We now want to check that codimension-$k$ skeletons of and Psi-classes in
$\M_{0,n}$ are in fact tropical fans, i.e.\ that they satisfy the balancing
condition. So let us fix $ k \in \lk 0,\ldots, n-3\rk $ and a weight function
$w: \M_{0,n}^{(n-3-k)} \to \Z_{>0}$ on the codimension-$k$ skeleton of $
\M_{0,n} $. Moreover, let $ \tau \in \M_{0,n}^{(n-3-k-1)}$ be a cone,
corresponding by Remark \ref {combtypes} to a certain combinatorial type of
$n$-marked curves. In order to verify the balancing condition for $w$ at $ \tau
$, the following toolkit will be useful.

\begin{remark} [About the balancing condition in $ \M_{0,n} $]
    \label{splitting}
  The cones $\sigma \in \M_{0,n}^{(n-3-k)}$ containing $\tau$ can be obtained
  by resolving one chosen vertex $V$ of valence $r$ at least $4$ in $ \tau $ in
  the same way as it can be resolved in $ \M_{0,r} $ by adding one
  bounded edge. More precisely, this means that we replace $V$ by two vertices
  joined by a bounded edge, with the $r$ edges of $V$ split up in every
  possible way onto the two new vertices such that there are at least two such edges
  on each side of the bounded edge. The following picture shows these types of
  resolutions; it can either be seen as a picture of curves in $ \M_{0,r} $ or
  as a \textit {local} picture of curves in $ \M_{0,n} $ around $V$.

  \pict{star}

  In order to check the balancing condition at $ \tau $ it therefore suffices
  to split the total sum $\sum_{\sigma > \tau} w(\sigma) \, v_{\sigma/\tau}$
  into parts, where each part corresponds to the resolution of one vertex $V$,
  and verify it for each part separately. This leads to the following lemma
  which states that the balancing condition for codimension-$k$ cells in $
  \M_{0,n} $ can be split up into several conditions in lower-dimensional
  moduli spaces.
\end{remark}

\begin{lemma} [Splitting the balancing condition] \label{local}
  With notations being as above, let $\lk V_1,\ldots,V_m \rk$ be the set of
  $4$- or higher valent vertices of a curve in $\tau$, and let $ r_1,\dots,r_m
  $ be their respective valences. Then to verify the balancing condition
  at $ \tau $ in $ \M_{0,n} $ it suffices to check the balancing at all
  $0$-dimensional cones $ \tau_i $ in $ \M_{0,r_i}$ for $i \in \lk 1,\ldots,
  m\rk $ (corresponding to curves (``stars'') having only one vertex of valence
  $r_i$).

  \pict{local}
\end{lemma}

\begin{proof}
  Consider a 4- or higher valent vertex $ V \in \{ V_1,\dots,V_m \} $ of $ \tau
  $, let $r$ be its valence, and denote by $ A_i $ for $ i=1,\dots,r $ the set
  of marked edges behind the $i$-th edge of $V$. Thus we have $ \sqcup_i A_i= \lk
1,\ldots,n \rk$. Define a linear map $\phi$ by
    \[ \phi: \R^{\binom r2} \to \R^{\binom n2}, \quad
         (x_{i,j})_{i<j} \mapsto (\tilde x_{k,l})_{k<l} \]
  where
    \[ \tilde x_{k,l} = \begin {cases}
                          x_{i,j} &
                            \mbox {if $ (k,l) \in A_i \times A_j $ or
                                      $ (l,k) \in A_i \times A_j $ for some
                                      $ 1 \le i < j \le r $,} \\
                          0 & \mbox {otherwise} \\
                        \end {cases} \]
  for $ 1 \le k < l \le n $. Then by construction the distance vectors of
  Construction \ref {moduli} are transformed by $ \phi $ as
    \[ \phi ( \tilde v(I)) = \tilde v \Big( \bigcup_{i \in I} A_i \Big)
       \tag {1} \]
  for all subsets $ I \subset \{ 1,\dots,r \} $. In particular, for $ i=1,
  \dots,r $ the $i$-th basis vector $ d_i $ of the lineality space of
  $\M_{0,r}$ is mapped to $ \phi(d_i)=\tilde{v}(A_i) $.

  Let $ \tau_\star $ be the $0$-dimensional cone in $ \M_{0,r} $,
  corresponding to the star curve with one vertex. By Remark \ref {splitting}
  a $1$-dimensional cone $ \sigma_\star $ around $ \tau_\star $ corresponds to
  a cone $ \sigma $ around $ \tau $ in $ \M_{0,n} $ describing the same local
  resolution. Let us assume that the balancing condition holds at $
  \tau_\star $, i.e.\ that
    \[ \sum_{\sigma_\star > \tau_\star}
         w(\sigma) \, \tilde v_{\sigma_\star/ \tau_\star}=\sum_i a_i \, d_i
       \qquad \in \R^{\binom r2}
       \tag {2} \]
  for some $ a_i \in \R $. By Remark \ref {combtypes} the normal vectors
  $ \tilde v_{\sigma_\star/ \tau_\star} $ in $ \M_{0,r} $ are exactly $ \tilde
  v(I) $ for the corresponding subset $ I \subset \{ 1,\dots,r \} $ with $ 2
  \le |I| \le r-2 $, and by (1) these vectors are mapped by $ \phi $ to the
  corresponding normal vectors $ \tilde v_{\sigma/\tau} $ in $ \M_{0,n} $. So
  applying $ \phi $ to (2) we get
    \[ \sum_{\sigma_* > \tau_*}
         w(\sigma) \, \tilde v_{\sigma/\tau}=\sum_i a_i \, \tilde v(A_i)
       \qquad \in \R^{\binom n2}. \]
  The vectors $ \tilde v(A_i) $ lie in the lineality space for $ |A_i|=1 $
  and in $ V_\tau $ otherwise, so taking the quotient by these spaces this sum
  reduces to zero. The claim of the lemma thus follows with the second part of
  Remark \ref {splitting}.
\end{proof}

\begin{proposition} [Codimension-$k$ skeleton of $\M_{0,n}$] \label {comod}
  Let $k \in \lk 0,\ldots, n-3 \rk$. Then the codimension-$k$ skeleton
  ${\M_{0,n}^{(n-3-k)}}_\tn{sk}$ of $\M_{0,n}$ with the weight function $
  w:\M_{0,n}^{(n-3-k)} \to \lk 1\rk $ is balanced. Hence, it is a tropical fan.
\end{proposition}

\begin{proof}
  By Lemma \ref {local} we can reduce the proof to the local situation of a
  star curve with a vertex of some valence $ r \in \{ 4,\dots,k+3 \} $, i.e.\
  to the balancing condition around the vertex $ \tau_\star $ in $ \M_{0,r} $.

  Let us think of this balancing condition in terms of coordinate vectors
  in $ \R^{\binom r2} $. We have to compute the sum $ \tilde v $ of all normal
  vectors $ \tilde{v}_{\sigma_\star / \tau_\star} $ arising from resolving the
  vertex of the star. Consider the first entry of this vector: here, we sum up
  $1$ a number of times, a $1$ for each type where the marked edges $1$ and $2$
  lie on opposite sides of the bounded edge. By symmetry (i.e.\ no marked edge
  is distinguished), this sum is the same in each other entry of $ \tilde v $.
  So $ \tilde v $ is a multiple of the vector $ (1,\ldots,1)^\top $. Consider
  now the lineality space: summing up all vectors $ d_1,\dots,d_r $ gives $ 2
  \cdot (1,\ldots,1)^\top$. So $ \tilde v $ is the zero vector modulo the
  lineality space.
\end{proof}

\begin{proposition} [Codimension-$k$ skeleton of a Psi-class] \label {copsi}
  Let $k \in \lk 0,\ldots, n-4 \rk$ and $i \in \lk 1,\ldots,n \rk$. Then
  the codimension-$k$ skeleton ${\psi_i^{(n-4-k)}}_{\tn{sk}}$ of the $i$-th
  Psi-class $\psi_i$ of $\M_{0,n}$ is balanced for the weight function
  $w:\psi_i^{(n-4-k)} \to \lk 1\rk$. Hence, it is a tropical fan.   	
\end{proposition}

\begin{proof}
  Again we can use Lemma \ref {local} to reduce the proof to the local
  situation of a star with a vertex of some valence $ r \in \{ 4,\dots,k+4 \}
  $. If the $i$-th unbounded edge is not adjacent to the chosen
  vertex then the computation is exactly the same as in the proof of
  Proposition \ref {comod}, so let us assume that it is adjacent to the chosen
  vertex. Then, as in the picture of Example \ref {ex-psi} below, we only have
  to consider resolutions of the star in which the $i$-th edge remains adjacent
  to a vertex of valence at least $4$.

  Think again in terms of coordinate vectors in $ \M_{0,r} $. Again, denote the
  sum of all normal vectors of these resolutions by $ \tilde v $. Then the $
  (j,k) $-coordinates of $ \tilde v $ with $ j,k \neq i $ are all the same by
  symmetry, and likewise for the $ (j,k) $-coordinates where $ j=i $ or $ k=i
  $. So if we set w.l.o.g.\ $ i=1 $ then we can write $ \tilde v $ as
    \[ \tilde v = (\underbrace{M,\ldots,M}_{r-1 \tn{ times }},N,\ldots,N)^\top
                = M \, d_1 + \frac N2 \, (-d_1+d_2+\cdots+d_r) \]
  for suitable $ M,N \in \N $. As this is a vector in the lineality space, the
  balancing condition follows.
\end{proof}

\begin {example} [Codimension-$2$ skeleton of $\psi_1$ in $\M_{0,7}$]
    \label {ex-psi}
  The codimension-2 skeleton of $ \psi_1 $ in $ \M_{0,7} $ is 1-dimensional,
  and thus there is only one balancing condition to check, namely that around
  the 0-dimensional cell corresponding to the star curve with only one vertex.
  Allowed resolutions of this type are:

  \pict{psi1}

  As in the proof of Proposition \ref {copsi} let $ \tilde v $ be the sum of
  all normal vectors of these resolutions. Write $ \tilde v $ as $ \tilde v =
  \tilde v_{\textrm {(a)}} + \tilde v_{\textrm {(b)}} + \tilde v_{\textrm
  {(c)}} $, where the summands denote the parts of $ \tilde v $ arising from
  resolutions of type (a), (b), (c) as in the picture above, respectively. Note
  that each such type corresponds to various resolutions corresponding to the
  choice of labeling of the marked ends. In the array below, for each
  resolution type the coordinates of $ \tilde v $ are listed, where $ j,k \neq
  1 $. 
  \begin{center}
    \begin{tabular}{|c||c|c|c|c|} \hline
      coordinate & $ \tilde v_{\textrm {(a)}} $ &
                   $ \tilde v_{\textrm {(b)}} $ &
                   $ \tilde v_{\textrm {(c)}} $ & $ \tilde v $ \\ \hline \hline
      $ (1,j) $ & $\binom{5}{4}=5$ & $\binom{5}{3}=10$ & $\binom{5}{2}=10$ &
        $25$ \\	\hline
      $ (j,k) $ & $2 \cdot \binom{4}{3}=8$&$2 \cdot \binom{4}{2}=12$&$2
        \cdot \binom{4}{1}=8$ &$28$ \\ \hline
    \end{tabular}
  \end{center}
  For example, the $(1,j)$-coordinate $10$ of $ \tilde v_{\textrm {(b)}} $
  corresponds to the 10 choices of distributing the remaining labels on the
  ends in type (b) if the unbounded edge $j$ has been put at the left vertex.

  So we have
    \[ \tilde v_{\textrm {(a)}} = 4 \sum_{j=1}^7 d_j - 3 d_1, \qquad
       \tilde v_{\textrm {(b)}} = 6 \sum_{j=1}^7 d_j - 2 d_1, \qquad
       \tilde v_{\textrm {(c)}} = 4 \sum_{j=1}^7 d_j + 2 d_1. \]
  Note that it is not just the sum $ \tilde v $ that is zero modulo the
  lineality space, but also the individual vectors $ \tilde v_{\textrm {(a)}}
  $, $ \tilde v_{\textrm {(b)}} $, $ \tilde v_{\textrm {(c)}} $ corresponding
  to the resolution types themselves. In fact, the proof of Proposition \ref
  {copsi} shows that the analogous statement holds for the (one-dimensional)
  codimension-$ (n-5) $ skeleton of a Psi-class in $ \M_{0,n} $ for all $ n \ge
  5 $ since the symmetry argument given there also applies if we only consider
  a single resolution type. \hfill $\circ$
\end {example}

\begin {remark} [Choice of weight function]
  Of course, the computations above depend on the chosen weight function. The
  moduli spaces $ \M_{0,n} $ and the Psi-classes $ \psi_i $, considered as
  tropical cycles, have all weights of their facets equal to $1$, and thus
  it was natural in Propositions \ref {comod} and \ref {copsi} to also equip
  the $k$-skeletons of these cycles with the constant weight function 1. For
  other cycles such as for instance intersection products $ \psi_1^{m_1} \cdot \ldots
  \cdot \psi_n^{m_n} $ of Psi-classes (where $ m_1,\ldots,m_n \in \N_0 $)
  this is in general no longer the case, especially when they are of codimension $2$ or
  higher. 
\end {remark}

