Domain Stacks Main reference: moduli space of pre stable log curves
- \(\mathfrak M(G,\mathbf g) \): is the moduli space of \((G,\mathbf g)\)-marked pre-stable curves over the ground field \(k\) with its basic log structure as a nodal curve.
- \(\mathfrak{M}_B(G, \mathbf{g}) \): the base change of \((G, \mathbf g)\)-marked pre-stable curves to a base \(B\).
- \(\mathfrak M_B^\circ(G, \mathbf g)\): the stack of \((G, \mathbf g)\)-marked punctured curves over \(B\), admitting negative contact orders at the marked sections.
Logarithmic Maps Main references: moduli space of basic stable log maps, moduli space of basic stable log maps marked by a decorated type.
- \( \mathcal M(X, \beta) \): this is the moduli space of basic stable log maps of class \(\beta\). The maps are of a specified class, but the tropical type can vary. Definition 2.16 in “Decomposition of Degenerate Log GW”.
- \( \mathcal M(X, \tilde \tau) \): is the moduli space of basic stable log maps marked by decorated type \(\tilde \tau\). Definition 2.31 in “Decomposition of Degenerate Log GW”.
- \(\mathcal M(X/B, \beta, \sigma)\): the moduli space of stable log maps passing through a specifically chosen constrained set of sections \(\sigma\). Defined on page 54 of “Decomposition of Degenerate Log GW”.
Punctured Log Maps Here \(\boldsymbol{\tau} = (G, \mathbf g, \boldsymbol \sigma, \overline{\mathbf u}, \mathbf A) = (\tau, \mathbf A)\)
- \(\mathscr M(X/B, \boldsymbol \tau)\) and \(\mathscr M(X/B, \tau)\): basic punctured stable log maps to \(X\) over \(B\) marked by \(\boldsymbol \tau\) and \(\tau\) respectively, Definition 3.8 in “Punctured Logarithmic Maps”.
- \(\mathfrak M(\mathcal X/B, \boldsymbol \tau)\) and \(\mathfrak M(\mathcal X/B, \boldsymbol \tau)\): basic punctured pre-stable log maps to the Artin fan \(\mathcal X\) of \(X\) marked by \(\boldsymbol \tau\) and \(\tau\) respectively, also Definition 3.8 in “Punctured Logarithmic Maps”.
- We have variants for weakly marked curves which are adorned with primes, \(\mathfrak M'(\mathcal X/B, \boldsymbol \tau)\) for instance.