Let $\mathscr{M}(X^{\dagger}/S^{\dagger})$ denote the stack of basic stable log maps in $X^{\dagger}$ over $S^{\dagger}$. Then:

1. (1)

   $\mathscr{M}(X^{\dagger}/S^{\dagger})$ is a Deligne-Mumford stack.

2. (2)

   Let $\beta$ denote a choice of genus $g$, number of marked points $k$, homology class in $H_{2}(X,\mathbb{Z})$, along with a collection of tangency data for the marked points (this notion can be made precise). Let $\mathscr{M}(X^{\dagger}/S^{\dagger},\beta)$ denote the substack of $\mathscr{M}(X^{\dagger}/S^{\dagger})$ of basic stable log maps of curves of genus $g$ and $k$ marked points, representing the given homology class, and satisfying the given tangency conditions. Then modulo some technical hypotheses on $X^{\dagger}$, $\mathscr{M}(X^{\dagger}/S^{\dagger},\beta)$ is proper over $S$ if $X$ is proper over $S$.

3. (3)

   Assuming further that $X^{\dagger}\rightarrow S^{\dagger}$ is log smooth, $\mathscr{M}(X^{\dagger}/S^{\dagger},\beta)$ carries a virtual fundamental class, allowing for the definition of logarithmic Gromov-Witten invariants.