Basic Monoid Examples

Example

Take the following degeneration of a degree two map to \(X = \mathbb P^1\) with log structure given by \(D = \pt\).

Let’s find the basic monoid for the rightmost stable map. We have five generic points, with \(P_{\eta_i} = \mathbb N\) for \(i = 1,2,3\) and \(P_{\eta_i} = 0\) for \(i = 4,5\). Each node is mapped to \(D\), so \(P_q = \mathbb N\) for all the nodes as well. Number the four nodes as follows:

\begin{align*} q_1 &= D_4\cap D_1 \\ q_2 &= D_1\cap D_3 \\ q_3 &= D_3 \cap D_2 \\ q_4 &= D_2\cap D_5 \end{align*}

The basic monoid lives inside \[\left(P_{\eta_1}\oplus P_{\eta_2}\oplus P_{\eta_3}\oplus \mathbb N^4\right)^{\text{gp}}/R = \mathbb Z^7/R\] and is generated by

\begin{align*} a_{q_1}(1) &= (-1,0,0,1,0,0,0) \\ a_{q_2}(1) &= (1,0,-1,0,1,0,0) \\ a_{q_3}(1) &= (0,1,-1,0,0,1,0) \\ a_{q_4}(1) &= (0,1,0,0,0,0,1) \end{align*}

where the ordering has been assigned arbitrarily.

We can use the generators of \(R\) to kill the last four summands of \(\mathbb Z^7\), and thereby get \(Q\subset \mathbb Z^3\) generated by \(e_1 = (1,0,0)\), \(e_2 = (1,0,-1)\), \(e_3 = (0,1,-1)\) and \(e_4 = (0,1,0)\). These are subject to one relation, namely \[e_1 + e_3 = e_2 + e_4,\] so \(\Spec\mathbb C[Q]\) is the quadric cone in \(\mathbb A^4\).

Choosing a map \(Q \to \mathbb N = \Sigma(X)\) defines a tropical curve:

Why is this the case? Well, choosing a map \(\phi:Q\to \mathbb N \) is the same as picking

• a vector \(V_\eta \in \overline{\mathcal M}_{X, f(\eta)}^\vee = \phi_\eta\) for each generic point \(\eta\in C\)
• a non-negative integer \(e_q = \phi(1_q)\in \mathbb N\) for each node \(q\in C\).

Example

Let \(X\) be the dog bone space, and let