Introduction

• The tropical counting problem
• Nonarchimedean amoebae

The tropical counting problem

Question 1. Let $m \ge 1$ be an integer. What is the number $N(m)$ of rational curves $C \subseteq \mathbb{P}^2$ of degree $m$ that pass through $3m-1$ generic points? (Note that $C$ itself will be nodal.)

Konsevich gave a nice recursive answer to this problem. The goal of this seminar is to give a different proof using tropical geometry. This was first done by Mikhalkin in 2005, but we will follow the approach of Shustin in 2006.

Definition 2. A tropical curve is an equivalence class of $(\Gamma, w, h)$ where

• $\Gamma$ be a finite (simple) graph with no degree $2$ vertices, regarded as a topological space where the degree $1$ vertices are open,
• $w \colon E(\Gamma) \to \mathbb{Z}_ {\ge 1}$ is a weight function,
• $h \colon \Gamma \to \mathbb{R}^n$ is a proper map,
• $h$ restricted to each edge is either constant or a topological embedding with image contained in a line with rational slope,
• for every (non-leaf) vertex $v \in V(\Gamma)$, if $E_1, \dotsc, E_m$ are the edges adjacent to $v$ with weight $w_1, \dotsc, w_m$ and $v_1, \dotsc, v_m \in \mathbb{Z}^n$ are the primitive integer vectors in the direction of the edges, then $\sum_{i=1}^m w_i v_i = 0$

It follows from the axioms that if there are half-lines in the primitive integer directions $v_1, \dotsc, v_m$ with weights $w_1, \dotsc, w_m$, then we have $\sum_{i=1}^n w_i v_i = 0$. In this case, we say that $h \colon \Gamma \to \mathbb{R}^n$ has toric degree given by the multiset $$ \mathcal{T} = \lbrace v_1^{w_1}, \dotsc, v_n^{w_n} \rbrace. $$

Definition 3. We say that a tropical curve $h \colon \Gamma \to \mathbb{R}^n$ is a tropical projective curve of degree $d$ when its toric degree is $$ \lbrace (-1, 0, \dotsc, 0)^d, \dotsc, (0, \dotsc, 0, -1)^d, (1, \dotsc, 1)^d \rbrace. $$

Remark 4. One of the nice features of $\mathbb{R}^2$ is that given a combinatorial type of the tropical curve (meaning that we fix $\Gamma$ and prescribe the rational slopes of all the edges), the space of tropical curves almost always has the expected dimension. The way this fails is when there is a cycle in $\Gamma$ where all the edges have slope contained in a smaller-dimensional subspace in $\mathbb{R}^n$, but this is very degenerate in $2$-dimensions.

Example 5. We consider the following examples of tropical curves for $d = 3$.

• An example of a genus $1$ curve.
• An example of a genus $0$ curve where where $h$ is not injective.
• An example of a genus $0$ curve of weight $4$, where one edge has weight $2$.
• An example of a genus $0$ curve of weight $3$, where all edges have weight $1$.

For a tropical curve $T$ say in $\mathbb{R}^2$, we can consider its dual subdivision $S_T$, which is a subdivision of the triangle with vertices $$ (0, 0), (d, 0), (0, d). $$ We say that $T$ is simple when $S_T$ consists only of triangles and parallelograms, and we let $\mu(T)$ be the product of the areas of the triangles in $S_T$.

Theorem 6 (Mikhalkin, 2005). Let $p_1, \dotsc, p_{3m-1} \in \mathbb{R}^2$ be generic points, and let $C$ be the set of simple tropical projective curves passing through $p_1, \dotsc, p_{3m-1}$ with the property that $\Gamma$ is a tree. Then $$ N^\mathrm{trop}(m) = \sum_{T \in C} \mu(T) $$ agrees with $N(m)$.

Nonarchimedean amoebae

So how do we relate curves in $\mathbb{P}^2$ to tropical curves in $\mathbb{R}^2$? We consider an algebraically closed non-archimedean field $\mathbb{K}$ with a valuation $$ \mathrm{val} \colon \mathbb{K} \to \mathbb{R} \cup \lbrace -\infty \rbrace. $$ In practice, we will take the field of convergent Puiseux series $$ \mathbb{K} = \bigcup_m \mathbb{C}\lbrace\lbrace t^{1/m} \rbrace\rbrace, \quad \mathbb{C}\lbrace\lbrace t \rbrace\rbrace = \biggl\lbrace \sum_{n=-N}^\infty c_n t^n \text{ with positive radius of convergence} \biggr\rbrace, $$ where the valuation of $\sum_{n=-N}^\infty c_n t^{n/m}$ with $c_{-N} \neq 0$ is $N/m$. So $$ \operatorname{val}(xy) = \operatorname{val}(x) + \operatorname{val}(y), \quad \operatorname{val}(x+y) \le \max(\operatorname{val}(x), \operatorname{val}(y)). $$

Definition 7. Let $Z \subseteq (\mathbb{K}^\times)^n$ be a hypersurface. We define its amoeba $\mathcal{A}(Z) \subseteq \mathbb{R}^n$ as the closure of the image of $$ Z \subseteq (\mathbb{K}^\times)^n \xrightarrow{\mathrm{val}} \mathbb{R}^n. $$

Let us say that $Z$ is cut out by the Laurent polynomial $$ f(z_1, \dotsc, z_n) = \sum_{\omega \in I} c_\omega z_1^{\omega_1} \dotsm z_n^{\omega_n} $$ where $I \subseteq \mathbb{Z}^n$ and $c_\omega \in \mathbb{K}^\times$.

Theorem 8 (Kapranov). The amoeba $\mathcal{A}(Z)$ is the corner locus of the piecewise linear convex function on $\mathbb{R}^n$ given by $$ N_f(x_1, \dotsc, x_n) = \max_{\omega \in I}(\omega_1 x_1 + \dotsb + \omega_n x_n + \operatorname{val}(c_\omega)). $$

Now the rough strategy seems to be as follows.

1. Since these enumerative invariants are algebraic invariants, we can always base change to $\mathbb{K}$ and count them.
2. The advantage over $\mathbb{K}$ is now that we can pick generic points $x_1, \dotsc, x_m \in (\mathbb{K}^\times)^2$ with generic images $y_i = \operatorname{val}(x_i) \in \mathbb{R}^2$.
3. For points with generic enough images $y_1, y_2, \dotsc, y_m$, we relate tropical curves passing through $y_i$ with algebraic curves passing through $x_1, \dotsc, x_m$.

For our particular choice of $\mathbb{K}$, we can see that a curve over $\mathbb{K}$ is more or less a family of curves over a small punctured disk $D^\circ(\epsilon)$. So in this case, tropicalization, i.e., taking the non-archimedean amoeba and recording some extra information, is essentially degeneration of a family of curves.

The more nontrivial part is to deform a degenerated curve back into a family of curves. Here, we can’t take any deformation, because we want the family of curves to actually pass through these points $x_i$, which can be interpreted as section of $\mathbb{P}^2 \times D^\circ \to D^\circ$. This procedure goes by the name of “patchworking” and is something we will spend a lot of time on.