Moduli Space Of Basic Stable Log Maps Marked By A Decorated Type

Definition

Let \(\tilde \tau = (G, \mathbf g, \boldsymbol \sigma, \mathbf u, \mathbf A) = (\tau, \mathbf A)\) be the decorated type of a tropical map. A marking by \(\tilde \tau\) of a stable logarithmic map \((C/S, \mathbf p, f)\) to \(X_0\) over a logarithmic base scheme \(S\) over \(b_0\) is the following data:

1. An isomorphism \(\underline C/\underline S\) with a \((G, \mathbf g)\)-marked pre-stable curve, i.e. we demand that \(\underline C/\underline S\) has dual graph \(G\) and genera prescribed by \(\mathbf g\).
2. The restriction of \(\underline f\) to the closed subscheme \(Z\subset \underline C\) (i.e. a subcurve or a nodal or punctured section of \(C\)) defined by \(x \in V(G)\cup E(G)\cup L(G)\) factors through \(X_{\sigma(x)} \subset X_0\)
3. For each geometric point \(\overline s\to S\) with decorated type \(\tilde \tau_{\overline s} = (G_{\overline s}, \mathbf g_{\overline s}, \boldsymbol{\sigma}_{\overline s}, \mathbf u_{\overline s}, \mathbf A_{\overline s})\) of \(C/S, \mathbf p, f)\), the morphism \((G_{\overline s}, \mathbf g_{\overline s}) \to (G, \mathbf g)\) of decorated graphs defines a morphism \(\tilde \tau_{\overline s} \to \tilde \tau\) of decorated types of tropical maps. In particular, there is an associated localization map \(\chi_{\overline s}: Q_{\tau_{\overline s}} \to Q_{\tau}\) of corresponding basic monoids.
4. In the situation of (3) above, the preimage \(\mathcal K_{\overline s} \subset \mathcal M_{S,\overline s}\) of \(Q_\tau\setminus \{0\}\) under the composition \[\mathcal M_{S, \overline s}\to \overline {\mathcal M}_{S,\overline s} = Q_{\tau_{\overline s}} \xrightarrow{\chi_{\overline s}} Q_{\tau}\]maps to \(0\) under the structure morphism \(\mathcal M_{S, \overline s}\to \mathcal O_{S, \overline s}.\)

We then define the moduli space of basic stable log maps marked by decorated type \(\tilde \tau\) to be \(\mathcal M(X_0, \tilde \tau)\). It’s a moduli space of stable maps over \(b_0\), but this is supressed from the notation.