Let $k = \bar{k}$ be an algebraically closed field with a nontrivial valuation $k \to \mathbb{R} \cup \lbrace -\infty \rbrace$.
Theorem 1 (Kapranov). For $f = \sum_{u \in \mathbb{Z}^n} c_u x^u$, the following are equal:
- the closure of the image of $$ V(f) \subseteq (k^\times)^n \xrightarrow{\mathrm{val}} \mathbb{R}^n, $$
- the corner locus of $$ N_f(x) = \max(x \cdot \lambda + \operatorname{val}(c_\lambda)), $$ i.e., the locus where the maximum is achieved at least twice.
Example 2. For $k = \bar{\mathbb{Q}}_ 3$ and $f(x, y) = 3xy + 9x - 27y + 1$, we are looking at $$ N_f(x, y) = \max(x+y-1, x-2, y-3, 0). $$ Then we get a tropical curve that is a tree.
Example 3. Let us abuse notation and write things like $$ f = \varpi^0 x^2 + \varpi^{-3} xy + \varpi^0 y^2 + \varpi^{-1} x + \varpi^{-1} y + 0. $$ This defines a “degree $2$” tropical curve in $\mathbb{P}^2$. If we look at something like $$ f = \varpi^0 x^2 + \varpi^{-3} xy + \varpi^0 y^2 + \varpi^{-1} x + \varpi^{0} y + 0, $$ this has double edge.
Recall we had the field $$ k = \bigcup_m \mathbb{C} \lbrace\lbrace t^{1/m} \rbrace\rbrace $$ which is the field of locally convergent Puiseux series.
Example 4. If we look at the polynomial $$ f = xy^2 - tx^2 + (2+t^3)xy + (1+2t+t^3)x + t^3 y - (t+t^3) $$ then the point $p = (1, 1) \in (k^\times)^2$ is a node of $V(f)$. The tropicalization of this curve looks like a “dog” and the image of the node under the valuation map lies on an edge (and not a vertex).
Denote by $S$ the subdivision of $\Delta$ into $\Delta_1 \cup \dotsb \cup \Delta_k$. Then $S$ is dual to the tropical curve $T$ in the sense that
- the components of $\mathbb{R}^2 - T$ correspond to vertices of $S$,
- the edges of $T$ correspond to edges of $S$,
- the wedge of an edge of $T$ correspond to the “lattice length”, i.e., the number of lattice points it passes through plus one,
- the vertices of $T$ correspond to polygons in $S$,
- the degree/valence of a vertex of $T$ corresponds to the number of sides of the polygon in $S$.
On the $S$ side, every polygon has the property that the edges oriented counterclockwise add up to zero. This translates to the property on the $T$ side that for all vertices $v \in T$ we have $$ \sum_{v \in e} w(e) u(e,v) = 0, $$ where $w(e)$ is the weight of $e$ and $u(e, v)$ is the primitive (integral) vector starting at $v$ in the direction of $e$.
Definition 5. We say that a plane tropical curve $T$ dual to $(S, \Delta)$ is
- smooth when $S$ is a unimodular triangulation,
- nodal when $S$ only has triangles and parallelograms,
- simple when it is nodal and all integral vertices on $\partial \Delta$ are in $S$.
Definition 6. For a simple tropical curve $T$, we define the Mikhalkin multiplicity as $$ \mu(T) = \prod_{\Delta_k} (2\operatorname{Area}(\Delta_k)), $$ and also the Welschinger multiplicity as $$ W(T) = \begin{cases} 0 & \text{if there is an edge of even weight}, \br (-1)^{S(T)} & S(T) \text{ the number of integral points in the interior of triangles of } S. \end{cases} $$
Note that tropical curves dual to a fixed subdivision $(\Delta, S)$ are parametrized by a convex polyhedron $I(\Delta, S)$. The dimension of $I(\Delta, S)$ is called the rank of $(\Delta, S)$.
Lemma 7. For a plane tropical curve $T$, we have $$ \operatorname{rank}(T) \ge \lvert \operatorname{Vert}(S) \rvert - 1 - \sum_k (\lvert \operatorname{Vert}(\Delta_k) \rvert - 3). $$ If $T$ is nodal, then we have equality.
The idea is that for an $m$-valency vertex, we are imposing $m-3$ linear conditions. Then when $T$ is nodal, these conditions can be checked to be linearly independent.
Definition 8. We say that distinct points are in $(\Delta, S)$-general position if the condition of $T$ passing through these points gives the desired codimension inside $I(\Delta, S)$. We say that distinct point are in $\Delta$-general position if it is in $(\Delta, S)$-general position for all $S$.
Let $\Delta \subseteq \mathbb{R}^2$ be a non-degenerate convex lattice polygon. Then $\Delta$ defines a toric surface $\operatorname{Tor}(\Delta)$ together with a line bundle $\mathscr{L}_ \Delta$. This has the property that $\mathscr{L}_ \Delta$ is generated by monomials $x^i y^j$ for $(i, j) \in \Delta \cap \mathbb{Z}^2$. Let $n$ be an integer $0 \le n \le \lvert \operatorname{Int}(\Delta) \cap \mathbb{Z}^2 \rvert$. Then we have the Severi variety $$ \operatorname{Sev}_ n(\Delta) $$ of curves $C$ in the linear system $\mathscr{L}_ \Delta$ having $n$ nodes as their only singularities. This is smooth and quasi-projective of dimension $$ \zeta = \dim \lvert \mathscr{L}_ \Delta \rvert - n. $$ We also have the variety $\operatorname{Sev}_ n^\mathrm{irr}(\Delta)$ of irreducible curves.
Question 9. What is the degree of $\operatorname{Sev}_ n(\Delta)$ and $\operatorname{Sev}_ n^\mathrm{irr}(\Delta)$ inside $\mathbb{P}(H^0(\operatorname{Tor}(\Delta), \mathscr{L}_ \Delta))$?
What we want to do is to work in this field $K$ of locally convergent Puiseux series. Fix $\zeta$ general points $(K^\times)^2$, which we label $p_1, \dotsc, p_\zeta$. Then the degree is the number of curves in $\operatorname{Sev}_ n(\Delta)$ passing through all of these points $p_1, \dotsc, p_\zeta$. Let us denote the set of such curves by $\operatorname{Sev}_ n(\Delta; p)$.
Write $x_i = \operatorname{val}(p_i) \in \mathbb{Q}^2$. We will deal with the following two problems.
- Describe and enumerate the tropical curves which are projections of $C \in \operatorname{Sev}_ n(\Delta; p)$.
- For each such tropical curve $C^\prime$, describe the multiplicity of $C^\prime$ appearing as a tropicalization of a curve in $\operatorname{Sev}_ n(\Delta; p)$.
Consider a del Pezzo toric surface (which is either $\mathbb{P}^1 \times \mathbb{P}^1$ or $\mathbb{P}^2$ blown up at some collection of $T$-fixed points, so we that we are blowing up at at most three points) over $\mathbb{R}$. Given $\lvert \partial \Delta \cap \mathbb{Z}^2 \rvert - 1$ generic real points in $\operatorname{Tor}(\Delta)$, there are finitely many real rational curves $C \in \lvert \mathscr{L}_ \Delta \rvert$ passing through these points. We define the Welschinger invariant as $$ W_0(\Delta) = \sum_{C} (-1)^{s(C)} $$ where $s(C)$ is the number of real solitary nodes (i.e., those that locally look like $x^2 + y^2 = 0$).
Theorem 10. Given $\Delta \subseteq \mathbb{R}^2$ and an integer $0 \le n \le \lvert \operatorname{Int}(\Delta) \cap \mathbb{Z}^2 \rvert$, we have $$ \deg \operatorname{Sev}_ n(\Delta) = \sum_{T \in T_{\Delta,\zeta}(u)} \mu(T) $$ where $u \subseteq \mathbb{Q}^2$ is a $\Delta$-general collection of $\zeta = \lvert \Delta \cap \mathbb{Z}^2 \rvert - n - 1$ points, and $T_{\Delta,\zeta}(u)$ is the set of simple nodal tropical curves with Newton polytope $\Delta$ and rank $\zeta$ passing through $u$.
Theorem 11. Assume that $\Delta$ corresponds to a toric del Pezzo surface. Then we have $$ W_0(\Delta) = \sum_{T \in T_{\Delta,\zeta}(u)} W(T) $$ where $\zeta = \lvert \partial \Delta \cap \mathbb{Z}^2 \rvert - 1$.