\section {Introduction} \label{l1}

In classical algebraic geometry, a divisor on a nonsingular algebraic variety
$X$ over $\C$ is a (formal) finite $\Z$-linear combination of irreducible
subvarieties of codimension $1$. Interpreting a divisor as an algebraic variety
(with multiplicities attached to its components) it is of course irreducible if
and only if it consists of only one subvariety. In tropical geometry however,
we define a divisor in a tropical fan $X$ to be, roughly speaking, a weighted
subfan of codimension one --- so in the case of non-negative weights it is just
an embedded tropical variety. As there is no unique decomposition of a such a
variety into irreducible components (see Remark \ref{rem-class}), the situation
is much more complicated here. The notion of irreducible tropical fan was
introduced in \cite{GKM} and can easily be generalized to tropical divisors in
$X$ (Definition \ref{def-irred}). A divisor is then irreducible if and only if
its weight function is unique up to a global multiple (Lemma \ref{criterion}).
We explore this notion when the ambient space is $ X=\mathcal{M}_{0,n} $, the
parameter space of tropical rational $n$-marked tropical curves \cite{GKM,M},
or equivalently the tropical Grassmannian $\mathcal{G}(2,n)$ modulo its
lineality space \cite{SS}. We prove for example that tropical Psi-classes and
vital divisors are irreducible. Note that the corresponding statement is also
true in algebraic geometry for the vital divisors \cite{K}, where they form the
main objects in the intersection theory on the moduli spaces
$\overline{M}_{0,n}$ of stable curves. In contrast, Psi-classes in algebraic
geometry cannot be defined as divisors, but only as divisor classes on
$\overline{M}_{0,n}$ \cite{AC}, and hence the notion of irreducibility does not
make sense for them. They are useful for the construction of enumerative
invariants like descendant Gromov-Witten numbers as in \cite{KoM}. Furthermore
we show that for $ n \le 6 $ every divisor in $\mathcal{M}_{0,n}$ that is
locally irreducible is already globally irreducible (Proposition
\ref{prop-m06}).

The second part of this paper, which is almost independent of the first, deals
with the tropical notion of points in special position for enumerative
problems. If one wants to count for instance curves of degree $d$ in $ \PP^2 $
passing through $ n=3d-1 $ given points, these point conditions should be
generic enough to obtain at least a finite result that is independent of the
conditions. We call the set of these generic point configurations in
$(\PP^2)^n$ the set of \textit{points in general position}. There exist several
definitions of this notion --- we can for example just require the number of
curves to be finite, or in addition that the curves are all smooth. A common
feature of these definitions is that they lead to an open dense subset of
$(\PP^2)^n$ of points in general position. The complement of this set in
$(\PP^2)^n$ is called the set of \textit{points in special position}. We
explore this notion in the corresponding tropical context. It turns out that
this set is the support of a tropical fan in $(\R^2)^n$, more precisely of a
push-forward of a divisor in the tropical moduli space corresponding to the
classical moduli space $ \overline{\mathcal{M}}_{0,3d-1} (\PP^2,d)$ of stable
maps (Propositions \ref{v1} and \ref{v2}). Finally, we compute the weights of
the top-dimensional cones in this tropical fan by means of elementary divisors
of certain matrices (Corollary \ref {resum}).

We would like to thank Hannah Markwig for inspiring and helpful discussions.
Part of this work was done at the Mathematical Sciences Research Institute
(MSRI) in Berkeley, CA, USA, during the one-semester program on tropical
geometry in fall 2009. The authors would like to thank the MSRI for
hospitality and support. In particular, Andreas Gathmann was supported by
the Simons Professorship of the MSRI.        

