2·1. Algebraic moduli spaces

The moduli space $\mathcal{M}_{0,n}$ parameterises isomorphism classes of smooth, genus 0 curves with n marked points. A point of $\mathcal{M}_{0,n}$ is an isomorphism class of n ordered, distinct marked points on ${{{\mathbb{P}}}}^1$ which we denote $(p_1,\ldots,p_n)$ . Two points $({{{\mathbb{P}}}}^1,p_1,\ldots,p_n),({{{\mathbb{P}}}}^1,q_1,\ldots,q_n)\in{{{{{{\mathcal{M}}}}_{0,n}}}}$ are equal if there is $\Phi\in\textrm{Aut}({{{\mathbb{P}}}}^1)$ such that $\Phi(p_i)=(q_i)$ , for all i. Using cross ratios, we may assign any n-tuple $(p_1,\ldots,p_n)$ to $(0,1,\infty,\Phi_{CR}(p_4),$ $\ldots,\Phi_{CR}(p_n))$ where $\Phi_{CR}$ is the unique automorphism of ${{{\mathbb{P}}}}^1$ sending $p_1,$ $p_2,$ and $p_3$ to 0, 1, and $\infty$ . The first two nontrivial cases occur when $n=3$ and $n=4$ . As varieties, ${{{\mathcal{M}}}}_{0,3}$ is a point, as we send $(p_1,p_2,p_3)$ to $(0,1,\infty)$ and ${{{\mathcal{M}}}}_{0,4}={{{\mathbb{P}}}}^1\setminus\{0,1,\infty\}$ because the fourth point is free to vary as long as it doesn’t coincide with the other 3 markings. In general, this shows that $\mathcal{M}_{0,n}$ is an $n-3$ dimensional space and

\begin{equation*} {{{{{{\mathcal{M}}}}_{0,n}}}} = \overbrace{{{{\mathcal{M}}}}_{0,4}\times\cdots\times {{{\mathcal{M}}}}_{0,4}}^{n-3\textrm{ times}}\setminus\{\textrm{all diagonals}\}.\end{equation*}

From the $n=4$ example, we can see that $\mathcal{M}_{0,n}$ is not compact in general. The most notable compactification, ${{{\overline{{{\mathcal{M}}}}_{0,n}}}}$ , is due to Deligne and Mumford which allows nodal curves with finite automorphism group; such curves are called stable curves [Reference Deligne and Mumford4, Reference Knudsen13].

Stable nodal curves arise as the limit of a family of smooth curves where a number of points collide, e.g., $p_1\mapsto p_2$ . In Figure 1, we see an example of a nodal curve in $\overline{{{{\mathcal{M}}}}}_{0,4}$ where the marked points $p_3,$ and $p_4$ have collided. This curve also arises if $p_1$ and $p_2$ collide. The dual graph or combinatorial type of a stable curve in ${{{\overline{{{\mathcal{M}}}}_{0,n}}}}$ , is defined by assigning a vertex to each component, an edge to each node, and a half-edge to each marked point, as shown in Figure 1. An alternative definition of stability can be posed in terms of dual graphs.

Fig. 1. A marked algebraic curve and its dual graph.

We define the boundary of ${{{\overline{{{\mathcal{M}}}}_{0,n}}}}$ to be $\partial{{{\overline{{{\mathcal{M}}}}_{0,n}}}}={{{\overline{{{\mathcal{M}}}}_{0,n}}}}\setminus{{{{{{\mathcal{M}}}}_{0,n}}}}$ ; it consists of all points corresponding to nodal stable curves. We call the closure of a codimension one stratum a boundary divisor. The boundary is stratified by nodal curves of a given topological type with an assignment of marks to each component. In other words, $\partial{{{\overline{{{\mathcal{M}}}}_{0,n}}}}$ is stratified by dual graphs of stable nodal pointed curves. Dual graphs of boundary divisors partition the set of markings into two sets $I\sqcup I^c$ . We adopt the convention that the marking $1\in I^c$ ; therefore, a boundary divisor $D\,{:\def\negativespace{}=}\,D_I$ is uniquely identified by its index set I.

2·2. Tropical moduli spaces

We begin by introducing necessary background terminology on tropical moduli spaces. For a more thorough survey of tropical moduli spaces, see [Reference Maclagan and Sturmfels14]. Consider the space of genus 0, n-marked abstract tropical curves ${{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}}$ . Points of ${{{\mathcal{C}}}}\in{{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}}$ are in bijection with metrised trees with bounded edges having finite length and n unbounded labelled edges called ends. By forgetting the lengths of the bounded edges of ${{{\mathcal{C}}}}$ we get a tree with labelled ends called the combinatorial type of ${{{\mathcal{C}}}}$ . The space ${{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}}$ naturally has the structure of a cone complex where curves of a fixed combinatorial type with d bounded edges are parameterised by ${{{\mathbb{R}}}}_{\gt0}^{d}$ . We obtain ${{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}}$ by gluing several copies of ${{{\mathbb{R}}}}_{\geq0}^{n-3}$ via appropriate face morphisms, one for each trivalent combinatorial type.

The space ${{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}}$ may be embedded into a real vector space as a balanced, weighted, pure-dimensional polyhedral fan as in [Reference Gathmann, Kerber and Markwig6]. We briefly recall this construction. A weighted fan $(X,\omega)$ is a fan X in ${{{\mathbb{R}}}}^n$ where each top-dimensional cone $\sigma$ has a positive integer weight associated to it, denoted by $\omega(\sigma)$ . A weighted fan is balanced if for all cones $\tau$ of codimension one, the weighted sum of primitive normal vectors of the top-dimensional cones $\sigma_i\supset\tau$ is 0, i.e.,

\begin{equation*}\sum_{\sigma_i\supset\tau}\omega(\sigma_i)\cdot u_{\sigma_i/\tau}=0\in V/V_\tau,\end{equation*}

where $u_{\sigma_i/\tau}$ is the primitive normal vector, V is the ambient real vector space, and $V_\tau$ is the smallest vector space containing the cone $\tau$ . See [Reference Gathmann, Kerber and Markwig6, construction 2.3] for a construction of the primitive normal vectors $u_{\sigma_i/\tau}$ .

For a curve ${{{\mathcal{C}}}}$ , define ${{{\textrm{dist}}}}(i,\;j)$ as the sum of lengths of all bounded edges between the ends marked by i and j. Then the vector

(2·1) \begin{align} d({{{\mathcal{C}}}})=({{{\textrm{dist}}}}(i,\;j))_{i\lt j}\in{{{\mathbb{R}}}}^{\binom{n}{2}}/\Phi({{{\mathbb{R}}}}^n)=Q_n \end{align}

identifies ${{{\mathcal{C}}}}$ uniquely, where $\Phi\,:\,{{{\mathbb{R}}}}^n\rightarrow{{{\mathbb{R}}}}^{\binom{n}{2}}$ by $x\mapsto(x_i+x_j)_{i\lt j}$ .

In [Reference Fry5], an alternate stability condition using a combinatorial graph is introduced for rational pointed tropical curves. Let $\Gamma$ be a simple graph on $n-1$ vertices, labelled $2,...,n$ with at least one edge. Denote $E(\Gamma)$ as its edge set. The graph $ \Gamma $ controls which pairs of markings are allowed to collide. Edges in $ \Gamma $ correspond to pairs of points that cannot collide, while missing edges, with respect to the complete graph $K_{n-1}\supseteq\Gamma $ , correspond to pairs of points that can collide.

Note that there are two types of graphs in this definition. The graph ${{{\mathcal{C}}}}$ corresponds to a stable tropical curve, which we will reference using the terminology bounded edges, ends, unmarked vertices, and markings labelled $1,\ldots,n$ . Whereas the graph $\Gamma$ is a combinatorial graph with edges and vertices labelled $2,\ldots,n$ .

As a cone complex, ${{{{{{\mathcal{M}}}}_{0,\Gamma}^{\textrm{trop}}}}}$ is a subcomplex of ${{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}}$ . Using graphic stability, there exists a projection map from the vector space $Q_n={{{\mathbb{R}}}}^{\binom{n}{2}-n}$ to ${{{\mathbb{R}}}}^{\binom{n}{2}-n-N}$ that forgets the coordinates corresponding to the N edges removed from $K_{n-1}$ to obtain $\Gamma$ , see [Reference Fry5, equations 7, 8]. Although there are two projections defined, lemma 3·18 of [Reference Fry5] identifies them via a linear transformation, so we will use ${{{\textrm{pr}_{\Gamma}}}}$ to refer to both projection maps. We will see in lemma 3·3 that ${{{\textrm{pr}_{\Gamma}}}}$ is the tropicalisation of a regular map between algebraic tori.

2·3. Geometric tropicalisation for $\mathcal{M}_{0,n}$

Two theories, developed simultaneously, arise when dealing with tropicalisations of subvarieties of tori: tropical compactification and geometric tropicalisation. The former, introduced by Tevelev [Reference Tevelev17], describes a situation where the tropical variety determines a good choice of compactification. Specifically, the tropical compactification of $U\subset{{{\mathbb{T}}}}^r$ is its closure $\overline{U}$ in a toric variety $X(\Sigma)$ with $|\Sigma|=\textrm{trop}(U)$ . The latter, introduced by Hacking, Keel and Tevelev [Reference Hacking, Keel and Tevelev10] and further developed by Cueto [Reference Cueto3], explores the converse statement, how a nice compactification determines its tropicalisation.

We recall some useful definitions for geometric tropicalisation. We note that geometric tropicalisation can be completed with more relaxed conditions, such as replacing a smooth compactification with a normal, ${{{\mathbb{Q}}}}$ -factorial compactification and replacing simple normal crossing by combinatorial normal crossings. For explicit details, see [Reference Cueto3].

Let U be a smooth subvariety of a torus ${{{\mathbb{T}}}}^r$ and Y be a smooth compactification containing U as a dense open subvariety. The boundary of Y, $\partial Y=Y\setminus U$ , is divisorial if it is a union of codimension-1 subvarieties of Y. We say $(Y,\partial Y)$ is a simple normal crossings (snc) pair when the boundary of Y behaves locally like an arrangement of coordinate hyperplanes. In other words, $\partial Y$ is an snc divisor if a non-empty intersection of k irreducible boundary divisors is codimension k and the intersection is transverse. The boundary complex of Y, $\Delta(\partial Y)$ , is a simplicial complex whose vertices are in bijection with the irreducible divisors of the boundary divisor $\partial Y$ , and whose k-cells correspond to a non-empty intersection of k boundary divisors. The cells containing a face $\tau$ correspond to the boundary strata that lie in the closure of $\tau$ ’s stratum.

Let $\phi_1,\ldots,\phi_r\in {\mathcal{O}}^*(U)$ . The $\phi_i$ ’s define a morphism $\vec{\phi}$ from U to a torus ${{{\mathbb{T}}}}^r$ , sending $u\in U$ to $(\phi_1(u),\ldots,\phi_r(u))$ . When there are enough invertible functions, this map is an embedding. Given an irreducible boundary divisor $D\subset\partial Y$ we can compute the order of vanishing of each $\phi_i$ on D, $\textrm{ord}_D(\phi_i)$ , yielding an r-dimensional integer vector $\vec{v}_D=(\textrm{ord}_D(\phi_1),\ldots,\textrm{ord}_D(\phi_r))$ living inside the cocharacter lattice of ${{{\mathbb{T}}}}^r$ , $N_{{{{\mathbb{T}}}}^r}\subseteq N_{{{\mathbb{R}}}}={{{\mathbb{R}}}}^r$ . Let $\pi:\Delta(\partial Y)\rightarrow N_{{{\mathbb{R}}}}$ be the map defined by sending a vertex $v_i$ to $\vec{v}_{D_i}$ and extending linearly on every simplex. We call $\pi$ a divisorial valuation map. Geometric tropicalisation says precisely that the support of the tropical fan is the cone over this complex and this result is independent of our choice of compactification Y, i.e., $\textrm{trop}(U)=\textrm{cone}(\textrm{Im}(\pi))$ . As we will see later, $\pi$ is not necessarily injective, so $\textrm{trop}(U)$ may not be the cone over $\Delta(\partial Y)$ .

Tevelev [Reference Tevelev17, theorem 5·5] first computes the tropicalisation of $\mathcal{M}_{0,n}$ via geometric tropicalisation by combining results of [Reference Kapranov12, Reference Speyer and Sturmfels16]. This result is generalised by Gibney and Maclagan [Reference Gibney and Maclagan7, theorem 5·7]. They use the fact that $\mathcal{M}_{0,n}$ can be embedded into a torus of dimension $\binom{n}{2}-n$ using the Plücker embedding of the Grassmannian G(2,n) into ${{{\mathbb{P}}}}^{\binom{n}{2}-1}$ . For explicit details, see [Reference Gibney and Maclagan7, Reference Maclagan and Sturmfels14]. Comparing the algebraic Plücker embedding to the tropical distance coordinates we realise that the distance coordinates from equation (2·1) can be recovered from the tropicalisation of the Plücker coordinates, for details see [Reference Gross8, section 3·1].

Fig. 2. The boundary complex of $\overline{{{{\mathcal{M}}}}}_{0,5}$ with divisors (vertices) labelled by their index set.

We may define ${{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}}$ alternatively as the cone over $\Delta(\partial{{{\overline{{{\mathcal{M}}}}_{0,n}}}})$ . Geometric tropicalisation states precisely that $\textrm{cone}(\Delta(\partial{{{\overline{{{\mathcal{M}}}}_{0,n}}}}))=\textrm{trop}({{{{{{\mathcal{M}}}}_{0,n}}}})$ . The following theorem, due to Tevelev and Gibney-Maclagan, states that ${{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}}=\textrm{trop}({{{{{{\mathcal{M}}}}_{0,n}}}})$ .

Theorem 2·1 ([Reference Gibney and Maclagan7, Reference Tevelev17]). The geometric tropicalisation of ${{{\overline{{{\mathcal{M}}}}_{0,n}}}}$ via the embedding

\begin{equation*}{{{{{{\mathcal{M}}}}_{0,n}}}}\hookrightarrow T^{\binom{n}{2}-n}\end{equation*}

gives the fan $\textrm{trop}({{{{{{\mathcal{M}}}}_{0,n}}}})$ whose underlying cone complex is identified with ${{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}}$ . Furthermore, the tropical compactification of $\mathcal{M}_{0,n}$ in the toric variety $X({{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}})$ is ${{{\overline{{{\mathcal{M}}}}_{0,n}}}}$ .

It follows from the previous theorem that the divisorial valuation map is injective, and thus induces a bijective map of cone complexes from ${{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}}$ to $\textrm{trop}({{{{{{\mathcal{M}}}}_{0,n}}}})$ . This is an important fact that we will revisit when discussing $\Gamma$ -stability.

3·1. The moduli space of rational graphically stable curves

In [Reference Smyth15], Smyth gives a complete classification of all modular compactifications of $\mathcal{M}_{0,n}$ using combinatorial objects called extremal assignments (theorem 1.9). In this section, we prove that ${{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}$ is a modular compactification of $\mathcal{M}_{0,n}$ by showing that $\Gamma$ -stability, as in definition 2·3, is an extremal assignment over $\mathcal{M}_{0,n}$ . In addition, we show that the pair $({{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}},\partial{{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}})$ is an snc pair. Before we define $\Gamma$ -stable curves and the moduli space ${{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}$ , we first introduce notation that tracks the markings that may collide.

We say a subcurve of a proper algebraic curve C over an algebraically closed field is a reduced closed subscheme of C. Let $(C,p_1,\ldots,p_n)$ be an n-marked curve, let x be a closed point of C, and Z be a subcurve of C. Denote the set of markings at x to be

\begin{equation*}\textrm{Mar}(x)\,{:\def\negativespace{}=}\,\{i\in[n]\ |\ p_i=x\}\end{equation*}

and the set of markings a subcurve Z to be

\begin{equation*}\textrm{Mar}(Z)\,{:\def\negativespace{}=}\,\{i\in[n]\ |\ p_i\in Z\} = \bigcup_{x\in Z}\textrm{Mar}(x).\end{equation*}

For a subset $I\subset\{2,\ldots,n\}$ let $\Gamma_I$ be the induced subgraph of $\Gamma$ containing the vertices indexed by I and all edges of $\Gamma$ between those vertices. We denote two special subgraphs of $\Gamma$ given by the markings at a closed point x and the markings in a subsub Z:

\begin{equation*}\Gamma_x\,{:\def\negativespace{}=}\,\Gamma_{\textrm{Mar}(x)} \textrm{ and } \Gamma_Z\,{:\def\negativespace{}=}\,\Gamma_{\textrm{Mar}(Z)}.\end{equation*}

Define ${{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}$ to be the parameter space of all rational $\Gamma$ -stable n-marked curves with the interior ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ to be all smooth rational $\Gamma$ -stable n-marked curves.

Consider the assignment defined by

(3·1) \begin{align} {{{\mathcal{Z}}}}(C) = \{Z\subset C \ |\ |Z\cap Z^c|=1,\ p_1\not\in Z,\ E(\Gamma_Z) = \emptyset \}\end{align}

If we call a subcurve $Z\subset C$ satisfying $|Z\cap Z^c|=1$ a tail, then the assignment ${{{\mathcal{Z}}}}$ is defined by picking out all tails of $(C,p_1,\ldots,p_n)$ not containing $p_1$ that have no edges in $\Gamma$ between vertices corresponding to the marked points on the tail. For the purposes of this document, we require $\Gamma$ to be a simple connected graph on $n-1$ vertices, in which case equation (3·1) is an extremal assignment.

By definition, the only components contracted by ${{{\mathcal{Z}}}}$ -stability are exactly those which are contracted by $\Gamma$ -stability. Every tail is contracted to a point of singularity type (0,1) which is a smooth point. Therefore, ${{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}=\overline{{{{\mathcal{M}}}}}_{0,n}({{{\mathcal{Z}}}})$ (as defined in [Reference Smyth15]) and so ${{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}$ is a modular compactification of $\mathcal{M}_{0,n}$ .

3·2. Geometric tropicalisation for ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$

For the remainder of the paper we add the condition that $\Gamma$ is connected. We also assume without loss of generality that $\Gamma$ contains the edge $e_{23}$ . The connected assumption is used in the proof of lemma 3·4.

The aim of this section is to walk through the process of geometric tropicalisation for the case of $\Gamma$ -stability and study the tropical compactification of ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ . We begin by investigating the projection of the Plücker embedding of $\mathcal{M}_{0,n}$ . Graphic stability defines a projection map that will give a torus embedding using the remaining Plücker coordinates. Next, we examine the divisorial valuation map from the boundary complex of ${{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}$ into the cocharacter lattice of the torus. Fixing $e_{23}\in E(\Gamma)$ prescribes a set of coordinates on the torus. The tropicalisation is a fan which coincides with the tropical moduli space ${{{{{{\mathcal{M}}}}_{0,\Gamma}^{\textrm{trop}}}}}$ if and only if $\Gamma$ is complete multipartite.

We may set up a torus embedding for ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ in the following way. Recall that the Plücker embedding is given by sending a $2\times n$ matrix, representing a choice of basis for a subspace V, to its vector of $2\times 2$ minors called the Plücker coordinates. Let $\textrm{Mat}^\Gamma(2,n)$ be the set of $2\times n$ matrices where the $ij^\textrm{th}$ minor, $z_{ij}$ , is nonzero whenever $i=1$ or $e_{ij}\in E(\Gamma)$ . Let $G^\Gamma(2,n)$ be the open subspace of G(2,n) given by $\textrm{Mat}^\Gamma(2,n)$ ; that is, the points of $G^\Gamma(2,n)$ are given by the subset of nonvanishing Plücker coordinates $z_{ij}$ whenever $i=1$ or $e_{ij}\in E(\Gamma)$ . Let k be an algebraically closed field and consider the action of the $(n-1)$ -dimensional torus $T^{n-1}=(k^*)^n/k^*$ on ${{{\mathbb{P}}}}^{\binom{n}{2}-1}$ given by

(3·2) \begin{align} (t_1,\ldots,t_n)\cdot[z_{ij}]_{1\leq i\lt j\leq n}=[t_it_jz_{ij}]_{1\leq i\lt j\leq n}.\end{align}

The $T^{n-1}$ torus action amounts to a nonzero scaling of the columns of the $2\times n$ matrices in $G^\Gamma(2,n)$ modulo diagonal scaling and $T^{n-1}$ acts freely on $G^\Gamma(2,n)$ .

An n-tuple of (potentially overlapping) points of ${{{\mathbb{P}}}}^1$ , $([x_1\,:\,y_1],\ldots,[x_n\,:\,y_n])$ , may be encoded into a $2\times n$ matrix where each point is a column of the matrix. Thus, $([x_1\,:\,y_1],\ldots,[x_n\,:\,y_n])$ is sent to $(z_{12}\,:\,\cdots\,:\,z_{n-1n})\in{{{\mathbb{P}}}}^{\binom{n}{2}-1}$ where $z_{ij}=x_iy_j-x_jy_i$ . The coordinates $z_{ij}$ are nonzero precisely when the ith and jth points are distinct; in this way, ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ is equal to the quotient $G^\Gamma(2,n)/T^{n-1}$ .

After applying ${{{\textrm{Pr}_{\Gamma}}}}$ , the images of the remaining Plücker coordinates are non-zero, meaning the image of $\textrm{Pl}\left(G^\Gamma(2,n)\right)$ via ${{{\textrm{Pr}_{\Gamma}}}}$ lives inside a torus in ${{{\mathbb{P}}}}^{\binom{n}{2}-1-N}$ . Recall that $\mathcal{M}_{0,n}$ lives inside an $(\binom{n}{2}-n)$ -dimensional torus, $T^{\binom{n}{2}-n}\subset {{{\mathbb{P}}}}^{\binom{n}{2}-1}$ . The projection map ${{{\textrm{Pr}_{\Gamma}}}}$ is regular on $T^{\binom{n}{2}-n}$ ; in fact, we prove in lemma 3·2 that the projection of $T^{\binom{n}{2}-n}$ via ${{{\textrm{Pr}_{\Gamma}}}}$ is an $\left(\binom{n}{2}-n-N\right)$ -dimensional torus containing ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ . Note the embedding into the quotient torus is given by $\textrm{Pl}$ , but then we may choose an isomorphism to a torus of correct dimension. The diagram in Figure 3 summarises the above conversation.

Fig. 3. Torus embedding of ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ via Plücker map.

Proof. Let $({{{\mathbb{P}}}}^1,(x_1\,:\,y_1),\ldots,(x_n\,:\,y_n))$ be a $\Gamma$ -stable curve in ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ . Using the discussion preceding definition 3·3, this marked curve, up to the $T^{n-1}$ action in equation 3·2, corresponds to a point $\vec{x}\,{:\def\negativespace{}=}\,(z_{12}\,:\,\cdots\,:\,z_{n-1n})\in{{{\mathbb{P}}}}^{\binom{n}{2}-1}$ where $z_{ij}=x_iy_j-x_jy_i$ . By $\Gamma$ -stability, the $z_{ij}$ coordinates are allowed to be zero when $e_{ij}\not\in E(\Gamma)$ , which are the same coordinates that are forgotten by ${{{\textrm{Pr}_{\Gamma}}}}$ . Since each coordinate of ${{{\textrm{Pr}_{\Gamma}}}}(\vec{x})\in{{{\mathbb{P}}}}^{\binom{n}{2}-1-N}$ is necessarily nonzero, ${{{\textrm{Pr}_{\Gamma}}}}(\vec{x})$ lies in the big open torus of ${{{\mathbb{P}}}}^{\binom{n}{2}-1-N}$ , denoted $T^{\binom{n}{2}-1-N}$ . The torus action in equation 3·2 is carried through the projection, therefore

\begin{equation*}({{{\textrm{Pr}_{\Gamma}}}}\circ\textrm{Pl})({{{{{{\mathcal{M}}}}_{0,\Gamma}}}})\subset T^{\binom{n}{2}-1-N}/T^{n-1}\,{:\def\negativespace{}=}\,T^{\binom{n}{2}-n-N}.\end{equation*}

Finally, we show that ${{{\textrm{Pr}_{\Gamma}}}}$ is injective on the image of ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ . Fix $\vec{r},\vec{s}\in {{{\textrm{Pl}}}}({{{{{{\mathcal{M}}}}_{0,\Gamma}}}})$ and suppose ${{{\textrm{Pr}_{\Gamma}}}}(\vec{r}) = {{{\textrm{Pr}_{\Gamma}}}}(\vec{s})$ . Then for some $\lambda\in{{{\mathbb{C}}}}^*$ , $r_{ij} = \lambda s_{ij}$ whenever $i=1$ or $e_{ij}\in E(\Gamma)$ . Fix such a $\lambda$ .

We wish to show $r_{kl}=\lambda s_{kl}$ whenever $e_{kl}\not\in E(\Gamma)$ . Using the automorphisms of ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ , we fix the first two marked points of the $\Gamma$ -stable curves sent to $\vec{r}$ and $\vec{s}$ to be $(0\;:\;1)$ and $(1\;:\;0)$ . We write the remaining marked points as $(1\;:\;t_i)$ and $(1\;:\;u_i)$ for $\vec{r}$ and $\vec{s}$ , respectively, so that

\begin{equation*}r_{ij}=\begin{cases} -1, & i=1\lt j\\[-3pt] t_j, & i=2\lt j\\[-3pt] t_j-t_i & 2\lt i\lt j\end{cases} \hspace{30pt} s_{ij}=\begin{cases} -1, & i=1\lt j\\[-3pt] u_j, & i=2\lt j\\[-3pt] u_j-u_i & 2\lt i\lt j\end{cases}.\end{equation*}

If $k=2$ , then $r_{2l}=t_l$ and $s_{2l}=u_l$ .

Consider the path from vertices 2 to l passing through vertices $i_1,\ldots,i_m$ (which exists because $\Gamma$ is a connected graph), as in Figure 4.

Fig. 4. Path for lemma 3·4.

This path gives us the following sequence of equations

\begin{equation*}r_{2i_1}=\lambda s_{2i_1},\ r_{i_1i_2}=\lambda s_{i_1i_2},\ \ldots,\ r_{i_ml}=\lambda s_{i_ml}\end{equation*}

yielding

\begin{equation*}t_{i_1}=\lambda u_{i_1},\ t_{i_2}-t_{i_1}=\lambda (u_{i_2}-u_{i_1}),\ \ldots,\ t_{l}-t_{i_m}=\lambda (u_{l}-u_{i_m}).\end{equation*}

By substituing the first in equation into the second we see that $t_{i_2}=\lambda u_{i_2}$ . Continuing the substitution process eventually yields $t_{l}=\lambda u_{l}$ , and hence $r_{2l} =\lambda s_{2l}$ .

On the other hand, if $k\neq2$ , we can repeat the above strategy by picking separate paths from 2 to k and 2 to l to get $t_{k}=\lambda u_{k}$ and $t_{l}=\lambda u_{l}$ , respectively. This completes the proof as

\begin{equation*}r_{kl} = t_l-t_k = \lambda (u_{l}-u_{k}) = \lambda s_{kl}.\end{equation*}

The boundary of ${{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}$ is divisorial in the same way that $\partial{{{\overline{{{\mathcal{M}}}}_{0,n}}}}$ is divisorial, except that there are fewer irreducible divisors. The ratios $z_{ij}/z_{23}$ , for $2\leq i\lt j\leq n$ , $(i,\;j)\neq(2,3)$ and $e_{ij}\in E(\Gamma)$ , are rational functions on ${{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}$ and act as a choice of coordinates on the torus $T^{\binom{n}{2}-n-N}$ .

Define the divisorial valuation map $\pi_\Gamma:\Delta(\partial{{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}})\rightarrow N_{{{\mathbb{R}}}}$ by assigning the vector $\vec{v}_{D_I}=(\textrm{ord}_{D_I}(z_{24}/z_{23}),\ldots,\textrm{ord}_{D_I}(z_{n-1n}/z_{23}))$ to a divisor $D_I$ where

\begin{equation*}\textrm{ord}_{D_I}(z_{ij}/z_{23})=\begin{cases} 1 & \{2,3\}\not\subset I,\ \{i,\;j\}\subset I, \textrm{ and } e_{ij}\in E(\Gamma); \\[3pt] -1 & \{2,3\}\subset I,\ \{i,\;j\}\not\subset I, \textrm{ and } e_{ij}\in E(\Gamma); \\[3pt] 0 & \textrm{else}.\end{cases}\end{equation*}

This means

(3·3) \begin{align}\pi_\Gamma(D_I)=\vec{v}_{D_I}=\begin{cases} \displaystyle\sum_{\substack{i,\;j\in I;\\ e_{ij}\in E(\Gamma)}}\vec{e}_{ij} & \{2,3\}\not\subset I;\\[2.5pc] \displaystyle-\sum_{\substack{i,\;j\not\in I;\\ e_{ij}\in E(\Gamma)}}\vec{e}_{ij} & \{2,3\}\subset I.\end{cases}\end{align}

The standard basis vectors of $T^{\binom{n}{2}-n-N}$ are given by $\vec{v}_{D_{\{i,\;j\}}}$ , where $e_{ij}\in E(\Gamma)\setminus\{e_{23}\}$ . Additionally we have $\vec{v}_{D_{\{2,3\}}}=-\vec{1}$ . For a divisor $D_I$ with $|I|\geq3$ ,

(3·4) \begin{align} \vec{v}_{D_I}&=\sum_{\substack{\{i,\;j\}\subset I;\\ e_{ij}\in E(\Gamma)}} \vec{v}_{D_{\{i,\;j\}}}.\end{align}

From [Reference Fry5], we know that ${{{{{{\mathcal{M}}}}_{0,\Gamma}^{\textrm{trop}}}}}={{{\textrm{pr}_{\Gamma}}}}({{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}})$ if and only if $\Gamma$ is a complete multipartite graph. Tropically, this characterisation comes from studying the injectivity of a restriction morphism on graphic matroids. Algebraically, we study the injectivity of the divisorial valuation maps. Unlike in the $\mathcal{M}_{0,n}$ case (discussed in Section 2·3), the map $\pi_\Gamma$ may not be injective. There is a similar relation to equation (3·4) for ${{{\overline{{{\mathcal{M}}}}_{0,n}}}}$ that we may use to demonstrate the simplest case of non-injectivity: consider the divisor $D_{\{i,\;j,k\}}$ in ${{{\overline{{{\mathcal{M}}}}_{0,n}}}}$ and its image under the divisorial valuation map

(3·5) \begin{align} \vec{v}_{D_{\{i,\;j,k\}}}=\vec{v}_{D_{\{i,\;j\}}}+\vec{v}_{D_{\{i,k\}}}+\vec{v}_{D_{\{\;j,k\}}}.\end{align}

If exactly two of the vectors on the right correspond to $\Gamma$ -unstable divisors, then $\pi_\Gamma$ cannot be injective. This case does not happen when $\Gamma$ is complete multipartite. Example 3·7 demonstrates the failure of injectivity for $\pi_\Gamma$ , while example 3·8 exhibits a case where $\pi_\Gamma$ is injective.

Example 3·7. Let $\widetilde{\Gamma}$ be the subgraph of $K_4$ with edges $e_{35}$ and $e_{45}$ removed; see Figure 5. Then we have ${{{{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}}}}\hookrightarrow T^{\binom{5}{2}-5-2}=T^3$ with coordinates $z_{24}/z_{23},\ z_{25}/z_{23},$ and $z_{34}/z_{23}$ . In ${{{\overline{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}}}}$ , there are 8 irreducible boundary divisors, labelled in Figure 6(a). Comparing the cone complexes of ${{{{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}^{\textrm{trop}}}}}$ and $\textrm{trop}({{{{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}}}})$ , we can see that the cones associated to the boundary strata $D_{\{3,4\}}$ , $D_{\{3,4,5\}}$ , and $D_{\{3,4\}}\cap D_{\{3,4,5\}}$ in ${{{{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}^{\textrm{trop}}}}}$ are all mapped to the ray given by $D_{\{3,4\}}$ in $\textrm{trop}({{{{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}}}})$ . Explicitly, $\pi_{\widetilde{\Gamma}}\,:\,\Delta(\partial{{{\overline{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}}}})\rightarrow {{{\mathbb{R}}}}^{3}$ where the divisors have been mapped to the following primitive vectors:

\begin{align*} \vec{v}_{D_{\{2,4\}}}&=(1,0,0) & \vec{v}_{D_{\{2,5\}}}&=(0,1,0) &\vec{v}_{D_{\{3,4\}}}&=(0,0,1)&\\ \vec{v}_{D_{\{2,3\}}}&=(-1,-1,-1) & \vec{v}_{D_{\{2,3,4\}}}&=(0,-1,0) &\vec{v}_{D_{\{2,3,5\}}}&=(-1,0,-1)&\\ \vec{v}_{D_{\{2,4,5\}}}&=(1,1,0) & {\vec{v}_{D_{\{3,4,5\}}}}&{=(0,0,1)} & &&\end{align*}

Fig. 5. The graph $\widetilde{\Gamma}$ in example 3·7.

Fig. 6. Cone complexes for example 3·7.

Example 3·8. Let $\Gamma=K_{2,2}$ be the complete bipartite graph obtained by removing edges $e_{25}$ and $e_{34}$ from $K_4$ , as shown in Figure 7(a). Then we have ${{{\mathcal{M}}}}_{0,K_{2,2}}\hookrightarrow T^{\binom{5}{2}-5-2}=T^3$ with coordinates $z_{24}/z_{23},\ z_{35}/z_{23},$ and $z_{45}/z_{23}$ . In $\overline{{{{\mathcal{M}}}}}_{0,K_{2,2}}$ , there are 8 irreducible boundary divisors, labelled in Figure 7(b). Explicitly, $\pi_\Gamma:\Delta(\partial\overline{{{{\mathcal{M}}}}}_{0,K_{2,2}})\rightarrow {{{\mathbb{R}}}}^{3}$ where the divisors have been mapped to the following primitive vectors:

\begin{align*} \vec{v}_{D_{\{2,4\}}}&=(1,0,0) & \vec{v}_{D_{\{3,5\}}}&=(0,1,0) &\vec{v}_{D_{\{4,5\}}}&=(0,0,1).&\\[5pt] \vec{v}_{D_{\{2,3\}}}&=(-1,-1,-1) & \vec{v}_{D_{\{2,3,4\}}}&=(0,-1,-1) &\vec{v}_{D_{\{2,3,5\}}}&=(-1,0,-1)&\\[6pt] \vec{v}_{D_{\{2,4,5\}}}&=(1,0,1) & {\vec{v}_{D_{\{3,4,5\}}}}&{=(0,1,1)} & &&\end{align*}

Fig. 7. Graph and cone complexes for example 3·8.

Lemma 3·9 gives useful characterisations of a complete multipartite graph also used in [Reference Fry5].

Proof. We begin by proving the forwards direction by contradiction. Suppose $\pi_\Gamma$ is injective and $\Gamma$ is not complete multipartite. Using lemma 3·9, fix three vertices $v_i, v_j,$ and $v_k$ where $e_{ij}\in E(\Gamma)$ but $e_{ik},e_{jk}\not\in E(\Gamma)$ . We have the following contradiction

\begin{equation*}\pi_\Gamma(D_{\{i,\;j,k\}})=\vec{v}_{D_{\{i,\;j,k\}}}=\vec{v}_{D_{\{i,\;j\}}}=\pi_\Gamma(D_{\{i,\;j\}}).\end{equation*}

For the backwards direction, assume $\Gamma$ is complete multipartite. Let $D_I$ and $D_J$ be two $\Gamma$ -stable divisors such that $\vec{v}_{D_I}=\vec{v}_{D_J}.$ By equation (3·3), $\{2,3\}\subseteq I$ if and only if $\{2,3\}\subseteq J$ . If $\{2,3\}\subseteq I,J$ , then we have

\begin{equation*}-\vec{1} + \sum_{\substack{\{i,\;j\}\subset I,\ \{i,\;j\}\neq\{2,3\};\\ e_{ij}\in E(\Gamma)}}\vec{e}_{ij} = -\vec{1} + \sum_{\substack{\{i,\;j\}\subset J,\ \{i,\;j\}\neq\{2,3\};\\ e_{ij}\in E(\Gamma)}}\vec{e}_{ij}.\end{equation*}

If $\{2,3\}\not\subseteq I,J$ , then we have

\begin{equation*}\sum_{\substack{\{i,\;j\}\subset I;\\ e_{ij}\in E(\Gamma)}}\vec{e}_{ij} = \sum_{\substack{\{i,\;j\}\subset J;\\ e_{ij}\in E(\Gamma)}}\vec{e}_{ij}.\end{equation*}

In either case, this implies that the induced subgraphs $\Gamma_I$ and $\Gamma_J$ of $\Gamma$ have the same edge sets, $E(\Gamma_I)=E(\Gamma_J)$ . If $I\neq J$ , then there exists $i\in I\setminus J$ . But $v_i\in\Gamma$ must be isolated in $\Gamma_I$ , otherwise it would be contained in an edge in $E(\Gamma_I)$ and thus $i\in J$ . However, $\Gamma_I$ is a complete multipartite graph, so it cannot have any isolated vertices. Therefore, $I=J$ , concluding the proof that $\pi_\Gamma$ is injective.

Additionally, the map $\pi_\Gamma$ induces a map of cone complexes from ${{{{{{\mathcal{M}}}}_{0,\Gamma}^{\textrm{trop}}}}} = \textrm{cone}(\Delta(\partial{{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}))$ to $\textrm{trop}({{{{{{\mathcal{M}}}}_{0,\Gamma}}}}) = \textrm{cone}(\textrm{Im}(\pi_\Gamma))$ which is an isomorphism if and only if $\Gamma$ is complete multipartite. Furthermore, we know that ${{{{{{\mathcal{M}}}}_{0,\Gamma}^{\textrm{trop}}}}}={{{\textrm{pr}_{\Gamma}}}}({{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}})$ is a balanced fan (with constant weight function 1) by [Reference Fry5, theorem 29].

In ${{{{{{\mathcal{M}}}}_{0,n}^{\textrm{trop}}}}}$ , $\tau$ is contained in three facets spanned by the rays of $\tau$ and an additional ray coming from one of the tropical curves $D_{\{i,\;j\}}$ , $D_{\{i,k\}}$ , or $D_{\{\;j,k\}}$ . However, the tropical curves $D_{\{i,k\}}$ and $D_{\{\;j,k\}}$ are not $\Gamma$ -stable, and thus their rays are contracted in ${{{{{{\mathcal{M}}}}_{0,\Gamma}^{\textrm{trop}}}}}$ . In ${{{{{{\mathcal{M}}}}_{0,\Gamma}^{\textrm{trop}}}}}$ , the image of $\tau$ is a condimension-one cone contained in only a single facet, and thus cannot be balanced in any embedding.

Before the main result of the paper, we prove a lemma that identifies the units of ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ as forgetful morphisms to ${{{\mathcal{M}}}}_{0,4}$ by extending cross ratios on $\mathcal{M}_{0,n}$ to ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ . Not all forgetful morphisms can be considered as units of ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ because the space may contain points corresponding to curves that are not mapped to ${{{\mathcal{M}}}}_{0,4}$ by every forgetful morphism. For example, if $\Gamma=K_{2,3}$ , ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ contains a point corresponding to the curve with three distinct marked points $p_1,$ $p_2=p_3,$ and $p_4=p_5=p_6$ . If $p_2$ and $p_3$ are forgotten, the point of ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ doesn’t land in ${{{\mathcal{M}}}}_{0,4}$ . Thus we want to only keep the extended forgetful morphisms for which this doesn’t happen.

Proof. The space $\mathcal{M}_{0,n}$ can be viewed as the subset of $({{{\mathbb{C}}}}^*\setminus\{1\})^{n-3}$ minus the hyperplanes $x_i-x_j=0$ . The functions which don’t vanish on $\mathcal{M}_{0,n}$ are rational functions that have zeros and poles on the hyperplanes, i.e. Laurent monomials in $x_i$ , $x_i-1$ , and $x_i-x_j$ . We claim that we can write any monomial function as a product of cross ratios:

\begin{equation*}\frac{(P_1-P_2)(P_3-P_4)}{(P_1-P_3)(P_2-P_4)}\end{equation*}

for marked points $P_1, P_2, P_3, P_4$ .

For $x_i$ , let $P_1=x_i$ , $P_2=0$ , $P_3=\infty$ , and $P_4=1$ .

For $x_i-1$ , let $P_1=x_i$ , $P_2=1$ , $P_3=\infty$ , and $P_4=0$ .

For $x_i-x_j$ , take a product of $x_i$ and $P_1=x_i$ , $P_2=x_j$ , $P_3=0$ , and $P_4=\infty$ .

Consider the embedding of $\mathcal{M}_{0,n}$ into ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ in the diagram below where $\phi$ is a unit of ${\mathcal{O}}^*({{{{{{\mathcal{M}}}}_{0,\Gamma}}}})$ .

From arguments above, $\tilde{\phi}$ must be a product of cross ratios. Because $\mathcal{M}_{0,n}$ is dense in ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ , $\phi$ must be the extension of $\tilde{\phi}$ which does not vanish, nor acquire poles at the extra interior points in ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}\backslash{{{{{{\mathcal{M}}}}_{0,n}}}}$ . Indeed, $\phi$ is also a product of cross ratios.

Theorem 3·13. For $\Gamma$ compelte multipartite, there is a torus embedding

\begin{equation*}{{{{{{\mathcal{M}}}}_{0,\Gamma}}}}\hookrightarrow T^{\binom{n}{2}-n-N}=T_\Gamma \end{equation*}

whose tropicalisation $\textrm{trop}({{{{{{\mathcal{M}}}}_{0,\Gamma}}}})$ has underlying cone complex ${{{{{{\mathcal{M}}}}_{0,\Gamma}^{\textrm{trop}}}}}$ . Furthermore, the tropical compactification of ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ is ${{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}$ , i.e, the closure of ${{{{{{\mathcal{M}}}}_{0,\Gamma}}}}$ in the toric variety $X({{{{{{\mathcal{M}}}}_{0,\Gamma}^{\textrm{trop}}}}})$ is ${{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}$ .

We note that condition (1) occurs if and only if $\Gamma$ is a complete multipartite graph, but condition (2) does not force $\Gamma$ to be complete multipartite.

For condition (1), recall that the general element of a boundary divisor $D_I$ has exactly one node and may be described by I, the set of marked points on a component. By lemma 3·12, the units in ${\mathcal{O}}^*({{{{{{\mathcal{M}}}}_{0,\Gamma}}}})$ are generated by forgetful morphisms to ${{{\mathcal{M}}}}_{0,4}$ using cross ratios. Such a forgetful morphism has valuation 1 on D if the image of the general element of D is nodal and valuation 0 on D if the image of the general element of D is smooth. We show forgetful morphisms with this property exists if and only if $\Gamma$ is a complete multipartite graph.

We prove the forwards direction by way of contradiction. Assume $\Gamma$ is not complete multipartite. Using lemma 3·9, fix three vertices $v_i, v_j,$ and $v_k$ where $e_{ij}\in E(\Gamma)$ but $e_{ik},e_{jk}\not\in E(\Gamma)$ . Consider the divisors $D_{\{i,\;j,k\}}$ and $D_{\{i,\;j\}}$ whose intersection yields the stratum S whose dual graph is shown in Figure 8. Every forgetful morphism that has valuation 1 on $D_{\{i,\;j,k\}}$ must not forget i and j, otherwise, the image of the general element of $D_{\{i,\;j,k\}}$ is smooth. However, any such morphism also has valuation 1 on $D_{\{i,\;j\}}$ , a contradiction.

Fig. 8. Dual graph of the stratum S contained in $D_{\{i,\;j\}}$ and $D_{\{i,\;j,k\}}$ .

Now suppose $\Gamma$ is complete multipartite. Fix a stratum S and a divisor $D_I$ containing S, as shown in Figure 9. Our aim is to find four markings, $\{a,b,c,d\}$ , such that the general element of $D_I$ remains nodal and the general element of all other divisors containing S become smooth in ${{{\mathcal{M}}}}_{0,4}$ on $\{a,b,c,d\}$ . We proceed by fixing $d=1$ so that a,b,c correspond to vertices in $\Gamma$ and without loss of generality, let $a,b\in I$ and $c\in I^c\setminus\{1\}$ . To ensure the image of the general element of $D_I$ remains nodal, we must pick a,b such that $e_{ab}\in E(\Gamma)$ , though, we have no additional restrictions of c because of the “global stability” the marking 1 carries with it.

Fig. 9. Dual graphs of the stratum S and divisor $D_{I}$ from the proof of theorem 3·3, where dashed edges represent potential extra components.

Consider the subcurves Z and $Z^c$ of S and $\overline{Z}$ and $\overline{Z}^c$ of $D_I$ that share a node, as illustrated by their dual graphs in Figure 9. We partition I in the following way. Split S into connected components by separating S at the nodes of Z. Now, S has been deconstructed into several connected components: Z with its markings and a connected component for each node on Z. Let $\lambda_I$ be the partition of I given by the markings on Z and the components previously attached to Z, excluding the component with $Z^c$ .

For brevity, we highlight that many facts in this paragraph are justified by stability in some way. If Z has only one node, then $\lambda_I=I$ and we need only choose $a,b\in I$ so that $e_{ab}\in E(\Gamma)$ . Suppose Z has more than one node. There exists at least one part $A\in\lambda_I$ with $|A|\geq2$ , fix such a part A. Let $B=I\setminus A$ , which is necessarily nonempty. Fix an edge $e_{a_1a_2}\in E(\Gamma)$ for markings $a_1,a_2\in A$ . Lemma 3·4 says that for a marking $b\in B$ , either $e_{a_1b}$ or $e_{a_2b}$ is in $E(\Gamma)$ . Hence, there exists $a,b\in I$ so that $e_{ab}\in E(\Gamma)$ when Z has multiple nodes.

Indeed, in each of the above cases the forgetful morphism which remembers $\{1,a,b,c\}$ has valuation 1 on $D_I$ since $e_{ab}\in E(\Gamma)$ . Let $D_J$ be any other divisor containing S. From our choices of a,b,c, we know $|J\cap\{1,a,b,c\}|\neq2$ . This means that the image of the general element $D_J$ under the forgetful morphism which keeps $\{1,a,b,c\}$ will be smooth, and thus $D_J$ has valuation 0.

For condition (2), a stratum S is very affine because it can be viewed as a product of ${{{\mathcal{M}}}}_{0,\Gamma'}$ s. Each component of the universal curve over a point S contains at least one node which acts as the ‘special’ marking 1; the marked points behave under $\Gamma'$ -stability, since any subgraph of a complete multipartite graph is complete multipartite; and any extra node serves as a marked point whose vertex in $\Gamma'$ is connected to all other vertices, which keeps $\Gamma'$ as a complete multipartite graph since a fully connected vertex serves as its own independent set. Finally, since the boundary of ${{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}$ is a simple normal crossings divisor, as in the case of ${{{\overline{{{\mathcal{M}}}}_{0,w}}}}$ , the surjectivity of the restriction map follows the same proof outline as in [Reference Cavalieri, Hampe, Markwig and Ranganathan2, theorem 3·9]. The local structure of $\partial{{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}$ is an intersection of coordinate hyperplanes and restricting the coordinates is surjective.

Many statements remain true when $\Gamma$ is not complete multipartite. The geometric tropicalisation of ${{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}$ using the embedding in lemma 3·4 still equals ${{{\textrm{pr}_{\Gamma}}}}({{{{{{\mathcal{M}}}}_{0,\Gamma}^{\textrm{trop}}}}})=\textrm{trop}({{{{{{\mathcal{M}}}}_{0,\Gamma}}}})$ . However, not all cones are mapped injectively. On the algebraic side, we still have a map from ${{{\overline{{{\mathcal{M}}}}_{0,\Gamma}}}}$ to the toric variety $X(\textrm{trop}({{{{{{\mathcal{M}}}}_{0,\Gamma}}}}))$ , but it does not map all boundary strata injectively. Example 3·14 highlights this observation.

There are only 8 2D cones and 7 rays in $\textrm{trop}({{{{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}}}})$ while the cone over $\partial{{{\overline{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}}}}$ has 9 2D cones and 8 rays. This means $X(\textrm{trop}({{{{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}}}}))$ isn’t large enough to contain ${{{\overline{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}}}}$ . In other words, the locus of smooth curves ${{{{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}}}}$ is missing the limit as the marked points 3, 4, and 5 collide. The modular compactification of ${{{{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}}}}$ assigns a ${{{\mathbb{P}}}}^1$ to the limit but $X(\textrm{trop}({{{{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}}}}))$ doesn’t have enough coordinates to include a ${{{\mathbb{P}}}}^1$ . Rather, this limit gets closed with a single point in $X(\textrm{trop}({{{{{{\mathcal{M}}}}_{0,\widetilde{\Gamma}}}}}))$ (which is the intersection of two smooth curves where the marked points 3 and 4 and 4 and 5, have collided).