Today Abhishek Koparde spoke on the tropicalization of abstract curves. These are some rough notes taken during the talk.
Main reference: Tropical geometry and correspondence theorems via toric stacks. Link to Abhishek's notes: not yet obtained.
\(K\) is algebraicaly closed, \(R\) is a complex DVR with residue field \(k\) and fraction field \(F\). We let \(\overline F\) be the separable closure of \(F\) and \(v\) is a normalized valuation on \(\overline F\) so that \(v(F^*) = \mathbb Z\).
Graphs are finite connected with the usual notations.
Curves are denoted by \((C,D)\), are assumed to be complete with marked points \((q_1,...,q_{|D|})\) over \(\overline F\), and \((C_{R_L}, D_{R_L})\) is the nodal model (\(F \subset L\subset \overline F\) is an intermediate field extension). \(C_{R_L} \to \operatorname{Spec} R_{L}\) is a proper curve and \(L/F\) is finite separable. \(D_{R_L}\) is a finite set of \(R_L\) points and the total space \(C_{R_L}\) is normal. \((C_{R_L},D_{R_L}) \times_{\operatorname{Spec} R_{L}} \operatorname{Spec} k\) is nodal and \((C_{R_L}, D_{R_L})\times_{\operatorname{Spec}_{R_L}} \operatorname{Spec} \overline F \cong (C,D)\).
A tropical curve \(\Gamma\) is a topological graph with a complete (possibly degenerate) metric. (Note that a degenerate metric is one where \(d(x,y) = 0\) does not necessarily imply that \(x = y\). I think this is sometimes called a pseudo-metric.)
(s1) Vertices of \(\Gamma\) are divided into finite vertices and infinite vertices
(s2) \(V^\infty(\Gamma)\) is equipped with a total order and \(V^f(\Gamma)\) is just a set.
(p1) \(\Gamma\) has finitely many vertices and edges
(p2) any infinite vertex has valency 1 and is connected to a finite vertex by an edge called an "unbounded edge". We denote by \(E^b(\Gamma)\) those edges between finite vertices and by \(E^\infty(\Gamma)\) those edges between a finite and infinite vertex.
(p3) Any bounded edge \(e\) is isometric to the interval \([0,|e|]\) (read \(|e|\) as "the length of \(e\)") where \(|e| \in \mathbb R_{\geq 0}\) and an unbounded edge is isometric to \([0,\infty]\) where \(0\) maps to the finite vertex and \(\infty\) maps to the infinite vertex.
These axioms describe a general tropical curve. We say
a \(\mathbb Q\)-tropical curve if \(|e| \in \mathbb Q\) for any \(e\in \Gamma^b(\Gamma)\)
a tropical curve is irreducible if \(\Gamma\) is connected
the genus of a tropical curve is \(g(\Gamma) = 1 - |V(\Gamma)| + |E(\Gamma)|\)
we say a curve is stable if all finite vertices have valency at least \(3\)
An isomorphism of tropical curves is an isomorphism of the underlying metric graphs.
The following modifications made to a general tropical graph will not change the genus:
Operation 1: Divide each bounded edge \(e\) into finitely many pieces
Operation 2: Subdivide each unbounded edge into finitely many pieces
Operation 3: Attach rooted metric trees at some finite vertices such that all edges (but maybe not some leaves of that metric tree) are bounded
Claim: Let \(\Gamma\) be an irreducible tropical curve satisfying \(g(\Gamma) + \frac{|V^\infty(\Gamma)| + 1}{2} \geq 2\). Then there exists a unique stable tropical curve \(\Gamma^{st}\) such that \(V^\infty(\Gamma) = V^\infty(\Gamma^{st})\) and \(\Gamma\) can be obtained from \(\Gamma^{st}\) and \(\Gamma\) can be obtained from \(\Gamma^{st}\) by using finitely many of the above steps. We call \(\Gamma^{st}\) the stabilization of \(\Gamma\).
Consider a curve \((C,D)\) and remember that \((C_{R_L},D_{R_L})\) is the nodal model and \(C_{R_L} \times_{\operatorname{Spec} R_L} \operatorname{Spec} k\) is the fiber over the base. Irreducible components will correspond to finite vertices, nodes will correspond to edges between finite vertices and \(D_{R_L}'s\) which specialize to \(C_V\) will correspond to infinite edges. This should determine a tropical curve, and we call this tropical curve the tropicalization of the curve.
At this point Abhishek included a picture of a sphere identified with a torus at a point corresponding a tropical curve with two finite vertices, one finite edge and three infinite edge. One finite vertex has valence 3 and the other has valence 2. In particular, this means the tropicalization is not stable.
There was some ambiguity about what it means to say "\((C,D)\) is stable". At first, we thought it might mean that there exists a nodal model \((C_{R_L},D_{R_L})\) such that its special fiber is stable, but this is true of any curve because of the stable reduction theorem, so it must be something else.
Let \(N\) be a lattice, \(N_\mathbb R = N\otimes \mathbb R\). A parameterized tropical curve is a pair \((\Gamma, h_\Gamma)\) where \(h_\Gamma:V(\Gamma) \to N_\mathbb R\) is a map such that \(h_\Gamma(v) \in N\) for \(v\in V^\infty(\Gamma)\) and
\[\begin{aligned} \frac{1}{|e|}(h_\Gamma(v) - h_\Gamma(v')) \in N. \end{aligned}\]We also require that it satisfies the balancing condition:
\[\begin{aligned} v \in V^f(\Gamma) \implies \sum_{v'\in V^f(\Gamma), e\in E_{vv'}(\Gamma)} \frac{1}{|e|}(h_\Gamma(v) - h_\Gamma(v')) ~+~ \sum_{v' \in V^\infty(\Gamma), e\in E_{v,v'}(\Gamma)} h_\Gamma(v') ~=~ 0. \end{aligned}\]If \(h_\Gamma(v) \in N_\mathbb Q\) for all \(v\) then we say \(\Gamma\) is a \(N_\mathbb Q\)-parameterized \(\mathbb Q\)-tropical curve. For the duration of today we'll attempt to justify the balancing condition.
Suppose that you have a map \(f:C\setminus D \to T_{N, \overline F}\) and let \((C_{R_L}, D_{R_L})\) be a nodal model. Here \(T_{N,\overline F}\) is the algebraic torus of dimension equal to the rank of \(N\). There's a diagram that I'd like to draw but can't because of tikzcd support in markdown, but its rows look like \(C\setminus D \to T_{N,\overline F}\) and \(C_{R_L}\setminus D_{R_L} \to T_{N,R_L}\).
In this picture, given a character \(\chi^m\) on \(T_{N,R_L}\), I can pull back to obtain a character on \(C_{R_L}\setminus D_{R_L}\). I can evaluate the order of this on an irreducible component \(V\) of \(C\), and arrive at the following definition for \(h_\Gamma\):
\[ h_\Gamma(V) := \frac{1}{|e|} \operatorname{ord}_{V} f^*(\chi^m). \]The character \(\chi^m\) is a linear function of \(m\) for some \(m\in M = N^\vee\), so in particular, \(\frac{1}{|e|} \operatorname{ord}_{V} f^*(\chi^m) \in N_\mathbb Q\). Furthermore,
\[ \operatorname{ord}_{V} f^*(\chi^{m + m'}) = \operatorname{ord}_{V} f^*(\chi^m) + \operatorname{ord}_{V} f^*(\chi^{m'}), \]so this definition of \(h_\Gamma(V)\) is linear. We also need that \(\frac{1}{|e|}(h_\Gamma(v) - h_\Gamma(v')) \in N_\mathbb Q\) for any bounded edge \(v\).
We completed the proof that \(h_\Gamma\) does indeed satisfy the balancing condition, but I wasn't able to get all of it down. Please see the end of Abhishek's notes.