Tropical Curve

Definition

Let \(\overline \Gamma\) be a connected graph and let \(\Gamma\) be the topological space obtained by removing from \(\overline \Gamma\) a set of univalent vertices of \(\overline \Gamma\) so that \(\Gamma\) has both compact and non-compact edges (these vertices to be removed are part of the data). We assume that \(\Gamma\) has at least one vertex. Let \(N \cong \mathbb Z^n\) and \(N_{\mathbb R} = N\otimes_\mathbb Z\mathbb R\).

A tropical curve in \(N_{\mathbb R}\) with domain \(\Gamma\) consists of the following data:

1. For each flag \((v,E)\) of \(\Gamma\), where \(v\) is a vertex of \(\Gamma\) and \(E\) an edge containing \(v\), we are given a weight vector \(u_{(v,E)} \in N\). If \(E\) has two vertices \(v_1\) and \(v_2\), then \(u_{(v_1,E)} = - u_{(v_2,E)}\) and if \(E\) is a loop, then \(u_{(v,E)} = 0\). The graph \(\Gamma\) along with the weight vectors is called the type of the tropical curve.
2. A map \(h:\Gamma \to N_{\mathbb R}\) with the following properties:
   • For any edge \(E\) of \(\Gamma\) with vertex \(v\), \(h|_E\) is constant if \(u_{(v,E)} = 0\), and otherwise \(h|_E\) is proper and identifies \(E\) with an affine line segment or ray. Furthermore, \(u_{(v,E)}\) is a tangent vector to \(h(E)\) pointing away from \(h(v)\).
   • For each vertex \(v\), we have the balancing condition \[\sum_{E} u_{(v,e)} = 0\] where the sum is over all edges \(E\) with vertex \(v\).

Definition

A tropical curve \((G, g, \ell)\) over a cone \(\omega\) is a graph \(G\) consisting of vertices \(V(G)\) edges \(E(G)\) and legs/half-edges/unbounded edges \(L(G)\) together with an ordering of the legs and two maps \[g:V(G)\to \mathbb N\] the genus function and \[\ell:E(G)\to \Hom(\omega \cap N_\omega, \mathbb N) \setminus \{0\}\] the length map. For \(v\in V(G)\) and \(E\in E(G)\), \(g(v)\) is the genus of \(v\) and \(\ell(E)\) is the length function of \(E\). Here \(N_\omega\) is the cocharacter lattice of the cone \(\omega\), and so \(\ell(E)\) is a character of \(N_\omega\) which is strictly positive on the cone \(\omega\). If you choose primitive vectors of \(\omega\) then their images under \(\ell(E)\) are “lengths”.

The genus of a family of tropical curves \((G, g, \ell)\) is \[|g| = b_1(G) + \sum_{v\in V(G)} g(v).\]