\section{(Irreducible) tropical subfans and cycles in $\M_{0,n}$} \label{l2}

In this section, we want to verify that the codimension-$k$ skeletons of the
moduli spaces $\M_{0,n}$ of rational $n$-marked abstract tropical curves are
tropical subvarieties, for suitable weight functions. Furthermore, it turns out
that the same holds for codimension-$k$ skeletons of Psi-classes $\psi_i$
living in $\M_{0,n}$. This will be done by checking the balancing condition
explicitly in terms of coordinate vectors of $\M_{0,n}$. Inspired by this
method, we will show in the subsequent part that tropical Psi-classes $\psi_i$
are irreducible. In addition, we will prove that the tropical vital divisors
$D^S$ in $\M_{0,n}$ are also irreducible. As a by-product we will give a new
characterization of irreducible tropical fans and cycles.

Let us start by reviewing the definitions and notions indispensable below and
by fixing the notation. More details about the following notions can be found in
\cite[Definitions 2.4, 2.6. and 2.8]{GKM}.

\begin{definition}[Tropical (sub-)fans, codimension-$k$ skeleton] \label {defo}
   Pick some $r \geq 0$ and a lattice $ \Lambda
  \cong \Z^r $; set $V:=\Lambda \otimes_\Z \R$.
  \begin {enumerate}
  \item Let $X$ be a fan of pure dimension $N$ in $V$. For $ 0 \le k \le N $ we
    denote the collection of its $k$-dimensional cones by $ X^{(k)} $. The
    vector subspace of $V$ spanned by a cone $ \sigma $ of $X$ will be denoted
    $ V_\sigma $. We call $X$ a \textit {tropical fan} if it is equipped with a
    weight function $ w: X^{(N)} \to \N $ such that the balancing condition
      \[ \sum_{\sigma>\tau} w(\sigma) \, v_{\sigma/\tau} = 0 \quad \in V/V_\tau
      \]
    holds for all $ (N-1) $-dimensional cones $ \tau $, where $ v_{\sigma/\tau}
    $ is the primitive normal vector of $ \sigma $ relative to $ \tau $. The
    union of all cones of $X$ will be written as $ |X| \subset V $.
  \item A \textit{(tropical) subfan} $Y$ of a (tropical) fan $X$ in $V$ is
    itself a (tropical) fan $Y$ in $V$ with the property that each cone of $Y$
    is contained in a cone of $X$. Note that the weight function of $Y$ is not
    necessarily inherited by $X$.
  \item The \textit{codimension-$k$ skeleton} $X^{(N-k)}_{\tn{sk}}$ of a fan
    $X$ in $V$ of pure dimension $N$ (for $ 0 \leq k \leq N$) consists of all
    cones of dimension at most $ N-k $ in $X$. It is a pure-dimensional fan of
    dimension $N-k$, however with no canonical weight function associated to
    it.
  \end {enumerate}
\end{definition}

\begin{construction}[Moduli space $\M_{0,n}$ of rational $n$-marked abstract
    tropical curves] \label {moduli}
  A \textit {rational $n$-marked (abstract) tropical curve} is a tuple
  $ (\Gamma;x_1,\ldots,x_n) $, where $ \Gamma $ is a connected metric tree
  graph all of whose vertices have valence at least 3, and where $ x_1,\dots,
  x_n $ is a labeling of its unbounded edges. In the following, we will sometimes
  write marked edge $x_i$ or marked end $x_i$ for the unbounded edge with label $x_i$. An
  \textit{isomorphism} of rational tropical curves $(\Gamma;x_1,\ldots,x_n) $ and
  $(\tilde{\Gamma};\tilde{x}_1,\ldots,\tilde{x}_n)$ is a homeomorphism $\Gamma \to
  \tilde{\Gamma}$ sending $x_i$ to $\tilde{x}_i$ and mapping the edges of
  $\Gamma$ bijectively onto edges of $\tilde{\Gamma}$ by affine maps of slope
  $\pm 1$. As in \cite{GKM}, we will denote by $\M_{0,n}$ the parameter space
  of all such curves modulo isomorphisms. For many purposes it plays the
  tropical role of the classical moduli space of rational $n$-marked stable
  curves --- even if it is not compact.

  The moduli space $ \M_{0,n} $ can be given the structure of a tropical fan
  (see Remark \ref{combtypes} for the fan structure) of dimension $n-3$ in a
  quotient space of $ \R^{\binom{n}{2}}$ (\cite[Theorem 3.7]{GKM});
  in fact, it can be described as the tropical Grassmannian
  $\mathcal{G}(2,n)$ modulo its lineality space (see \cite[Theorem 3.4]{SS} or
\cite[Remark 3.9]{GKM}).
  More precisely, for each $n$-marked tropical curve $ (\Gamma;
  x_1,\dots,x_n) $ and $ 1 \le i < j \le n $ let $ \tn{dist}_\Gamma(x_i,x_j) $
  be the distance between the unbounded edges $x_i$ and $x_j$ in $\Gamma$. We
  thus get a map
  \begin {align*}
    \tilde v: \qquad\qquad \M_{0,n}  &\to     \R^{\binom n2} \\
              (\Gamma;x_1,\dots,x_n) &\mapsto
                (\tn{dist}_\Gamma(x_i,x_j))_{i<j}
  \end {align*}
  where we choose the lexicographic ordering of the pairs $ (i,j) $ for the
  coordinates in $ \R^{\binom n2} $. We will call $ \tilde v(\Gamma;
  x_1,\dots,x_n) $ the \textit {distance vector} of $ (\Gamma;x_1,\dots,x_n) $.

  The following vectors in $ \R^{\binom n2} $ will be of particular importance:
  let $ I \subset \{1,\dots,n\} $ be any subset, and denote by $ \tilde v(I)
  \in \R^{\binom n2} $ the vector whose $ (i,j) $-coordinate is equal to 1
  if $I$ contains exactly one of the numbers $i$ and $j$, and 0 otherwise. Note
  that $ \tilde v(I^c) = \tilde v(I) $, where $ I^c $ denotes the complement of
  $I$ in $ \{1,\dots,n\} $.

  The vectors $ d_i := \tilde v(\{i\}) $ for $ i=1,\dots,n $ form a basis of
  the so-called lineality space mentioned above; i.e.\ by taking the quotient
  by this subspace we obtain a map
  \begin {align*}
    v: \qquad\qquad \M_{0,n}  &\to
         \R^{\binom n2}/\langle d_1,\dots,d_n\rangle \\
       (\Gamma;x_1,\dots,x_n) &\mapsto
         \overline {(\tn{dist}_\Gamma(x_i,x_j))_{i<j}}
  \end {align*}
  that embeds $ \M_{0,n} $ as a tropical fan in $ \R^{\binom n2}/\langle
  d_1,\dots,d_n\rangle $ \cite[Theorem 3.7]{GKM}. For this structure of a tropical
fan the weights of
  all top-dimensional cones are chosen to be $1$, and the underlying lattice
  $\Lambda$ is taken to be the one generated by the classes $ v(I) $ of the
  vectors $ \tilde v(I) $ for all $ I \subset \{1,\dots,n\} $ modulo the
  lineality space. In the remainder, we will always view $ \M_{0,n} $ as a
  tropical fan with this embedding.
\end {construction}

\begin {remark} \label {combtypes}
  The cones of the tropical fan $\M_{0,n}$ correspond to curves of the same
  \textit{combinatorial type}, i.e.\ curves which differ just by the lengths of their
  bounded edges. They are open convex polyhedra of dimension
    $$ n-3- \sum_{\tn{ vertices } V \tn{ in } \Gamma}
       (\tn{val } V-3), $$
  where $\tn{val } V$ is the valence of the vertex $V$ of any curve $ \Gamma $
  of the given type. The one-dimensional cones of $\M_{0,n}$ are generated by
  the vectors $ v(I) $ of Construction \ref {moduli} for all $ I \subset
  \{1,\dots,n\} $ with $ 2 \le |I| \le n-2 $; by construction these are just the
  distance vectors of curves $ (\Gamma;x_1,\ldots,x_n)$ in $\M_{0,n}$
  having exactly one bounded edge of length $1$, with the markings $I$ to one
  and $ I^c $ to the other side. Hence there is only a finite number of
  combinatorial types of curves in $\M_{0,n}$. Note that the fan structure of
  $\M_{0,n}$ as in Construction \ref{moduli} coincides with the fan structure
  of the quotient of $\mathcal{G}(2,n)$ by its lineality space, see
  \cite[Theorem 4.2 and proof of Theorem 3.4]{SS}.
\end {remark}

\begin{remdef} [Notions of intersection theory for our purpose, see \cite {AR}
    for more details] \label {intersection}
  A \textit{tropical (affine) $k$-cycle} is a weighted fan of pure
  dimension $k$ in $V$ satisfying the balancing condition but where the range of
  the weight function is $\Z$. Hence the difference between a tropical fan of
  dimension $k$ and a tropical $k$-cycle is just the range of the weight
  function. A tropical $k$-cycle in a tropical fan $X$ is simply a $k$-cycle
  that is a subfan of $X$. Typically a tropical $k$-cycle in $X$ is an
  equivalence class of weighted fans of pure dimension $k$, where two fans are
  equivalent if they have a common refinement \cite[Definition 2.8]{AR}.
  However, in this paper we will not distinguish between equivalence
  classes and representatives of the equivalence class as the difference does
  not concern us here.

  As before, we will denote by $ \bet{X} $ the subset of $V$ of all cones of $X$
  (with non-zero weight). A \textit{(Weil) divisor} $D$ on $X$ is a
  cycle in $X$ of codimension 1.

  Let $X$ be a $k$-cycle. A (non-zero) \textit{rational function on} $X$ is a
  continuous function $ f: \bet{X} \to \R $ which is integer affine linear on
  each cone $ \sigma \subset \bet{X} $. We denote the linear part of the
  restriction of $f$ to $ \sigma $ by $ f_\sigma $.

  Let $X$ be a $k$-cycle and $f: \bet{X} \to \R$ a rational function on $X$.
  The \textit{Weil divisor $D(f)$ associated to} $f$ is the divisor in $X$
  consisting of the codimension-$1$ cones $ \tau \in X^{(k-1)} $ with weights
    \[ w(\tau) = \sum_{\sigma>\tau} w(\sigma) \, f_\sigma (v_{\sigma/\tau})
               - f_\tau \Big(
                 \sum_{\sigma>\tau} w(\sigma) \, v_{\sigma/\tau}
                 \Big), \]
  where the sum runs over all cones $\sigma$ s.t.\ $\tau \subsetneq \sigma$ is a face of
  $\sigma$, and $v_{\sigma/\tau} $ denotes the primitive normal vector of $ \sigma $
  relative to $ \tau $. It has been shown in \cite[Proposition 3.7]{AR} that this is
  indeed a tropical cycle.
  \end{remdef}

\begin{remdef} [Tropical Psi-classes $\psi_i$] \label {psi}
  Fix $n >2$ and $i \in \lk 1,\ldots,n\rk$. The \textit{Psi-class} $\psi_i
  \subset \M_{0,n}$ is the subfan of $ \M_{0,n} $ consisting of all cones of
  $\M_{0,n}$ of curves $(\Gamma; x_1,\ldots,x_n)$ such that the marked edge $ x_i$ is
  adjacent to a vertex of
  valence at least $4$ \cite{M}. Giving each top-dimensional cone the weight
  $1$ it has the structure of a tropical subfan of $\M_{0,n}$ of codimension 1.
  Using the language of \cite{AR}, we can rephrase this as: $\psi_i$ is a
  tropical Weil divisor associated to a rational function as proven in
  \cite[Proposition 3.5]{KM}. Note that a Psi-class is not defined up to
  rational equivalence as it is the case in classical geometry (and should
  therefore better be named Psi-divisor).  
\end{remdef}

\begin {convention} \label{conv-sk}
  When we talk in the following of ``the'' codimension-$k$ skeleton of $
  \M_{0,n}$ respectively $\psi_i$, we mean this w.r.t.\ the fan structure of
  Remarks \ref{combtypes} and \ref{psi}. 
\end {convention}

\begin{remark} [Vital divisors]
  Going back to Sean Keel \cite{K}, in the classical world the vital divisors
  generate the divisor class group of the moduli spaces $\overline{M}_{0,n}$ of
  rational $n$-marked stable curves and are sometimes called \textit{boundary
  divisors of} $\overline{M}_{0,n}$. The tropical analogue has first been
  studied by Matthias Herold in his diploma thesis \cite{H}. The results are
  also partly mentioned in \cite[Definition 2.4 and Lemma 2.5]{R}.
  We will introduce them here carefully.
\end{remark}

\begin{definition} [Tropical vital divisors in $\M_{0,n}$] \label {vital}
  Let $ n \ge 4 $ and $ S \subset \lk 1,\ldots,n \rk $ with $ 2 \le \bet{S} \le
  n-2 $. As $ \M_{0,n} $ is a simplicial fan with rays generated by the vectors
  $ v(I) $ of Remark \ref {combtypes} there is a unique piecewise linear
  function $ \phi_S $ on $ \M_{0,n} $ linear on each cone that takes the value
  $1$ on the vector $ v(S) $ and $0$ on all other rays. The associated Weil
  divisor $ D(\phi_S) $ to this function is called \textit{vital divisor
  w.r.t.\ $S$} and is denoted by $ D^S $.
\end{definition}

As $ v(S)=v(S^c) $ it is clear from the definition that $ D^S = D^{S^c} $.
Moreover, the following proposition taken from \cite[proof of Theorem 3.7]{H}
computes the weight function of the divisor $ D^S $ explicitly:

\begin{proposition} [Weight functions of the vital divisors] \label {vital-weight}
  Let $ n \ge 4 $ and $ S \subset \{1,\dots,n\} $ with $ 2 \le |S| \le n-2 $.
  Let $ \tau \in \M_{0,n}^{(n-4)} $ be a cone of codimension $1$; 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. Then
  the weight of the vital divisor $ D^S $ on the cone $ \tau $ is
    $$ \qquad\quad
       w(\tau) = \left\{ \begin{array}{ll}
         1 & \tn{ if } S=I \cup J, \tn{ where }
           I \neq J \tn{ and } I,J \in \lk A,B,C,D \rk, \\
         -1 & \tn{ if } S=I \tn{ or } S^c=I \tn{ with }
           I \in \lk A,B,C,D \rk, \\
         0 & \tn{ otherwise}.
       \end{array} \right. $$   
\end{proposition}

\begin{proof}
  Adjacent to our given codimension-$1$ cone $ \tau $ of $ \M_{0,n} $ there are
  three top-dimen\-sional cones $ \sigma_1,\sigma_2,\sigma_3 $ corresponding to
  resolving the $4$-valent vertex in the three ways shown in the picture below.

  \pict {types}
	
  The primitive normal vectors $ v_{\sigma_i/\tau} $ for $ i=1,2,3 $ as in
  Remark \ref {intersection} are then just the distance vectors of the three
  curves shown above on the right, with the newly added edge having length $1$
  and all other edges shrunken to zero, i.e.\ we have
    \[ v_{\sigma_1/\tau} = v(A \cup B), \quad
       v_{\sigma_2/\tau} = v(A \cup C), \mbox { and} \quad
       v_{\sigma_3/\tau} = v(A \cup D) \]
  with the vectors $ v(\,\cdot\,) $ as defined in Remark \ref {combtypes}. As
  $\M_{0,n}$ is balanced \cite[proof of Theorem 3.7]{GKM} we know moreover that
    \[ v(A \cup B) + v(A \cup C) + v(A \cup D) = v(A) + v(B) + v(C) + v(D). \]
  Using Definition \ref {vital} of the function $ \phi_S $, an application of
  the formula of Definition \ref {intersection} for the weight $ w(\tau) $ of
  the divisor $ D^S = D(\phi_S) $ on $ \tau $ thus yields (note that the
  weights of $ \M_{0,n} $ are all $1$)
  \begin{align*}
    w(\tau)\! &= \sum_{i=1}^3 (\phi_S)_{\sigma_i} (v_{\sigma_i/\tau})
               - (\phi_S)_\tau \Big(
                 \sum_{i=1}^3 v_{\sigma_i/\tau}
               \Big) \\
            &= \phi_S(v(A \cup B))+\phi_S(v(A \cup C))+\phi_S(v(A \cup D))
             - \phi_S(v(A)+v(B)+v(C)+v(D)) \\
            &= \begin{cases}
               1 & \tn{ if } S=I \cup J \tn{ with } I \neq J \mbox { and } I,J
                   \in \lk A,B,C,D \rk, \\
               -1 & \tn{ if } S=I \tn{ or } S^c=I \tn{ with } I \in \lk
                   A,B,C,D \rk, \\
               0 & \tn{otherwise}.
               \end{cases}
  \end{align*}
\end{proof}

\begin{example} [Vital divisors in $\M_{0,5}$] \label {ex-vital-5}
  By symmetry, in $ \M_{0,5} $ we just have to consider sets $S$ with
  $\bet{S}=2$. There are $10$ such sets and thus $10$ vital divisors. Set for example
  $S=\lk 1,2 \rk$. Curves of codimension 1 in $ \M_{0,5} $ have one bounded
  edge on exactly one of whose ends lies a 4-valent vertex. So in the notation
  of Proposition \ref {vital-weight} we have w.l.o.g.\ $\bet{A}=2$;
  $A$ determines the type of the curve $\M_{0,5}$ uniquely. Using the
  weight function above, we get the following weights:
  \begin{center}
    \begin{tabular}{|c|c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}%
                    c@{\;\;}c@{\;\;}c@{\;\;}c|} \hline
      $A=$ & $\lk 1,2\rk$ & $\lk 1,3 \rk$ & $\lk 1,4 \rk$ & $\lk 1,5 \rk$ &
             $\lk 2,3\rk$ & $\lk 2,4 \rk$ & $\lk 2,5 \rk$ & $\lk 3,4 \rk$ &
             $\lk 3,5\rk$ & $\lk 4,5 \rk$ \\ \hline
      weight & $-1$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$1$ \\ \hline
    \end{tabular}
  \end{center} \hfill $\circ$
\end{example}

