Now we want to explore the question of uniqueness of the weight function for
certain tropical subfans of $\M_{0,n}$. This question is strongly related to
the one of irreducibility introduced in \cite {GKM}. To simplify notations, we
will assume from now on that all weights of the cones in tropical cycles are
non-zero.

\begin {definition} [Irreducible tropical cycles] \label {def-irred}
  A tropical cycle $X$ in $ V=\Lambda \otimes_\Z \R $ is said to be
  \textit{(globally) irreducible} if there does not exist a tropical cycle $Y$
  of the same dimension in $V$ such that $ |Y| \subsetneq |X| $. Of course,
  this definition then applies to tropical fans as well.
\end {definition}

\begin{remark} \label{rem-class}
  Lemma \ref{criterion} implies that, just as in the classical situation, the
  support of a tropical cycle can always be written as the union of the
  supports of irreducible tropical cycles. However, such a decomposition is in
  general not unique \cite[Remark 2.19]{GKM}.
\end{remark}

\begin {remark} \label {rem-irred}
  In Definition \ref {def-irred} the cones of $Y$ are just required to be
  contained in the union $ |X| $ of the cones of $X$. However, the definition
  does not change if one requires all cones of $Y$ to actually be cones of $X$,
  just with possibly different weights. To see this, assume there is a tropical
  cycle $Y$ satisfying $ |Y| \subsetneq |X|$. By passing to a common refinement
  with $X$, we can then first of all make sure that every cone of $Y$ is
  contained in a cone of $X$. But then all cones of $Y$ contained in the same
  cone of $X$ must have the same weight due to the balancing condition, and
  hence can be made into a single cone.
\end {remark}

\begin{lemma} \label {criterion}
  A tropical cycle $X$ is irreducible if and only if ``its weight function
  is unique up to a global multiple'', i.e.\ if and only if for every cycle $Y$
  of the same dimension and consisting of at most the cones of $X$ there is a
  rational number $ \lambda \in \Q $ such that $ w_Y(\sigma) = \lambda \,
  w_X(\sigma) $ for every cone $ \sigma $ of $X$.
\end{lemma}

\begin{proof}
  ``$ \Rightarrow $'': This is \cite[Lemma 2.21]{GKM}.

  ``$ \Leftarrow $'': Let $Y$ be a cycle with $ |Y| \subsetneq |X| $. By Remark
  \ref {rem-irred} we can assume that each cone of $Y$ is a cone of $X$, so
  there must be a cone $ \sigma $ of $X$ with $ w_Y(\sigma) = 0 $. But this
  requires $ \lambda=0 $ in our assumption, so $Y$ would have to be the zero
  cycle.
\end{proof}

Using this lemma the problem of checking irreducibility can often be reduced to
a local computation. For this we need the concept of a cycle that is connected
in codimension 1.

\begin {definition} [Locally irreducible cycles, connected in codimension 1]
    \label {def-locirr}
  Let $X$ be a tropical $k$-cycle.
  \begin {enumerate}
  \item \label {def-locirr-a}
    The cycle $X$ is called \textit {locally irreducible} if for each $
    (k-1) $-dimensional cone $ \tau $ there is up to a common multiple only
    one choice of weights $ w(\sigma) $ for the cones $ \sigma $ adjacent to $
    \tau $ such that the balancing condition $ \sum_{\sigma>\tau} w(\sigma) \,
    v_{\sigma/\tau} = 0 \mod V_\tau $ holds at $ \tau $, see \cite[Section 1.6]{R}.
  \item \label {def-locirr-b}
    The cycle $X$ is called \textit{connected in codimension 1} if any two
    $k$-dimensional cones $ \sigma,\sigma' $ in $X$ can be connected by a
    sequence $ \sigma=\sigma_0,\ldots,\sigma_m=\sigma'$ of $k$-dimensional
    cones in $X$ such that $ \sigma_{i-1} \cap \sigma_i $ is of dimension $ k-1
    $ for all $ i=1,\dots,m $ \cite[Section $3$]{BJSST}.
  \end {enumerate}
\end {definition}

\begin{remark} \label{rem-crit}
  A criterion similar to Definition \ref{def-irred} holds also for locally
  irreducible cycles: we just have to apply Lemma \ref{criterion} to the stars
  corresponding to each $\tau$ to see that a $k$-dimensional cycle is locally
  irreducible if and only if its stars at all $ (k-1) $-dimensional cones are
  irreducible in the sense of Definition \ref{def-irred}.
\end{remark}


\begin{lemma} [Criterion for irreducibility of a tropical cycle]
    \label {uniqueness}
  Every locally irreducible tropical cycle that is connected in codimension 1
  is also globally irreducible.
\end{lemma}

\begin{proof}
  Let $ \sigma, \sigma'\in X^{(k)} $ be two top-dimensional cones in a
  $k$-dimensional cycle $X$. As $X$ is connected in codimension 1 there exist
  cones $ \sigma_i \in X^{(k)}$ for $i \in \lk 1,\ldots,m \rk$ forming a
  sequence $ \sigma=\sigma_0,\ldots,\sigma_m=\sigma'$ such that each
  intersection $ \sigma_{i-1} \cap \sigma_i $ is a cone of dimension
  $k-1$. As $X$ is locally irreducible, the weight $ w(\sigma_i) $ determines
  the weight $ w(\sigma_{i-1}) $ uniquely. Hence, running through the sequence,
  we can conclude that $ w(\sigma) $ determines $ w(\sigma') $ uniquely and
  thus that $X$ is globally irreducible by Lemma \ref {criterion}.
\end{proof}

The converse of this lemma does not hold, see Example \ref {ex-m06}.

\begin{lemma} \label{mcon}
  $\M_{0,n}$ is connected in codimension $1$ for each $n >2$.
\end{lemma}

\begin{proof}
  Let $ \sigma $ be a top-dimensional cone of $ \M_{0,n} $, corresponding by
  Remark \ref {combtypes} to a combinatorial type of 3-valent tropical curves.

  First we show that $ \sigma $ can be connected in codimension 1 (as in
  Definition \ref {def-locirr} \ref {def-locirr-b}) to a top-dimensional
  cone corresponding to a \textit {caterpillar tree}, i.e.\ to a type of 3-valent curves
  for which the number $m$ of 3-valent vertices that are not adjacent to any
  end is zero. So assume that $ m>0 $. As in the picture below we can then pick
  such a vertex $V$ that is not adjacent to any end but adjacent to at least
  one further vertex $W$ with two ends attached to it.

  \pict{subtree}
  
  The picture then shows how $ \sigma $ is connected in codimension 1 to
  another type $ \sigma' $ for which the value of $m$ is reduced by 1 --- by
  contracting the edge joining $V$ and $W$ to a 4-valent vertex, and then
  resolving this 4-valent vertex again in a different way. Continuing this
  process, we can connect $ \sigma $ in codimension 1 to a caterpillar tree.

  But now any two caterpillar trees are connected in codimension 1 by a sequence of
  exchanges of neighboring ends as in the following picture:

  \pict {inter}

  Hence we conclude that any two top-dimensional cones of $ \M_{0,n} $ are
  connected in codimension 1.
\end{proof}

\begin{remark}
 The statement of Lemma \ref{mcon} is not new. It is noticed in
 \cite[Section $3$]{SS} that the Pl\"{u}cker ideal $I_{2,n}$ is prime and that
 $\mathcal{G}(2,n)$ is the tropical variety defined by $I_{2,n}$. By
 \cite[Theorem 3.1]{BJSST} it follows that  $\mathcal{G}(2,n)$ is connected in
 codimension one. When taking the quotient by its lineality space in order to
 obtain $\M_{0,n}$ this property is conserved.   
\end{remark}

We are now going to prove that for all $n>2$ the tropical moduli space
$\M_{0,n}$ and its vital divisors, respectively Psi-classes are irreducible.
These results also hold in classical algebraic geometry except for Psi-classes
which are not divisors but only divisor classes: for $\overline{M}_{0,n}$ one
can use the quotient description of \cite[Theorem 6.1]{K2}, and the classical
vital divisor $D^S$ is isomorphic to the product $\overline{M}_{0,|S|+1}\times
\overline{M}_{0,|S^c|+1}$ of irreducible varieties \cite[Corollary 3.9]{K1},
and hence is irreducible itself.

\begin{proposition} \label {mir}
  $\M_{0,n}$ is irreducible for each $n>2$.
\end{proposition}

\begin{proof}
  By Lemmata \ref{uniqueness} and \ref{mcon} it suffices to check local
  irreducibility. We have to look at the weights of cones corresponding to the
  resolving curves of a curve with one $4$-valent vertex. By Lemma \ref {local}
  we can assume that this curve is just a star with one $4$-valent vertex in
  the middle, living in $\M_{0,4}$.

  \pict{localm}

  So the task translates to finding non-trivial solutions $(a_1,\ldots,a_7)$
  for the linear system of equations
    $$  a_1 \, \tilde{v}(\lk 1,2\rk)
      + a_2 \, \tilde{v}(\lk 1,3\rk)
      + a_3 \, \tilde{v}(\lk 1,4\rk)
      = a_4 \, d_1 + a_5 \, d_2 + a_6 \, d_3 + a_7 \, d_4 $$
  in $ \R^{\binom 42} = \R^6 $. Solving this system gives $a_i=1$ for all $i\in
  \lk 1,\ldots,7\rk$ up to a multiple. Hence $ \M_{0,n} $ is locally
  irreducible and thus also globally irreducible.
\end{proof}

Next, we want to study the irreducibility of codimension-1 cycles in $ \M_{0,n}
$ --- in particular of Psi-classes and vital divisors. In this case we can
often not use Lemma \ref {uniqueness} as the cycles in question are not locally
irreducible. Let us start by illustrating this fact with an example that will
also be needed later.

\begin {example}[Difference between local and global irreducibility]
    \label {ex-m06}
  The following picture shows a type of curves corresponding to a
  codimension-2 cell $ \tau $ in $ \M_{0,6} $ with the types of its six
  adjacent codimension-1 cells $ \sigma_1,\dots,\sigma_6 $:

  \pict {123}

  By Lemma \ref {local} and the balancing condition in $ \M_{0,4} $ we have
  $ v_{\sigma_1/\tau} + v_{\sigma_2/\tau} + v_{\sigma_3/\tau} = 0 $ and
  $ v_{\sigma_4/\tau} + v_{\sigma_5/\tau} + v_{\sigma_6/\tau} = 0 $. In fact,
  an easy computation in coordinates shows that the linear equation
  $ \sum_{i=1}^6 a_i \, v_{\sigma_i/\tau} = 0 $ is satisfied in $ \M_{0,6} $ if
  and only if $ a_1=a_2=a_3 $ and $ a_4=a_5=a_6 $. All six resolutions above
  contribute to the vital divisor $ D^{\{1,2,3\}} $ with the same non-zero weight $-1$ by
  Proposition \ref {vital-weight}. This means that we can scale $ a_1,a_2,a_3 $
  by one multiple and $ a_4, a_5, a_6 $ independently by another one in order to
  satisfy the equation $ \sum_{i=1}^6 a_i \, v_{\sigma_i/\tau} = 0 $. Therefore $
  D^{\{1,2,3\}} $ is not locally irreducible
  at $ \tau $. We will see in Proposition \ref {prop-vital} however that vital
  divisors are always globally irreducible. \hfill $\circ$
\end {example}

\begin {lemma} \label {lem-divisor-set}
  Let $Y$ be a codimension-1 cycle in $ \M_{0,n} $. Let $ \sigma $ be a
  top-dimensional cone of $Y$; by Remark \ref {combtypes} its corresponding
  curves have exactly one $4$-valent vertex. Denote by $ A,B,C,D \subset
  \{1,\dots,n\} $ the sets of marked ends behind each of the four edges
  adjacent to this vertex (in any order). Then the weight $ w(\sigma) $ in $Y$
  depends only on the sets $ A,B,C,D $ (and not on the subtrees behind the
  $4$-valent vertex).
\end {lemma}

\begin {proof}
  Let $ \tau' $ be a codimension-1 cell of $Y$, i.e.\ a codimension-2 cell in $
  \M_{0,n} $. By Remark \ref {combtypes} the corresponding curves have either
  two 4-valent vertices or one 5-valent vertex, with all other vertices being
  3-valent. The case of a 5-valent vertex is irrelevant for the claim of the lemma and
  will be treated in Proposition \ref{pir}. Let us consider the case when $ \tau' $ has
two  4-valent vertices.
  Without loss of generality we may then assume that the ends are labeled in
  such a way that there is a forgetful map $ \M_{0,n} \to \M_{0,6} $ sending $
  \tau' $ to the cone $ \tau $ in Example \ref {ex-m06} above. By Remark \ref
  {splitting} the top-dimensional cones $ \sigma'_1,\dots,\sigma'_6 $ in $Y$
  adjacent to $ \tau' $ are in one-to-one correspondence with the cones $
  \sigma_1,\dots,\sigma_6 $ in $ \M_{0,6} $ of Example \ref {ex-m06}. Now let $
  a_i $ be the weight of $ \sigma'_i $ in $Y$ for $ i=1,\dots,6 $. By the
  balancing condition we have $ \sum_i a_i \, v_{\sigma'_i/\tau_i} = 0 $ in $
  \M_{0,n} $. Applying the (linear) forgetful map $ \M_{0,n} \to \M_{0,6} $ we
  conclude that $ \sum_i a_i \, v_{\sigma_i/\tau_i} = 0 $ in $ \M_{0,6} $ as
  well, which by Example \ref {ex-m06} means that $ a_1=a_2=a_3 $ and $
  a_4=a_5=a_6 $. In particular, the types $ \sigma'_1 $, $ \sigma'_2 $, $
  \sigma'_3 $ have the same weight in $Y$. Applying this to the labeling where
  the forgetful map sends the marked ends in $B$, $C$, $D$ to $ x_4 $, $ x_5 $,
  $ x_6 $, respectively, we see by the argument of the proof of Lemma \ref
  {mcon} that any two top-dimensional cells of $Y$ that just differ by the
  arrangement of the markings in $A$ behind one of the edges of the 4-valent
  vertex must have the same weight. As the same holds for $B$, $C$, $D$ as
  well, the lemma follows.
\end {proof}

\begin{lemma} \label {lem-psi-conn}
  $\psi_i$ is connected in codimension 1 for each $i \in \lk 1, \ldots,n \rk$.
\end{lemma}

\begin{proof}
  Let $ \sigma $ be a top-dimensional cone in $ \psi_i $. As in the picture
  below on the left it corresponds to a combinatorial type of tropical curves
  with exactly one 4-valent vertex $V$ adjacent to the marked point $ x_i $,
  all other vertices being 3-valent.

  By the proof of Lemma \ref {lem-divisor-set} we see that $ \sigma $ can be
  connected in codimension 1 to any other such type where the trees behind the
  edges of $V$ have been replaced by caterpillars, and where within each such
  caterpillar the marked points have been ordered in a given way. The picture
  below then shows that it is also possible to connect such a cone in codimension 1
  to the one where a marked point $ x_j $ with $ j \neq i $ changes its place
  from behind one of the edges of $V$ to another --- through a codimension-1
  cone in $ \psi_i $ corresponding to curves with a 5-valent vertex.

  \pict{psic} 

  Using these steps we see that $ \sigma $ can be connected in codimension 1
  to any other top-dimensional cone of $ \psi_i $.
\end{proof}

\begin{proposition} \label{pir}
  Each Psi-class $\psi_i$ in $\M_{0,n}$ is irreducible. 
\end{proposition}

\begin{proof}
  Assume w.l.o.g.\ $i=1$. As in the proof of Proposition \ref{mir} it suffices
  by Lemmata \ref {uniqueness} and \ref {lem-psi-conn} to check local
  irreducibility at each codimension-1 cone of $ \psi_1 $. There
  are two types of these cones: those with two 4-valent vertices (one of them
  adjacent to $ x_1 $) and those with one 5-valent vertex (adjacent to $ x_1
  $). In the former case only the vertex that is not adjacent to $ x_1 $ can be
  resolved in $ \psi_1 $, and the computation to check local irreducibility is
  exactly the same as in Proposition \ref {mir}. In the latter case we again
  apply Lemma \ref {local} to reduce the situation to the resolutions of a
  star in $\M_{0,5}$ such that the marking $ x_1 $ remains adjacent to a
  $4$-valent vertex. There are six such resolving curves of the $5$-valent
  vertex, corresponding to a choice of two of the marked ends $ x_2,\dots,
  x_5 $ to put at the 4-valent vertex together with $ x_1 $ and the bounded
  edge. Hence, this time the linear system of equations reads
  \begin {align*}
    &   a_1 \, \tilde{v}(\lk 2,3 \rk)
      + a_2 \, \tilde{v}(\lk 2,4 \rk)
      + a_3 \, \tilde{v}(\lk 2,5 \rk)
      + a_4 \, \tilde{v}(\lk 3,4 \rk)
      + a_5 \, \tilde{v}(\lk 3,5 \rk)
      + a_6 \, \tilde{v}(\lk 4,5 \rk) \\
    & \qquad \qquad
      = \sum_{j=7}^{11} a_j \, d_{j-6}
  \end {align*}
  in $ \R^{\binom 52} = \R^{10} $. Solving this system gives up to a
  multiple $ a_j=1 $ for $ j=1,\dots,7 $ and $ a_j=2 $ for $ j=8,\dots,11 $ as
  the only solution. Hence $ \psi_i $ is locally irreducible and thus also
  globally irreducible.
\end{proof}

\begin {remark}
  Using similar arguments as in Lemmata \ref {mcon} and \ref {lem-psi-conn} one
  can show that the codimension-$k$ skeletons of the moduli spaces $ \M_{0,n} $
  as well as of the Psi-classes $ \psi_i $ (see Propositions \ref {comod} and
  \ref {copsi}) are connected in codimension 1 as well. However, they are in
  general not irreducible, as the following example shows.
\end {remark}

\begin{example} [Irreducibility of other tropical fans in $\M_{0,n}$]
  Consider the (one-dimen\-sional) codimension-$ (n-5) $ skeleton of the
  Psi-class $ \psi_1 $ in $ \M_{0,n} $ for $ n \ge 6 $, corresponding to
  resolutions of the star curve with one $n$-valent vertex that keep $ x_1 $ at
  a vertex of valence at least 4. We have seen at the end of Example \ref{ex-psi} 
 that in this case it is not only the sum of all normal vectors of
  the possible resolutions that sum up to zero modulo the lineality space, but
  also the vectors of each individual resolution type, i.e.\ when fixing how
  many marked ends should be adjacent to $ x_1 $. In particular, as there is
  more than one such resolution type if $ n \ge 6 $, this means that there are
  several choices of weights satisfying the balancing condition and thus that
  the codimension-$ (n-5) $ skeleton of $ \psi_1 $ in $ \M_{0,n} $ is not
  locally irreducible. As for this cycle there is only one balancing condition
  to check it follows that it is not globally irreducible either.

  A similar argument can be used to show that other codimension-$k$ skeletons
  of a Psi-class or of $ \M_{0,n} $ also fail to be locally irreducible.
  In general this does not imply however that these cycles are not globally
  irreducible either (e.g.\ we will see in the proof of Proposition \ref{prop-vital} 
  that the vital divisors of $ \M_{0,n} $ are globally but not
  locally irreducible). \hfill $\circ$
\end{example}

The following example will be of use in the remainder.

\begin{example} [Irreducible divisors in $\M_{0,5}$] \label {m05}
  In $\M_{0,5}$ the problem of finding irreducible divisors is equivalent
  to determine all minimal linear dependent sets of the 10 vectors $v(I)$
  where $I \subset \lk 1,\ldots,5\rk$ and $\bet{I}=2$, or in other words to
  find all circuits of the vector matroid given by these vectors. An explicit
  computation shows that there are two kinds of irreducible divisors: those
  containing $4$ cones in their support and those with 6 cones. Up to
  multiples, these divisors are the following:
  \begin {itemize}
  \item 4 cones: The only effective divisors with four cones are the cycles $
    \psi_i^\natural $ for $ i=1,\dots,5 $ given as the difference of the
    1-skeleton and $ \psi_i $ --- for example $\psi_1^\natural $ consists of the
    cones $v(\lk 1,2\rk)$, $v(\lk 1,3\rk)$, $v(\lk 1,4\rk)$, and
    $v(\lk 1,5\rk)$, each with weight 1. The only other divisors with four
    cones are the ten vital divisors $ D^S $ of $\M_{0,5}$. They have
    respectively three cones of weight $1$ and one of weight $-1$ (see
    Proposition \ref {vital-weight}).
  \item 6 cones: The Psi-classes $\psi_i$ are the only effective irreducible
    divisors with six cones. There are ten more irreducible divisors with six
    cones, given by $ \psi_i^\natural - \psi_j^\natural =: V^{\lk i, j\rk} $
    for $i>j$. \hfill $\circ$
  \end {itemize} 
\end{example}

\begin{proposition} \label {prop-vital}
  For $n>3$ all vital divisors $D^S$ in $\M_{0,n}$ are irreducible.
\end{proposition}

\begin{proof}
  The case $n=4$ is clear as the only vital divisor corresponds to the
  star with a $4$-valent vertex. So let $ n \ge 5 $, and let $ S \subset
  \{1,\dots,n\} $ be a subset with $ 2 \le |S| \le n-2 $, giving rise to a
  vital divisor $ D^S $ in $ \M_{0,n} $. By Proposition \ref {vital-weight}
  the cones occurring in $ D^S $ correspond to curves with one 4-valent vertex
  such that, if $A$, $B$, $C$, $D$ denote the sets of marked points behind
  the 4-valent vertex, the given set $S$ can be written as a union of some of
  the sets $A$, $B$, $C$, $D$. We have to show that up to a global multiple
  there is only one weight distribution on these cones giving rise to a
  tropical cycle. By relabeling the subsets (and possibly replacing $S$ by
  $S^c$) we may assume w.l.o.g.\ that $ S=A $ or $ S=A \cup B $.

  By Lemma \ref {lem-divisor-set} we know already that the weights of these
  cones can only depend on the sets $A$, $B$, $C$, $D$; we denote these weights
  by $ w(A,B,C,D) $. So to prove irreducibility it suffices to show that
  \begin {enumerate}
  \item $ w(A,B,C_1,D \cup C_2) = w(A,B,C,D) $ if $ S=A \cup B $, $ C = C_1 \sqcup C_2 $,
and $ C_1,C_2 \neq \emptyset $,
  \item $ w(A \cup B, C_1,C_2, D) = -w(A,B,C,D) $ if $ S=A \cup B $, $ C= C_1 \sqcup C_2
$, and $ C_1,C_2 \neq \emptyset $,
  \end {enumerate}
  since these two operations can transform any partition of $ \{1,\dots,n\} $
  into four subsets occurring in $ D^S $ into any other.

  To prove this, consider a codimension-1 cone in $ D^S $ having one 5-valent
  vertex, with sets of marked points in $ A,B,C_1,C_2,D $ behind its adjacent
  edges. Since $ S=A \cup B $, only $4$ of the $10$ resolutions of this cone in
  the codimension-1 skeleton of $ \M_{0,n} $ occur in $ D^S $:
  \pict {vital}
  Now we map these cones to $ \M_{0,5} $ by the forgetful map that sends the
  marked ends in $ A,B,C_1,C_2,D $ to $ x_1,x_2,x_3,x_4,x_5 $, respectively.
  The $4$ cones occurring above are then exactly mapped to the four cones in
  the vital divisor $ D^{\{1,2\}} $ in $ \M_{0,5} $ of Example \ref{ex-vital-5}.
  As this divisor is irreducible (see Example \ref {m05}), we
  conclude that there is up to a global multiple only one choice of weights for
  the four cones above. By the explicit computation of Example \ref{ex-vital-5}
  these weights satisfy $ -w(A \cup B,C_1,C_2,D) = w(A,B,C_1,D
  \cup C_2) = w(A,B,C,D) $, proving both (a) and (b).
\end{proof}

\begin {remark} \label {rem-behavior}
  The proofs of Propositions \ref{pir} and \ref{prop-vital} suggest that, once
  we know that a divisor in $\M_{0,n}$ is locally irreducible, we might be able
  to conclude by some ``connectivity'' argument that the divisor is also
  (globally) irreducible. Unfortunately, the proof of this connectivity is very
  complicated for increasing $n$ since the number of possible cones contained
  in the support of such a divisor increases exponentially. So far, we are just
  able to show the equivalence of local and global irreducibility of a divisor
  in the case $n=6$ (for $n=5$ this is trivial since there is only one
  balancing condition):
\end {remark}

\begin {proposition} \label {prop-m06}
  In $ \M_{0,6} $ every locally irreducible divisor is also globally
  irreducible.
\end {proposition}

\begin{proof}
  Let $D$ be a locally irreducible divisor. We have to show that its weight
  function is unique up to a global multiple in the sense of 
  Lemma \ref{criterion}.

  To prove this, let us introduce the following notation. A codimension-$2$
  cone $\tau$ of $D$ will be called \textit {relevant} if it corresponds to
  curves with one $5$-valent vertex (rather than to curves with two $4$-valent
  vertices). Note that $D$ is determined uniquely by the weights of all cells
  adjacent to relevant cones (hence the name) since every codimension-$1$ cell
  is adjacent to such a relevant codimension-$2$ cone. Moreover, in $ \M_{0,6}
  $ all relevant cones correspond to curves having two markings $i$ and $j$ on
  one side of the bounded edge and four on the other. We will use the notation
  $ij$ for this cone.

  As $D$ is locally irreducible we know that the weight combination of the
  codimension-$1$ cells around a relevant cone $ \tau $ must be a multiple of
  the irreducible types of divisors in $ \M_{0,5} $ listed in Example \ref{m05}.
  If the weights around $ \tau $ are not all $0$ we will call the choice
  of type the \textit {resolution behavior} at $ \tau $, and the corresponding
  multiple the \textit {weight} of $ \tau $. However, in $\M_{0,6}$, where each
  curve has $6$ markings, we have to be careful how we split up the six
  markings over the subtrees behind the five bounded edges adjacent to the
  $5$-valent vertex. This means that all types of Example \ref {m05} correspond
  to several resolution behaviors. The following list shows all possible
  resolution behaviors at the cone $12$ (up to permutation of the labels of the
  markings $ 3,4,5,6 $). In this list, we understand by \textit{new bounded
  edge} the bounded edge appearing when resolving the $5$-valent vertex.
  \begin {itemize}
  \item $\psi_3$, where we require the marking $3$ to be at the $4$-valent
    vertex in the resolving curves;
  \item $\psi_{12}$ (not really a Psi-class!) corresponding to resolving the
    $5$-valent vertex where the subtree consisting of the markings $1$ and $2$
    is required to stay at the $4$-valent vertex;
  \item $V^{\lk 5,6 \rk}$, where either the marking $5$ or $6$ should lie at
    the $4$-valent vertex (but not both);
  \item $V^{\lk 12,3 \rk}$, where the marking $3$ or the subtree consisting of
    the markings $1$ and $2$ should be at the $4$-valent vertex, but not both;
  \item $\psi_3^\natural$, where the marking $3$ is not allowed to lie at the
    $4$-valent vertex;
  \item  $\psi_{12}^\natural$, where the subtree consisting of the markings $1$
    and $2$ is not allowed to lie at the $4$-valent vertex;
  \item  $D^{\lk 5, 6\rk}$, where the markings $5$ and $6$ lie both on the side
    of the new bounded edge without $4$-valent vertex, or both on the side
    of the new bounded edge with the $4$-valent vertex;
  \item $D^{\lk 12, 3\rk}$, where the marking $3$ and the subtree consisting of
    the markings $1$ and $2$ lie both on the side of the new bounded edge
    without $4$-valent vertex, or both on the side of the new bounded edge with
    the $4$-valent vertex.
  \end {itemize}
  For some resolution behaviors, we show the corresponding adjacent
  codimension-$1$ cells occurring in $D$ with non-zero weight in the picture
  below, where unmarked ends are understood to be labeled with the remaining
  markings in any possible way.

  \pict {resolv}

  The main idea of the proof is now the following: pick a relevant
  codimension-$2$ cone, say $12$, such that the weights of $D$ around it are
  not all $0$. If we prescribe the resolution behavior and the weight at $12$
  this will determine the weight (not necessarily the resolution behavior
  however) of all other relevant codimension-$2$ cones adjacent to
  codimension-$1$ cells occurring in the chosen resolution behavior at $12$.
  For example, if we choose the resolution behavior $ \psi_3 $ at $12$ as
  above, the picture below shows the type $12$ (on the left) and an adjacent
  codimension-$1$ cell with non-zero coefficient in $ \psi_3 $ in the middle.
  As this cell is also adjacent to the relevant codimension-$2$ cone $56$ (on
  the right), this means that the weight of $56$ is fixed by that of $12$.

  \pict {res-1}

  This argument can be extended slightly by using in addition the result of
  Lemma \ref {lem-divisor-set} that the weights of codimension-$1$ cells in
  divisors depend only on the distribution of the marked points behind the
  $4$-valent vertex. For example, the following picture shows that the weight
  of $14$ is also fixed by the weight (and the resolution behavior) of $12$.

  \pict {res-2}

  Using these arguments, the following list shows, when fixing a certain
  resolution behavior at $12$, the weights of which other relevant
  codimension-$2$ cones are fixed by the weight of $12$. The column on the
  right shows the number of relevant codimension-$2$ cones whose weight is
  fixed by that of $12$.
    \[ \begin{array}[c]{|l|l|r|} \hline
         \psi_3 & 12, 14,15,16,24,25,26,45,46,56&10\\ 
         \psi_{12} & 12, 34,35,36,45,46,56&7\\ 
         V^{\lk 5,6 \rk} & 12,15,16,25,26,35,36,45,46&9\\ 
         V^{\lk 12,3 \rk} & 12,14,15,16,24,25,26,34,35,36&10\\ 
         \psi_3^\natural & 12,13,23,34,35,36&6\\ 
         \psi_{12}^\natural & 12,13,14,15,16,23,24,25,26&9\\ 
         D^{\lk 5,6 \rk} & 12, 13, 14,23,24,34,56&7\\ 
         D^{\lk 12,3 \rk} & 12,13,23,45,46,56&6\\ \hline 
       \end{array} \]
  Note that there are $ \binom 62=15 $ relevant codimension-$2$ types in total.
  So if we assume that $D$ is reducible, this means that a resolution behavior of 
  $\psi_3 $ or $ V^{12,3} $ as in the table above is impossible since this fixes
  $10$ weights already, leaving only $5$ of the weights of relevant cones
  undetermined --- which is not enough for another independent weight since
  each weight has to occur in at least $6$ relevant cones by the table above.

  To rule out the other resolution behaviors we cannot use a pure counting argument
  alone, but rather look in detail at the precise cones whose weights are
  determined by a fixed weight in a certain resolution behavior. A straightforward
  but tedious case-by-case analysis using the table above shows that it is not
  possible to fix the resolution behaviors and independent weights at two relevant
  cones without contradictions, i.e.\ such that the sets of relevant cones
  whose weights are determined by the two initial ones are disjoint. This shows
  that $D$ must in fact be irreducible.
\end{proof}

Since the techniques used in Proposition \ref {prop-m06} are limited to $n=6$,
the following question remains open:

\begin{conj}
  Let $n>6$. Is any locally irreducible divisor in $ \M_{0,n} $ also globally
  irreducible?
\end{conj}

