
av (v) is deg(D) =∑
av .
We set
Div0(G ) = {D ∈ Div(G ) : deg(D) = 0}.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Divisors
The group Div(G ) of divisors on G is the free abeliangroup on V (G ).
We write elements of Div(G ) as formal sums
D =∑
v∈V (G)
av (v)
with av ∈ Z.
A divisor D is effective if av ≥ 0 for all v .
The degree of D =∑
av (v) is deg(D) =∑
av .
We set
Div0(G ) = {D ∈ Div(G ) : deg(D) = 0}.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Rational functions and principal divisors
The group of rational functions on G is
M(G ) = {functions f : V (G ) → Z}.
The Laplacian operator ∆ : M(G ) → Div0(G ) is definedby
∆f =∑
v∈V (G)
(∑e=vw
(f (v)− f (w))
)(v).
The group of principal divisors on G is the subgroup
Prin(G ) = {∆f : f ∈M(G )}
of Div0(G ).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Rational functions and principal divisors
The group of rational functions on G is
M(G ) = {functions f : V (G ) → Z}.
The Laplacian operator ∆ : M(G ) → Div0(G ) is definedby
∆f =∑
v∈V (G)
(∑e=vw
(f (v)− f (w))
)(v).
The group of principal divisors on G is the subgroup
Prin(G ) = {∆f : f ∈M(G )}
of Div0(G ).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Rational functions and principal divisors
The group of rational functions on G is
M(G ) = {functions f : V (G ) → Z}.
The Laplacian operator ∆ : M(G ) → Div0(G ) is definedby
∆f =∑
v∈V (G)
(∑e=vw
(f (v)− f (w))
)(v).
The group of principal divisors on G is the subgroup
Prin(G ) = {∆f : f ∈M(G )}
of Div0(G ).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
The Jacobian
Divisors D,D ′ ∈ Div(G ) are linearly equivalent, writtenD ∼ D ′, if D − D ′ is principal.
The Jacobian (or Picard group) of G is
Jac(G ) = Div0(G )/ Prin(G ).
This is a finite abelian group whose cardinality is thenumber of spanning trees in G (Kirchhoff’s Matrix-TreeTheorem).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
The Jacobian
Divisors D,D ′ ∈ Div(G ) are linearly equivalent, writtenD ∼ D ′, if D − D ′ is principal.
The Jacobian (or Picard group) of G is
Jac(G ) = Div0(G )/ Prin(G ).
This is a finite abelian group whose cardinality is thenumber of spanning trees in G (Kirchhoff’s Matrix-TreeTheorem).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
The Jacobian
Divisors D,D ′ ∈ Div(G ) are linearly equivalent, writtenD ∼ D ′, if D − D ′ is principal.
The Jacobian (or Picard group) of G is
Jac(G ) = Div0(G )/ Prin(G ).
This is a finite abelian group whose cardinality is thenumber of spanning trees in G (Kirchhoff’s Matrix-TreeTheorem).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
The Riemann-Roch theorem
The canonical divisor on G is
KG =∑
v∈V (G)
(deg(v)− 2) (v).
Its degree is 2g − 2, where g = dimR H1(G , R) is thegenus of G .
Define r(D) to be −1 iff D is not equivalent to an effectivedivisor, and to be at least k iff D − E is equivalent to aneffective divisor for every effective divisor of degree k.
Theorem (“Riemann-Roch for graphs”, B.–Norine)
For every D ∈ Div(G ), we have
r(D)− r(KG − D) = deg(D) + 1− g .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
The Riemann-Roch theorem
The canonical divisor on G is
KG =∑
v∈V (G)
(deg(v)− 2) (v).
Its degree is 2g − 2, where g = dimR H1(G , R) is thegenus of G .
Define r(D) to be −1 iff D is not equivalent to an effectivedivisor, and to be at least k iff D − E is equivalent to aneffective divisor for every effective divisor of degree k.
Theorem (“Riemann-Roch for graphs”, B.–Norine)
For every D ∈ Div(G ), we have
r(D)− r(KG − D) = deg(D) + 1− g .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
The Riemann-Roch theorem
The canonical divisor on G is
KG =∑
v∈V (G)
(deg(v)− 2) (v).
Its degree is 2g − 2, where g = dimR H1(G , R) is thegenus of G .
Define r(D) to be −1 iff D is not equivalent to an effectivedivisor, and to be at least k iff D − E is equivalent to aneffective divisor for every effective divisor of degree k.
Theorem (“Riemann-Roch for graphs”, B.–Norine)
For every D ∈ Div(G ), we have
r(D)− r(KG − D) = deg(D) + 1− g .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
A word about the proof
The proof of the Riemann-Roch theorem for graphs is based ona combinatorial study of reduced divisors, which aredistinguished coset representatives for the elements ofDiv0(G )/ Prin(G ). They are also known in the literature asG -parking functions.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Outline
1 The secret life of graphs
2 Tropical curves and their Jacobians
3 Graphs and arithmetic surfaces
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Tropical geometry
Let K be an algebraically closed field which is completewith respect to a (non-trivial) non-archimedean valuationval.
Examples: K = Cp or K = the Puiseux series field C{T}.If X is a d-dimensional irreducible algebraic subvariety ofthe torus (K ∗)n, then
Trop(X ) = val(X ) ⊆ Rn
is a connected polyhedral complex of pure dimension d .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Tropical geometry
Let K be an algebraically closed field which is completewith respect to a (non-trivial) non-archimedean valuationval.
Examples: K = Cp or K = the Puiseux series field C{T}.If X is a d-dimensional irreducible algebraic subvariety ofthe torus (K ∗)n, then
Trop(X ) = val(X ) ⊆ Rn
is a connected polyhedral complex of pure dimension d .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Tropical geometry
Let K be an algebraically closed field which is completewith respect to a (non-trivial) non-archimedean valuationval.
Examples: K = Cp or K = the Puiseux series field C{T}.If X is a d-dimensional irreducible algebraic subvariety ofthe torus (K ∗)n, then
Trop(X ) = val(X ) ⊆ Rn
is a connected polyhedral complex of pure dimension d .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
A tropical cubic curve in R2
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Metric graphs
A weighted graph is a graph G together with anassignment of a “length” `(e) > 0 to each edge e ∈ E (G ).A (compact) metric graph Γ is just the “geometricrealization” of a weighted graph: it is obtained from aweighted graph G by identifying each edge e with a linesegment of length `(e). In particular, Γ is a compactmetric space.A weighted graph G whose geometric realization is Γ willbe called a model for Γ.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Metric graphs
A weighted graph is a graph G together with anassignment of a “length” `(e) > 0 to each edge e ∈ E (G ).A (compact) metric graph Γ is just the “geometricrealization” of a weighted graph: it is obtained from aweighted graph G by identifying each edge e with a linesegment of length `(e). In particular, Γ is a compactmetric space.A weighted graph G whose geometric realization is Γ willbe called a model for Γ.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Metric graphs
A weighted graph is a graph G together with anassignment of a “length” `(e) > 0 to each edge e ∈ E (G ).A (compact) metric graph Γ is just the “geometricrealization” of a weighted graph: it is obtained from aweighted graph G by identifying each edge e with a linesegment of length `(e). In particular, Γ is a compactmetric space.A weighted graph G whose geometric realization is Γ willbe called a model for Γ.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Abstract tropical curves
Following Mikhalkin, an abstract tropical curve is just a“metric graph with a finite number of unbounded ends”.
Convention
We will ignore the unbounded ends and use the terms “tropicalcurve” and “metric graph” interchangeably.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Abstract tropical curves
Following Mikhalkin, an abstract tropical curve is just a“metric graph with a finite number of unbounded ends”.
Convention
We will ignore the unbounded ends and use the terms “tropicalcurve” and “metric graph” interchangeably.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Divisors
For a tropical curve Γ, we make the following definitions:
Div(Γ) is the free abelian group on Γ.
M(Γ) consists of all continuous piecewise affine functionsf : Γ → R with integer slopes.
The Laplacian operator ∆ : M(Γ) → Div0(Γ) is defined by−∆f =
∑p∈Γ σp(f )(p), where σp(f ) is the sum of the
slopes of f in all tangent directions emanating from p.
Prin(Γ) = {∆f : f ∈M(Γ)}.Jac(Γ) = Div0(Γ)/ Prin(Γ). This can be naturallyidentified with a g -dimensional real torus (‘Tropical Abeltheorem’).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Divisors
For a tropical curve Γ, we make the following definitions:
Div(Γ) is the free abelian group on Γ.
M(Γ) consists of all continuous piecewise affine functionsf : Γ → R with integer slopes.
The Laplacian operator ∆ : M(Γ) → Div0(Γ) is defined by−∆f =
∑p∈Γ σp(f )(p), where σp(f ) is the sum of the
slopes of f in all tangent directions emanating from p.
Prin(Γ) = {∆f : f ∈M(Γ)}.Jac(Γ) = Div0(Γ)/ Prin(Γ). This can be naturallyidentified with a g -dimensional real torus (‘Tropical Abeltheorem’).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Divisors
For a tropical curve Γ, we make the following definitions:
Div(Γ) is the free abelian group on Γ.
M(Γ) consists of all continuous piecewise affine functionsf : Γ → R with integer slopes.
The Laplacian operator ∆ : M(Γ) → Div0(Γ) is defined by−∆f =
∑p∈Γ σp(f )(p), where σp(f ) is the sum of the
slopes of f in all tangent directions emanating from p.
Prin(Γ) = {∆f : f ∈M(Γ)}.Jac(Γ) = Div0(Γ)/ Prin(Γ). This can be naturallyidentified with a g -dimensional real torus (‘Tropical Abeltheorem’).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Divisors
For a tropical curve Γ, we make the following definitions:
Div(Γ) is the free abelian group on Γ.
M(Γ) consists of all continuous piecewise affine functionsf : Γ → R with integer slopes.
The Laplacian operator ∆ : M(Γ) → Div0(Γ) is defined by−∆f =
∑p∈Γ σp(f )(p), where σp(f ) is the sum of the
slopes of f in all tangent directions emanating from p.
Prin(Γ) = {∆f : f ∈M(Γ)}.Jac(Γ) = Div0(Γ)/ Prin(Γ). This can be naturallyidentified with a g -dimensional real torus (‘Tropical Abeltheorem’).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Divisors
For a tropical curve Γ, we make the following definitions:
Div(Γ) is the free abelian group on Γ.
M(Γ) consists of all continuous piecewise affine functionsf : Γ → R with integer slopes.
The Laplacian operator ∆ : M(Γ) → Div0(Γ) is defined by−∆f =
∑p∈Γ σp(f )(p), where σp(f ) is the sum of the
slopes of f in all tangent directions emanating from p.
Prin(Γ) = {∆f : f ∈M(Γ)}.Jac(Γ) = Div0(Γ)/ Prin(Γ). This can be naturallyidentified with a g -dimensional real torus (‘Tropical Abeltheorem’).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Tropical Riemann-Roch
As before, define KΓ =∑
p∈Γ (deg(p)− 2) (p).For D ∈ Div(Γ), define r(D) to be −1 iff D is not equivalent toan effective divisor, and to be at least k iff D − E is equivalentto an effective divisor for every effective divisor of degree k.
Theorem (“Riemann-Roch for tropical curves”,Gathmann–Kerber, Mikhalkin–Zharkov)
For every D ∈ Div(Γ), we have
r(D)− r(KΓ − D) = deg(D) + 1− g .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Tropical Riemann-Roch
As before, define KΓ =∑
p∈Γ (deg(p)− 2) (p).For D ∈ Div(Γ), define r(D) to be −1 iff D is not equivalent toan effective divisor, and to be at least k iff D − E is equivalentto an effective divisor for every effective divisor of degree k.
Theorem (“Riemann-Roch for tropical curves”,Gathmann–Kerber, Mikhalkin–Zharkov)
For every D ∈ Div(Γ), we have
r(D)− r(KΓ − D) = deg(D) + 1− g .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Tropical Riemann-Roch
As before, define KΓ =∑
p∈Γ (deg(p)− 2) (p).For D ∈ Div(Γ), define r(D) to be −1 iff D is not equivalent toan effective divisor, and to be at least k iff D − E is equivalentto an effective divisor for every effective divisor of degree k.
Theorem (“Riemann-Roch for tropical curves”,Gathmann–Kerber, Mikhalkin–Zharkov)
For every D ∈ Div(Γ), we have
r(D)− r(KΓ − D) = deg(D) + 1− g .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Outline
1 The secret life of graphs
2 Tropical curves and their Jacobians
3 Graphs and arithmetic surfaces
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Notation and terminology
K : a field which is complete with respect to a (nontrivial)non-archimedean valuation
R: the valuation ring of K
k: the residue field of K (which we assume to bealgebraically closed)
X: an arithmetic surface, i.e., a flat proper scheme over Rwhose generic fiber X is a smooth (proper) geometricallyconnected curve over K . We call X a model for X .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Notation and terminology
K : a field which is complete with respect to a (nontrivial)non-archimedean valuation
R: the valuation ring of K
k: the residue field of K (which we assume to bealgebraically closed)
X: an arithmetic surface, i.e., a flat proper scheme over Rwhose generic fiber X is a smooth (proper) geometricallyconnected curve over K . We call X a model for X .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Notation and terminology
K : a field which is complete with respect to a (nontrivial)non-archimedean valuation
R: the valuation ring of K
k: the residue field of K (which we assume to bealgebraically closed)
X: an arithmetic surface, i.e., a flat proper scheme over Rwhose generic fiber X is a smooth (proper) geometricallyconnected curve over K . We call X a model for X .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Notation and terminology
K : a field which is complete with respect to a (nontrivial)non-archimedean valuation
R: the valuation ring of K
k: the residue field of K (which we assume to bealgebraically closed)
X: an arithmetic surface, i.e., a flat proper scheme over Rwhose generic fiber X is a smooth (proper) geometricallyconnected curve over K . We call X a model for X .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Semistability and dual graphs
An arithmetic surface X/R is called semistable if itsspecial fiber Xk is reduced and all singularities of Xk areordinary double points. It is called strongly semistable if inaddition every irreducible component of Xk is smooth.
By the Semistable Reduction Theorem, there exists a finiteextension L/K such that XL := X ×K L has a stronglysemistable model.
The dual graph of a strongly semistable arithmetic surfaceX is the weighted graph G = GX whose verticescorrespond to the irreducible components of Xk and whoseedges correspond to the singular points of Xk , with `(e)equal to the thickness of the corresponding singularity.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Semistability and dual graphs
An arithmetic surface X/R is called semistable if itsspecial fiber Xk is reduced and all singularities of Xk areordinary double points. It is called strongly semistable if inaddition every irreducible component of Xk is smooth.
By the Semistable Reduction Theorem, there exists a finiteextension L/K such that XL := X ×K L has a stronglysemistable model.
The dual graph of a strongly semistable arithmetic surfaceX is the weighted graph G = GX whose verticescorrespond to the irreducible components of Xk and whoseedges correspond to the singular points of Xk , with `(e)equal to the thickness of the corresponding singularity.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Semistability and dual graphs
An arithmetic surface X/R is called semistable if itsspecial fiber Xk is reduced and all singularities of Xk areordinary double points. It is called strongly semistable if inaddition every irreducible component of Xk is smooth.
By the Semistable Reduction Theorem, there exists a finiteextension L/K such that XL := X ×K L has a stronglysemistable model.
The dual graph of a strongly semistable arithmetic surfaceX is the weighted graph G = GX whose verticescorrespond to the irreducible components of Xk and whoseedges correspond to the singular points of Xk , with `(e)equal to the thickness of the corresponding singularity.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Thickness of singularities
If R is a DVR with maximal ideal (π), the thickness ofz ∈ X
singk ⊂ X is the unique natural number k such that z
has an analytic local equation of the form xy = πk .
z is a regular point of X iff its thickness is 1.
In general, the formal fiber red−1(z) ⊂ X (K ) isisomorphic (as a rigid analytic space) to an open annulus
A = {x ∈ K : |x | < R}\{x ∈ K : |x | ≤ r}
and we define the thickness of z to be the moduluslog R/ log r of A.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Thickness of singularities
If R is a DVR with maximal ideal (π), the thickness ofz ∈ X
singk ⊂ X is the unique natural number k such that z
has an analytic local equation of the form xy = πk .
z is a regular point of X iff its thickness is 1.
In general, the formal fiber red−1(z) ⊂ X (K ) isisomorphic (as a rigid analytic space) to an open annulus
A = {x ∈ K : |x | < R}\{x ∈ K : |x | ≤ r}
and we define the thickness of z to be the moduluslog R/ log r of A.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Thickness of singularities
If R is a DVR with maximal ideal (π), the thickness ofz ∈ X
singk ⊂ X is the unique natural number k such that z
has an analytic local equation of the form xy = πk .
z is a regular point of X iff its thickness is 1.
In general, the formal fiber red−1(z) ⊂ X (K ) isisomorphic (as a rigid analytic space) to an open annulus
A = {x ∈ K : |x | < R}\{x ∈ K : |x | ≤ r}
and we define the thickness of z to be the moduluslog R/ log r of A.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
The reduction graph
The reduction graph Γ = ΓX of X is the metric graphrealization of GX.
Remark: ΓX sits naturally inside the Berkovich analyticspace X an associated to X , and there is a canonicaldeformation retraction X an � ΓX. Berkovich calls ΓX theskeleton of the model X.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
The reduction graph
The reduction graph Γ = ΓX of X is the metric graphrealization of GX.
Remark: ΓX sits naturally inside the Berkovich analyticspace X an associated to X , and there is a canonicaldeformation retraction X an � ΓX. Berkovich calls ΓX theskeleton of the model X.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Models and their reductions
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Specialization of divisors
Suppose R is a DVR, and let X/R be a regular stronglysemistable arithmetic surface.
Let Z1, . . . ,Zn be the irreducible components of Xk ,corresponding to the vertices v1, . . . , vn of GX.
Given a Cartier divisor D ∈ Div(X), let L be theassociated line bundle, and define a homomorphismρX : Div(X) → Div(G ) by
ρX(D) =∑
i
deg(L|Zi)(vi ) ∈ Div(G ).
This extends to a homomorphism ρ : Div(X ) → Div(G ) bytaking Zariski closures.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Specialization of divisors
Suppose R is a DVR, and let X/R be a regular stronglysemistable arithmetic surface.
Let Z1, . . . ,Zn be the irreducible components of Xk ,corresponding to the vertices v1, . . . , vn of GX.
Given a Cartier divisor D ∈ Div(X), let L be theassociated line bundle, and define a homomorphismρX : Div(X) → Div(G ) by
ρX(D) =∑
i
deg(L|Zi)(vi ) ∈ Div(G ).
This extends to a homomorphism ρ : Div(X ) → Div(G ) bytaking Zariski closures.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Specialization of divisors
Suppose R is a DVR, and let X/R be a regular stronglysemistable arithmetic surface.
Let Z1, . . . ,Zn be the irreducible components of Xk ,corresponding to the vertices v1, . . . , vn of GX.
Given a Cartier divisor D ∈ Div(X), let L be theassociated line bundle, and define a homomorphismρX : Div(X) → Div(G ) by
ρX(D) =∑
i
deg(L|Zi)(vi ) ∈ Div(G ).
This extends to a homomorphism ρ : Div(X ) → Div(G ) bytaking Zariski closures.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Specialization of divisors
Suppose R is a DVR, and let X/R be a regular stronglysemistable arithmetic surface.
Let Z1, . . . ,Zn be the irreducible components of Xk ,corresponding to the vertices v1, . . . , vn of GX.
Given a Cartier divisor D ∈ Div(X), let L be theassociated line bundle, and define a homomorphismρX : Div(X) → Div(G ) by
ρX(D) =∑
i
deg(L|Zi)(vi ) ∈ Div(G ).
This extends to a homomorphism ρ : Div(X ) → Div(G ) bytaking Zariski closures.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Properties of the specialization map
deg(ρ(D)) = deg(D).
If D ∼X D ′ then ρ(D) ∼G ρ(D ′). Thus ρ induces ahomomorphism Jac(X ) → Jac(G ).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Properties of the specialization map
deg(ρ(D)) = deg(D).
If D ∼X D ′ then ρ(D) ∼G ρ(D ′). Thus ρ induces ahomomorphism Jac(X ) → Jac(G ).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Connection with Neron models
We continue to suppose that R is a DVR.
Let J /R be the Neron model of Jac(X ), and let ΦJ be thegroup of connected components of the special fiber of J .
The following is a reformulation of a theorem of Raynaud:
Theorem (Raynaud)
ΦJ is canonically isomorphic to Jac(GX) for any stronglysemistable regular model X/R for X .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Connection with Neron models
We continue to suppose that R is a DVR.
Let J /R be the Neron model of Jac(X ), and let ΦJ be thegroup of connected components of the special fiber of J .
The following is a reformulation of a theorem of Raynaud:
Theorem (Raynaud)
ΦJ is canonically isomorphic to Jac(GX) for any stronglysemistable regular model X/R for X .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Connection with Neron models
We continue to suppose that R is a DVR.
Let J /R be the Neron model of Jac(X ), and let ΦJ be thegroup of connected components of the special fiber of J .
The following is a reformulation of a theorem of Raynaud:
Theorem (Raynaud)
ΦJ is canonically isomorphic to Jac(GX) for any stronglysemistable regular model X/R for X .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Connection with Neron models
We continue to suppose that R is a DVR.
Let J /R be the Neron model of Jac(X ), and let ΦJ be thegroup of connected components of the special fiber of J .
The following is a reformulation of a theorem of Raynaud:
Theorem (Raynaud)
ΦJ is canonically isomorphic to Jac(GX) for any stronglysemistable regular model X/R for X .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
The specialization inequality
Suppose R is a DVR and X is a strongly semistable regulararithmetic surface over R with dual graph G .
Proposition (“Specialization Inequality”, B.)
For every D ∈ Div(X ), we have rG (ρ(D)) ≥ rX (D).
For general R, there is also a specialization mapτ : Div(XK ) → Div(Γ) which can be thought of in terms ofBerkovich spaces as ‘retraction to the skeleton’. The analogousspecialization inequality holds:
Proposition
For every D ∈ Div(XK ), we have r(τ(D)) ≥ rX (D).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
The specialization inequality
Suppose R is a DVR and X is a strongly semistable regulararithmetic surface over R with dual graph G .
Proposition (“Specialization Inequality”, B.)
For every D ∈ Div(X ), we have rG (ρ(D)) ≥ rX (D).
For general R, there is also a specialization mapτ : Div(XK ) → Div(Γ) which can be thought of in terms ofBerkovich spaces as ‘retraction to the skeleton’. The analogousspecialization inequality holds:
Proposition
For every D ∈ Div(XK ), we have r(τ(D)) ≥ rX (D).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
The specialization inequality
Suppose R is a DVR and X is a strongly semistable regulararithmetic surface over R with dual graph G .
Proposition (“Specialization Inequality”, B.)
For every D ∈ Div(X ), we have rG (ρ(D)) ≥ rX (D).
For general R, there is also a specialization mapτ : Div(XK ) → Div(Γ) which can be thought of in terms ofBerkovich spaces as ‘retraction to the skeleton’. The analogousspecialization inequality holds:
Proposition
For every D ∈ Div(XK ), we have r(τ(D)) ≥ rX (D).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
The specialization inequality
Suppose R is a DVR and X is a strongly semistable regulararithmetic surface over R with dual graph G .
Proposition (“Specialization Inequality”, B.)
For every D ∈ Div(X ), we have rG (ρ(D)) ≥ rX (D).
For general R, there is also a specialization mapτ : Div(XK ) → Div(Γ) which can be thought of in terms ofBerkovich spaces as ‘retraction to the skeleton’. The analogousspecialization inequality holds:
Proposition
For every D ∈ Div(XK ), we have r(τ(D)) ≥ rX (D).
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Tropical Brill-Noether theory I
Theorem (“Tropical Brill-Noether theorem Part I”, B.)
If g − (r + 1)(g − d + r) ≥ 0, then every tropical curve Γ ofgenus g has a special divisor (a divisor D with deg(D) ≤ d andr(D) ≥ r).
The proof uses classical algebraic geometry to prove a result incombinatorics!
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Tropical Brill-Noether theory I
Theorem (“Tropical Brill-Noether theorem Part I”, B.)
If g − (r + 1)(g − d + r) ≥ 0, then every tropical curve Γ ofgenus g has a special divisor (a divisor D with deg(D) ≤ d andr(D) ≥ r).
The proof uses classical algebraic geometry to prove a result incombinatorics!
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Sketch of proof
By a semicontinuity argument on the moduli space ofmetric graphs of genus g , we may assume without loss ofgenerality that Γ has rational edge lengths.
Rescaling, we may assume that Γ has a “regular model” G(i.e., a model with all edge lengths equal to 1).
By deformation theory, there exists a regular, totallydegenerate, strongly semistable arithmetic surface X withdual graph G .
By classical Brill-Noether theory, there is a special divisorD on XK .
By the specialization inequality, τ(D) is a special divisoron Γ.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Sketch of proof
By a semicontinuity argument on the moduli space ofmetric graphs of genus g , we may assume without loss ofgenerality that Γ has rational edge lengths.
Rescaling, we may assume that Γ has a “regular model” G(i.e., a model with all edge lengths equal to 1).
By deformation theory, there exists a regular, totallydegenerate, strongly semistable arithmetic surface X withdual graph G .
By classical Brill-Noether theory, there is a special divisorD on XK .
By the specialization inequality, τ(D) is a special divisoron Γ.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Sketch of proof
By a semicontinuity argument on the moduli space ofmetric graphs of genus g , we may assume without loss ofgenerality that Γ has rational edge lengths.
Rescaling, we may assume that Γ has a “regular model” G(i.e., a model with all edge lengths equal to 1).
By deformation theory, there exists a regular, totallydegenerate, strongly semistable arithmetic surface X withdual graph G .
By classical Brill-Noether theory, there is a special divisorD on XK .
By the specialization inequality, τ(D) is a special divisoron Γ.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Sketch of proof
By a semicontinuity argument on the moduli space ofmetric graphs of genus g , we may assume without loss ofgenerality that Γ has rational edge lengths.
Rescaling, we may assume that Γ has a “regular model” G(i.e., a model with all edge lengths equal to 1).
By deformation theory, there exists a regular, totallydegenerate, strongly semistable arithmetic surface X withdual graph G .
By classical Brill-Noether theory, there is a special divisorD on XK .
By the specialization inequality, τ(D) is a special divisoron Γ.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Sketch of proof
By a semicontinuity argument on the moduli space ofmetric graphs of genus g , we may assume without loss ofgenerality that Γ has rational edge lengths.
Rescaling, we may assume that Γ has a “regular model” G(i.e., a model with all edge lengths equal to 1).
By deformation theory, there exists a regular, totallydegenerate, strongly semistable arithmetic surface X withdual graph G .
By classical Brill-Noether theory, there is a special divisorD on XK .
By the specialization inequality, τ(D) is a special divisoron Γ.
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Tropical Brill-Noether theory II
Using purely combinatorial methods (the theory of reduceddivisors), the following result was recently proved:
Theorem (“Tropical Brill-Noether theorem Part II”,Cools-Draisma-Payne-Robeva)
If g − (r + 1)(g − d + r) < 0, then there exist (infinitely many)tropical curves Γ of genus g having no special divisor.
Combining this combinatorial result with our specializationinequality gives a new proof of the Brill-Noether-Griffiths-Harristheorem in classical algebraic geometry: Ifg − (r + 1)(g − d + r) < 0, then on a general smooth algebraiccurve X/C of genus g there is no divisor D with deg(D) ≤ dand r(D) ≥ r .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Tropical Brill-Noether theory II
Using purely combinatorial methods (the theory of reduceddivisors), the following result was recently proved:
Theorem (“Tropical Brill-Noether theorem Part II”,Cools-Draisma-Payne-Robeva)
If g − (r + 1)(g − d + r) < 0, then there exist (infinitely many)tropical curves Γ of genus g having no special divisor.
Combining this combinatorial result with our specializationinequality gives a new proof of the Brill-Noether-Griffiths-Harristheorem in classical algebraic geometry: Ifg − (r + 1)(g − d + r) < 0, then on a general smooth algebraiccurve X/C of genus g there is no divisor D with deg(D) ≤ dand r(D) ≥ r .
Specializationof Divisors
from Curvesto Graphs
MatthewBaker
The secret lifeof graphs
Tropicalcurves andtheirJacobians
Graphs andarithmeticsurfaces
Tropical Brill-Noether theory II
Using purely combinatorial methods (the theory of reduceddivisors), the following result was recently proved:
Theorem (“Tropical Brill-Noether theorem Part II”,Cools-Draisma-Payne-Robeva)
If g − (r + 1)(g − d + r) < 0, then there exist (infinitely many)tropical curves Γ of genus g having no special divisor.
Combining this combinatorial result with our specializationinequality gives a new proof of the Brill-Noether-Griffiths-Harristheorem in classical algebraic geometry: Ifg − (r + 1)(g − d + r) < 0, then on a general smooth algebraiccurve X/C of genus g there is no divisor D with deg(D) ≤ dand r(D) ≥ r .