Moduli Spaces of Rational Graphically Stable Curves

TL;DR

The paper classifies when the tropical compactification of a modular space $\mathcal{M}_{0,\Gamma}$, defined via graphic stability on a graph $\Gamma$, agrees with geometric tropicalization. It constructs $\overline{\mathcal{M}}_{0,\Gamma}$ and an associated torus embedding from Plücker coordinates, then applies geometric tropicalization to relate its tropicalization to $\mathrm{pr}_{\Gamma}(\mathcal{M}_{0,n}^{\textrm{trop}})$. The central result is that the tropicalization of $\mathcal{M}_{0,\Gamma}$ embeds as a balanced fan via the divisorial valuation map $\pi_{\Gamma}$ if and only if $\Gamma$ is a complete multipartite graph; in this case the tropical compactification coincides with the modular compactification $\overline{\mathcal{M}}_{0,\Gamma}$. When $\Gamma$ is not complete multipartite, injectivity fails and the tropical and algebraic pictures do not align, as demonstrated by explicit examples.

Abstract

We use a graph to define a new stability condition for algebraic moduli spaces of rational curves. We characterize when the tropical compactification of the moduli space agrees with the theory of geometric tropicalization. The characterization statement occurs only when the graph is complete multipartite.

Figures (8)

• Figure 1: A marked algebraic curve and it's dual graph
• Figure 2: The boundary complex of $\overline{\mathcal{M}}_{0,5}$ with divisors (vertices) labeled by their index set.
• Figure 3: Torus embedding of $\mathcal{M}_{0,\Gamma}$ via Plücker map.
• Figure 4: The graph $\widetilde{\Gamma}$ in Example \ref{['exam:ObstructionExampleDivisorialValuation']}
• Figure 5:

Theorems & Definitions (29)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Example 2.4
• Theorem 2.5: tevelev2007compactifications,gibney2011equations
• Definition 3.1
• Lemma 3.2
• proof
• Definition 3.3
• Lemma 3.4