specialization of divisors from curves to graphsdzb/conferenceslides/2010ams... · to graphs...

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

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Outline

1 The secret life of graphs

2 Tropical curves and their Jacobians

3 Graphs and arithmetic surfaces

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Graphs

By a graph G , we mean a connected, finite, undirectedmultigraph without loop edges.

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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) =∑

We set

Div0(G ) = {D ∈ Div(G ) : deg(D) = 0}.

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Divisors

D =∑

v∈V (G)

av (v)

with av ∈ Z.

We set

Div0(G ) = {D ∈ Div(G ) : deg(D) = 0}.

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Divisors

D =∑

v∈V (G)

av (v)

with av ∈ Z.

We set

Div0(G ) = {D ∈ Div(G ) : deg(D) = 0}.

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Divisors

D =∑

v∈V (G)

av (v)

with av ∈ Z.

We set

Div0(G ) = {D ∈ Div(G ) : deg(D) = 0}.

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Divisors

D =∑

v∈V (G)

av (v)

with av ∈ Z.

We set

Div0(G ) = {D ∈ Div(G ) : deg(D) = 0}.

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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))

The group of principal divisors on G is the subgroup

Prin(G ) = {∆f : f ∈M(G )}

of Div0(G ).

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∆f =∑

v∈V (G)

(∑e=vw

(f (v)− f (w))

Prin(G ) = {∆f : f ∈M(G )}

of Div0(G ).

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∆f =∑

v∈V (G)

(∑e=vw

(f (v)− f (w))

Prin(G ) = {∆f : f ∈M(G )}

of Div0(G ).

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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).

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The Jacobian

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The Jacobian

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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 .

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KG =∑

v∈V (G)

(deg(v)− 2) (v).

r(D)− r(KG − D) = deg(D) + 1− g .

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KG =∑

v∈V (G)

(deg(v)− 2) (v).

r(D)− r(KG − D) = deg(D) + 1− g .

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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.

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Outline

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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 .

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Tropical geometry

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Tropical geometry

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A tropical cubic curve in R2

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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.

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Metric graphs

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Metric graphs

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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.

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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.

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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’).

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Divisors

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Divisors

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Divisors

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Divisors

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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 .

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r(D)− r(KΓ − D) = deg(D) + 1− g .

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r(D)− r(KΓ − D) = deg(D) + 1− g .

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Outline

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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 .

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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.

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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.

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A = {x ∈ K : |x | < R}\{x ∈ K : |x | ≤ r}

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A = {x ∈ K : |x | < R}\{x ∈ K : |x | ≤ r}

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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.

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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.

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Models and their reductions

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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) =∑

deg(L|Zi)(vi ) ∈ Div(G ).

This extends to a homomorphism ρ : Div(X ) → Div(G ) bytaking Zariski closures.

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ρX(D) =∑

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ρX(D) =∑

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ρX(D) =∑

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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 ).

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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 ).

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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 .

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Theorem (Raynaud)

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Theorem (Raynaud)

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Theorem (Raynaud)

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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).

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Proposition

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Proposition

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Proposition

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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!

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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!

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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.

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Sketch of proof

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Sketch of proof

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Sketch of proof

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Sketch of proof

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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 .

MatthewBaker

specialization of divisors from curves to graphsdzb/conferenceslides/2010ams... · to graphs...

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