Recall we are trying to prove the following theorem.
Theorem 1 (Mikhalkin). Let $\Delta$ be an integer lattice polygon and consider the toric surface $\operatorname{Tor}(\Delta)$ with the ample divisor class $\lvert \mathcal{L}_ \Delta \rvert$. Fix an integer $0 \le n \le \lvert \operatorname{Int}(\Delta) \cap \mathbb{Z}^2 \rvert$, and consider $\zeta = \lvert \Delta \cap \mathbb{Z}^2 \rvert - 1 - n$. Then the number of curves $C \in \lvert \mathcal{L}_ \Delta \rvert$ with $n$ nodes passing through $\zeta$ generic points is the same as $$ \sum_T \mu(T) $$ the sum is over simple tropical curves $T$ of rank $\zeta$ passing through $\zeta$ generic points in $\mathbb{R}^2$.
Example 2. What is the number of rational $(1,1)$-curves passing through three generic points in $\mathbb{P}^1 \times \mathbb{P}^1$? This is $1$, because any $(1,1)$-curve is a graph of an automorphism of $\mathbb{P}^1$ and that action is simply triply transitive. On the tropical side, there are many combinatorial pictures that can occur, but it is still true that there is a unique tropical curve passing through three generic points.
Summary of the tropical limit
Our strategy was to try to solve this counting problem over $\mathbb{K} = \mathbb{C}\lbrace\lbrace t^{1/\infty} \rbrace\rbrace$. The thing we gain here is that when we say “generic points” $p_1, \dotsc, p_\zeta$, their valuations $\operatorname{val}(p_i)$ become generic in $\mathbb{R}^2$.
Question 3. Fix an integer $n$. Given a curve $C = V(f) = \lbrace C^{(t)} \rbrace \in \operatorname{Tor}(\Delta)_ \mathbb{K}$ that is generic with $n$ nodes in the divisor class $\lvert \mathcal{L}_ \Delta \rvert$, what does this curve $C^{(t)}$ look like as $t \to 0$?
Let us demonstrate this with the example of a cubic in $\mathbb{P}^2$. We will consider the polynomial $$ \begin{align} f(x, y) &= (2t^3+hot) y^3 \br &+ (-t+hot) y^2 &&+ (1+hot) xy^2 \br &+ (1+hot) y &&+ (-2+hot) xy &&+ (1+hot) x^2y \br &+ (2t^3+hot) &&+ (3t^2+hot) x &&+ (-t+hot) x^2 &&+ (t^{-1}+hot) x^3. \end{align} $$ We now look at what this does when we approach $t \to 0$.
The first observation is that if we want $f(x, y) = 0$, this separates into “different scales” of $x$ and $y$. That is, we have a region where $\lvert x \rvert \sim \lvert y \rvert \sim 1$, and there is a region where $\lvert x \rvert \sim \lvert t \rvert$ and $\lvert y \rvert \sim \lvert t \rvert^{-1}$. On each of these regions, we basically get a simpler toric variety after “renormalizing” $x$ and $y$. For example, when we renormalize $$ \tilde{x} = t^{-1} x, \quad \tilde{y} = ty, $$ then the relevant term in $f(x, y) = 0$ is $$ -\tilde{y} + \tilde{x} \tilde{y} + 1 = 0. $$ This is a line inside $\mathbb{P}^1$. That is, we end up obtaining a subdivision $$ \Delta = \Delta_1 \cup \dotsb \cup \Delta_N $$ together with a toric subvariety $$ C_i \subseteq \operatorname{Tor}(\Delta_i). $$
But this is not all we have. For each $C_i$ and $\sigma \in \Delta_i$, there is a point $p_\sigma = C_i \cap \operatorname{Tor}(\sigma)$ inside $\operatorname{Tor}(\sigma)$, which is common for polygons on the both sides of $\sigma$. The way these two toric varieties $C_i$ and $C_j$ are connected is around a small neighborhood of $p_\sigma$, but the question is what exactly this looks like.
Let us look at the above example again. At the scale $\lvert x \rvert \sim 1$ and $\lvert y \rvert \sim 1$, we have the toric hypersurface $$ C_i = V(xy^2 + y - 2xy + x^2 y), $$ and at the scale $\lvert x \rvert \sim 1$ and $\lvert y \rvert \sim \lvert t \rvert^2$, we have $$ C_j = V((y/t^2) - 2 x (y/t^2) + x^2 (y/t^2) + 3x). $$ Now the part that connects the two toric hypersurfaces will look like $$ x \approx 1, \quad 10^3 \lvert t \rvert \le \lvert y \rvert \le 10^{-3}. $$ But this doesn’t tell us about what it looks like topologically.
To understand that, we need to make another change of variables, $$ x = 1 + \delta_x. $$ Then the relevant term becomes $$ \begin{align} (1+hot)(1+\delta_x) y^2 &+ (1+hot) y + (-2+hot) (1+\delta_x) y \br &+ (1+hot) (1+\delta_x)^2 y + (3t^2+hot) (1+\delta_x) \br &= (1+hot) y^2 + (3t^2+hot) + (0+hot) y + (0+hot) \delta_x y + (1+hot) \delta_x^2 y. \end{align} $$ Here, we cannot ignore the $\delta_x^0$ and $\delta_x^1$ terms, because we are going to renormalize this as $$ y = t y^\prime, \quad \delta_x = t \delta_x^\prime. $$ To get a node, we see that the node has to appear at scale $\lvert y^\prime \rvert \sim 1$ and $\lvert \delta_x^\prime \rvert \sim 1$. Then this is the refined limit $C_\sigma$ we want to record.
The patchworking theorem
Theorem 4. Let $p_1, \dotsc, p_\zeta \in \mathbb{K}^{\times 2}$ be generic points whose projections $x_1, \dotsc, x_\zeta \in \mathbb{R}^2$ are $\Delta$-generic. Consider the data of
- a simple tropical curve $T$ of rank $\zeta$ with polygon $\Delta$ passing through $x_1, \dotsc, x_\zeta$, with corresponding subdivision $\Delta = \Delta_1 \cup \dotsb \cup \Delta_N$,
- curves $C_m \in \lvert \mathcal{L}_ {\Delta_m} \rvert$ for $1 \le m \le N$,
- satisfying the conditions from before (if $\Delta_m$ is a triangle then it is rational nodal and meets each codimension one boundary at a single point unibranch, and if $\Delta_m$ is a parallelogram then it is a union of lines),
- compatible with each other in the sense that $C_k \cap \operatorname{Tor}(\sigma) = C_l \cap \operatorname{Tor}(\sigma)$ if $\Delta_k \cap \Delta_l = \sigma$ is a shared edge, and also
- compatible with the points $p_1, \dotsc, p_\zeta$ in the sense that if $p_i$ is on the edge corresponding to $\sigma$ then this common intersection corresponds to $p_i$,
- curves $C_{[\sigma]}$ for each equivalence class $[\sigma]$ of edges,
- satisfying the condition that it is rational and cut out by a polynomial of the form $\alpha y + \beta y^{-1} + c_m x^m + \dotsb c_1 x + c_0$ where $c_{m-1} = 0$ and $\alpha, \beta$ are determined so that it is compatible with $C_1, \dotsc, C_N$.
Then there exists a unique nodal curve $C^{(t)} \in \operatorname{Tor}(\Delta)_ \mathbb{K}$ in $\lvert \mathcal{L}_ \Delta \rvert_ \mathbb{K}$ with $n$ nodes passing through $p_1, \dotsc, p_\zeta$, whose refined tropicalization recovers the above data.
Using this bijection between nodal curves passing through $p_1, \dotsc, p_\zeta$ and their refined tropicalizations, we can now attempt to count these.
Lemma 5. Assume $\sigma$ has lattice length $m = \lvert \sigma \rvert$. Given $\alpha$ and $\beta$, there exists $m$ choices for the rational curve $C_{[\sigma]}$.
Lemma 6. 1. Let $\Delta_m$ be a triangle with sides $\sigma$ and $\tau$. If we fix the points $p_\sigma = C_m \cap \operatorname{Tor}(\sigma)$ and $p_\tau = C_m \cap \operatorname{Tor}(\tau)$, the number of choices for $C_m$ is given by $$ \frac{\operatorname{Area}(\Delta_m)}{\lvert \sigma \rvert \cdot \lvert \tau \rvert}. $$
- Let $\Delta_m$ be a parallelogram with a side $\sigma$ and $\tau$. If we fix the points $p_\sigma$, then there exists a unique choice for the component of $C_m$ corresponding to $\sigma$.
Fact 7. Let $x_1, \dotsc, x_\zeta \in \mathbb{R}^2$ be $\Delta$-generic points, and let $T$ be a tropical curve passing through them. If we color the “maximally straight” edges passing through some $x_i$, then we can color all edges of $T$ by the following procedure: take a vertex $v \in T$ of degree three such that two of the adjacent edges are colored, and then color the uncolored edge.
Using this, we can count the number of choices of $C_1, \dotsc, C_N$ and $C_{[\sigma]}$, given a choice of a tropical curve.