Definition
Let \((C, f, p_1,...,p_n)\) be a stable log curve to a target \(X\). Denote by \(\eta\in C\) generic points of \(C\), \(p\in C\) marked points of \(C\) and \(q\in C\) nodes of \(C\). Let \(\{u_p, u_q\}\) be the contact order of \(C\) at the marks and nodes. We have two generization maps for every node \(q\), \(\chi_{\eta_i, q}:P_q\to P_{\eta_i}\) for \(i = 1,2\). For each \(q\) and \(m\in P_q\) we define an element
\begin{align*} a_q(m) &= \Big((0,...,\chi_{\eta_i, q}(m), ... , -\chi_{\eta_2, q}(m),...,0) ~ , ~ (0,...,u_q(m),...,0)\Big) \\ &\in \Bigg(\prod_{\eta\in C}P_\eta \times \prod_{q\in C}\mathbb N\Bigg)^{\text{gp}} \end{align*}which is \(0\) in every entry except the two indicated in \(P_\eta\) components of the product. Note here we’re assuming both \(X\) has an integral log structure so that the above product embedds in its groupificiation. We let \(R\) be the subgroup of the above group generated by the elements \(a_q(m)\) for all \(q\in C\). Then the basic monoid of \(C\) is \[Q:= \left[\iota\Bigg(\prod_{\eta\in C}P_\eta \times \prod_{q\in C}\mathbb N\Bigg)\Bigg/R\right]^{\text{sat}}\] where \(\iota\) is the embedding of the product monoid in its groupification.
- How to Find All Possible Basic Monoids
To find all possible basic monoids \(Q_\tau\) appearing in the moduli space \(\mathcal{M}(X, \beta)\), we construct them via a purely combinatorial optimization process over the intersection graph of the target’s tropicalization. Following the boundedness results in Section 3 of Gross & Siebert’s “Logarithmic Gromov-Witten Invariants”:
- Enumerate Combinatorial Graph Types: List all valid dual intersection graphs \(\Gamma_C\) whose total topological curve class matches \(\beta\).
- Assign Target Strata: Decorate each vertex \(v \in \Gamma_C\) by assigning it to a cone (stratum) \(\mathbf{G}_v\) of the target’s tropical cone complex \(\Sigma(X)\).
- Impose Tropical Balancing: For every internal edge and external leg, apply the required contact vectors \(u_q\). Filter for the discrete configurations \(\tau = (\Gamma_C, \{\mathbf{G}_v\}, \{u_q\})\) that strictly satisfy the tropical balancing condition.
- Compute via Construction 1.16: For every valid balanced type \(\tau\), form its relation group \(R_\tau\) out of its local generization and contact profiles. The basic monoid \(Q_\tau\) is exactly the saturation of the resulting quotient group. By the finiteness theorem (Theorem 3.14), this list of valid basic monoids is finite.