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Section 3.2 Divisors on tropical curves

Subsection 3.2.1 Definitions and examples

Definition 3.2.1.

A metric graph \(\Gamma\) is a graph in the combinatorial sense (multiple edges and loops are allowed) but modelled as a metric space by giving each edge \(e\) a positive length \(\ell(e) \in (0, \infty]\text{,}\) possibly restricted to some subgroup like \(\Q\) or \(\Z\text{.}\) In this way, we make each edge isometric to the interval \([0,\ell(e)]\text{.}\)

Sometimes, we will ask that the only vertices we consider are the ones that are required topologically. I.e. the ones of degree different from 2 since two edges meeting at a single vertex can just be joined up and their lengths added. Vertices of degree 2 will come and go as needed. The last requirement is that only leaf edges are allowed to be infinite and conversely, every leaf edge is required to be infinite.

We call the underlying combinatorial graph \(|\Gamma|\text{.}\)

Metric graph consisting of two circles joined by a bridging edge
Figure 3.2.2. Example of a metric graph (not drawn to any scale)

Mirroring Definition 3.1.1 the from the previous section, we define divisors and a divisor group.

Definition 3.2.3.

A divisor on \(\Gamma\) is a finitely-supported formal sum

\begin{equation*} D = \sum_{p \in \Gamma} a_p \cdot p \end{equation*}

with \(a_p \in \Z\) and \(p\) from anywhere in \(\Gamma\text{,}\) not just the topological vertices.

The set of all divisors is the group \(\Div(\Gamma)\) and the degree map is \(\deg(\sum a_p \cdot p) = \sum a_p\text{.}\)

Like with algebraic curves, we also have a notion of principal divisors. These are given as the divisors of piecewise-linear functions. Really these are piecewise affine-functions, but like with algebraic curves, the "scalar multiple" doesn't matter so everyone just calls them "linear."

In the next section, we will see why these are a natural choice and how they relate to rational functions on algebraic curves.

Definition 3.2.4.

A function \(f : \Gamma \to \R\) is piecewise linear, if on each edge it is a continuous function that can be broken up into a finite set of linear functions. By abuse of terminology, we will also call these rational functions.

The divisor of such an \(f\) is supported at the points \(p\) at which \(f\) is non-differentiable (including the topological vertices of \(\Gamma\)). We define \(\ord_p(f)\) to be the sum of the incoming slopes of \(f\) at \(p\) and define

\begin{equation*} \divisor(f) = \sum_{p \in \Gamma} \ord_p(f) \cdot p. \end{equation*}

As in the previous section, we define \(\Prin(\Gamma)\) as the set of all principal divisors and define \(\Jac(\Gamma) = \Div(\Gamma) / \Prin(\Gamma)\) and \(\Jac^0(\Gamma) = \Div^0(\Gamma)/\Prin(\Gamma)\text{.}\) Like before, if \(\Gamma\) is compact, then every principal divisor has degree \(0\text{.}\)

Example 3.2.5.

If \(|\Gamma|\) is at tree, then any divisor on \(\Gamma\) is principal. Trees are the tropical analogue of \(\A^1\text{.}\)

Example 3.2.6.

If \(\Gamma\) is a circle, then the rational functions on \(\Gamma\) look something like the following (absolute value of slopes indicated above)

Piecewise-linear function on a circle

On a circle, the endpoints necessarily have the same \(y\)-coordinate. The divisor associated to this piecewise-linear function is

\begin{equation*} (- \tfrac12 + 0)a + (0 - 2)b + (2 + 1)c + (-1 + 0)d + (0 + \tfrac12)e \end{equation*}

Remark 3.2.7. Divisors on \(G = |\Gamma|\).

We also consider the simpler model of a multigraph (with no metric). Here the divisors are supported only on the vertices of \(G\) and the principal divisors are the image of the Laplacian operator \(\Delta : V(G) \to \Div(G)\) given by

\begin{equation*} \Delta(v) = \sum_{w \sim v} (w - v) = \sum_{w \sim v} w - \val(v) \end{equation*}

(val = valence since we already have defined a "degree"). It's not too important whether one takes \(\Delta\) or \(-\Delta\text{.}\) This definition agrees with the intersection theory we do in the next section.

Let's finish this section by elaboration on the difference between a metric graph and a tropical curve.

Definition 3.2.8.

An abstract tropical curve or simply "tropical curve" is a metric graph \(\Gamma\) with non-negative weights at the vertices of \(|\Gamma|\text{.}\) Here we will also allow some vertices of degree 2 but require these to have a positive weight.

Subsection 3.2.2 From algebraic curves to tropical curves

Let us begin by explaining where tropical curves come from in the first place. We will need the following notation going forward.

\((K, v)\)

A field \(K\) with a non-trivial discrete valuation \(v : R \to \Z\)

\((\O_K, \mathfrak m_K)\)

The ring of integers and maximal ideal of \(K\)

\(k\)

The residue field \(k = \O_K/\mathfrak m_K\) of \(K\)

\(\X\)

A curve over \(\O_K\) (relative dimension 1 scheme)

\(X\)

The general fibre \(\mathfrak X_0\)—a curve over \(K\)

\(\X_k\)

The special fibre—a curve over \(k\)

So here's the idea: we have some equation with coefficients in \(\O_K\) and that gives us two curves—taking that equation mod \(\mathfrak m_K\) and viewing that equation in \(K\text{.}\) This gives us a degeneration of the curve \(X\) to the curve \(\X_k\)

Example 3.2.9. An elliptic curve degenerating into a collection of rational curves.

Consider the integer equation

\begin{equation*} y^2 = x(x - 7)(x + 1). \end{equation*}

This defines a smooth elliptic curve in \(\C\) and if we consider this mod 7, we get a nodal cubic \(y^2 = x^2(x + 1)\text{.}\) See Figure 3.2.10 for a picture

The curve y^2 = x(x - 7)(x + 1)
The curve y^2 = x^2(x + 1)
Figure 3.2.10. An elliptic curve degenerating to a nodal cubic

We now explain how to generate a metric graph out of this. To that end, we would like for \(\X_k\) to have several singularities but only nodal singularities.

Let \(V_1, \dots, V_k\) be the irreducible components of \(\X_k\text{.}\) Define a graph \(G\) with vertex set \(v_1, \dots, v_k\) and connect \(v_i\) to \(v_j\) if \(V_i\) and \(V_j\) intersect. We allow self-intersections and multiple intersections so \(G\) is a multi-graph.

This gets us the underlying graph. To get the metric, we define the length of an edge \(v_i v_j\) to be \(n\) if at that point of intersection, the variety is described locally by \(xy = f\) where \(f \in K\) has valuation \(n\text{.}\) This could be non-integral.

In the example above, the metric graph would be a circle of length 1 since locally the equation is \(xy = 7^1\text{.}\)

The weight of a vertex \(v_i\) is the genus of \(v_i\text{.}\) We will not do much with these weights in this discussion other than to observe that they dictate what sort of intersections we allow if we compare with Definition 3.2.8.

Definition 3.2.11.

We call this metric curve defined above the dual graph of \(\X_k\text{.}\)

A curve \(\X\) with the given notation is said to be a semistable model for \(X\) if if the dual graph of \(\X_k\) is a tropical curve (so an example requirement is that there are only nodal intersections).

Definition 3.2.12.

Given a point of \(X\) we can consider its Zariski closure in \(\X\) and extending this process linearly, we have a way of taking a divisor on \(X\) and getting a divisor on \(\X\text{.}\)

If \(V_1, \dots, V_k\) are the components of \(\X_k\) as before, then we have an intersection pairing taking a divisor \(\mathscr D\) on \(\X\) to

\begin{equation*} \mathscr D \cdot V_i = \deg(\L(\mathscr D)|_{V_i}). \end{equation*}

Putting this altogether we get a homomorphism \(\rho : \Div(X) \to \Div(G)\) where

\begin{equation*} \rho(D) = \sum_{i = 1}^k (\mathscr D \cdot V_i) v_i. \end{equation*}

Another way to describe this map is to take a point \(P \in X(K) = \X(\O_K)\) and then reduce mod \(\mathfrak m_K\) to get a point \(p \in \X_k\text{.}\) That point \(p\) belongs to one of the components \(V_1,\dots,V_k\) and that gives us a map from \(X(K)\) to the vertex set of \(G\text{.}\)

Remark 3.2.13.

Since the map \(X(K) \to \X_k\) is surjective, the map \(\rho : \Div(X) \to \Div(G)\) is also surjective.

Doing a bit of intersection theory, we see that for \(i \neq j\)

\begin{equation*} V_i \cdot V_j = |V_i \cap V_j| \end{equation*}

is the number of places components \(i\) and \(j\) intersect. Additionally, the self-intersection number is

\begin{equation*} V_i \cdot V_i = - \val(v_i) \end{equation*}

where valence of a vertex is the number of neighbours.

The reason why this self-intersection number is what it is is because the total degree needs to be \(0\text{.}\) Many self-intersection problems are computed this way. For instance, in the plane, two lines intersect in a single point. So \(\ell \cdot \ell = 1\text{.}\) But now if we blow up the point of intersection, the class \(\ell\) becomes \(\ell + E\)—an exceptional divisor. And here we still have \((\ell + E)(\ell + E) = 1\) except now \(\ell\) and \(\ell\) do not intersect but \(\ell \cdot E = 1\) and in order to make the equation come out right, \(E \cdot E = -1\text{.}\)

We point the reader to [3.2.4.2] (Chapter 9) for further explanation. The takeaway here is that divisors supported on \(\X_k\) always map to principal divisors in \(G\text{.}\)

Remark 3.2.14.

Now it's not true that elements of \(\Prin(X)\) remain principal in \(\X\text{.}\) However, as we have seen just above, the part of \(\divisor(f)\) which is supported on \(\X_k\) specializes to a principal divisor on \(G\text{.}\)

So if we start with a divisor \(\divisor(f)\) on \(X\) and take the closure in \(\X\text{,}\) the result differs from a principal divisor on \(\X\) by a divisor supported in \(\X_k\text{.}\)

Now two facts are true: first if \(\mathscr D\) is a principal divisor on \(\X\) then \(\O(\mathscr D) \cong \O\) so \(\rho(\mathscr D) = 0\) by definition. Second, if \(D \in \Prin(X)\) then \(\rho(D)\) differs from \(\rho(\mathscr D) = 0\) by \(\rho(\mathscr D|_{\X_k}) \in \Prin(G)\text{.}\)

Let's summarize what we have so far.

Remark 3.2.16.

There are three problems with all of this.

  1. We started with metric graphs but this whole last section was about non-metric graphs

  2. We have restricted ourselves to DVRs

  3. This construction is not functorial with respect to base change (next example)

All of this is fixed by looking at Berkovich spaces.

Example 3.2.17.

Let \(K = \C((t))\) and \(\X\) be the projective curve defined by \(xy = tz^2\text{.}\) This is a smooth conic which degenerates to the pair of lines \(V : x = 0\) and \(W : y = 0\text{.}\) So the dual graph consists of a single edge \(v \sim w\text{.}\)

Let \(D = P + Q\) with

\begin{equation*} P = (\sqrt t : \sqrt t : 1) \text{ and } Q = (-\sqrt t : - \sqrt t : 1). \end{equation*}

Then as \(t \to \infty\text{,}\) both \(P\) and \(Q\) converge to the node \((0:0:1)\text{.}\)

If we look at the solutions to \(x = y\) in \(\X\) these are exactly \(P + Q\) and we also have a couple of poles at infinity \((1:0:0) + (0:1:0)\text{.}\) Or in other words,

\begin{equation*} P + Q - (1:0:0) - (0:1:0) = \divisor(x - y) \end{equation*}

So \(P + Q \sim (1:0:0) + (0:1:0)\) and consequentially \(\rho(P + Q) = v + w\text{.}\)

On the other hand, if we base change to \(\C((t^{\frac12}))\) then we have to blow up at \(x = y = 0\) to regularize and this adds a new component \(U\) to \(\X'_k\) sitting exactly halfway between \(v\) and \(w\text{.}\) Now \(\rho(P + Q) = 2u\text{,}\) and while this is linearly equivalent to \(v + w\text{,}\) the map \(\rho\) has changed on the level of \(\Div(X)\text{.}\)

Note that \(x - y\) vanishes along this exceptional divisor and both \(P\) and \(Q\) degenerate to \(U\text{.}\)

Blow-up of \(\X_k\)

Subsection 3.2.3 The Baker Specialization Lemma

First we need a definition of \(r(D) = \dim \L(D) - 1\) that can be made sense of if \(D\) is a tropical divisor.

Definition 3.2.18.

If \(D\) is a divisor (tropical or algebraic), then recall that \(|D| = \{E \ge 0 : E \sim D\}\text{.}\) We define \(r(D) = -1\) if \(|D| = \varnothing\) and otherwise

\begin{equation*} r(D) = \max\{k : |D - E| \ne \varnothing, \text{ for all } E \in \Div_+^k(X)\}. \end{equation*}

(\(\Div_+^k(X) = \) effective and degree \(k\text{.}\))

This definition works the same if we replace \(X\) by \(G\) or \(\Gamma\text{.}\)

Let \(r = r(D)\text{.}\) If \(r(D) \le 0\) then this is evident.

Now suppose \(E = P_1 + P_2 + \dots + P_r\) is any rank \(k\) divisor then \(|D - P_1 - \dots - P_r| \neq \varnothing\) by definition. It follows that \(r(D - P_1) \ge r - 1\) for any point \(P_1\) since \(P_2, \dots, P_r\) are arbitrary.

Now suppose \(E = P_1 + \dots + P_r\) is the divisor that achieves the maximum. Then \(r(D - P_1) = r - 1\) since otherwise we could do better than \(E\text{.}\)

Now finally, the main event.

Let \(\bar D = \rho(D)\text{.}\) We will show by induction that if \(r_X(D) \ge r\) then \(r_G(\bar D) \ge r\) as well. The base case, \(r = -1\) is clear.

Suppose next \(r = 0\) and \(r_X(D) \ge 0\text{.}\) Then \(|D| \neq \varnothing\) so there exists an effective divisor \(E \sim D\text{.}\) Since \(\rho\) is a homomorphism, we have \(\bar D = \rho(D) \sim \rho(E) \ge 0\) and consequentially \(r_G(\bar D) \ge 0\text{.}\)

Now suppose \(r \ge 1\text{.}\) Let \(\bar P \in V(G)\) be any vertex. Since \(\rho\) is surjective, we can find some \(P \in X\) with \(\rho(P) = \bar P\text{.}\) Now by Theorem 3.2.20, \(r_X(D - P) \ge r - 1\) and by induction \(r_G(\bar D - \bar P) \ge r - 1\text{.}\) Since this holds for all \(\bar P\text{,}\) we conclude that \(r_G(\bar D) \ge r\text{.}\)

References 3.2.4 References

[2]
  
Qing Liu. Algebraic Geometry and Arithmetic Curves. Oxford Graduate Texts in Mathematics (Book 6). Oxford University Press, 2006.