\section {The set of points in special position for rational $n$-marked plane
  tropical curves} \label {l3}

One natural extension of the notion of rational $n$-marked abstract tropical
curve is the rational $n$-marked parametrized tropical curve introduced below,
which we consider to be the tropical analogue of a rational $n$-marked stable
map in classical algebraic geometry. Recall that for a given smooth complex
projective variety $X$, $n \in \N $ and a homology class $\beta \in H_2(X,\Z)$
a \textit{rational $n$-marked stable map (into $X$)} is a collection $(C;
x_1,\ldots,x_n,f)$ of a complete connected nodal curve $C$ of arithmetic genus
$0$, smooth points $x_i \in C$ which are pairwise disjoint, and a morphism $f:C
\to X$ satisfying $f_{∗}([C]) = \beta$ s.t.\ every irreducible component that
is mapped to a point in $X$ contains at least 3 \textit{special points}, i.e.\
markings or intersection points with other components (\textit{stability}). Two
stable $n$-marked maps $(C; x_1,\ldots,x_n, f)$ and $ (\tilde{C};
\tilde{x}_1,\ldots,\tilde{x}_n,\tilde{f}) $ are \textit{isomorphic} if there
exists an isomorphism $\varphi: C \to \tilde{C}$ s.t.\
$\varphi(x_i)=\tilde{x}_i$ and $\tilde{f} \circ \varphi = f$. The space
$\overline{M}_{0,n}(X, \beta)$ is the set of isomorphism classes of rational
$n$-marked stable maps; it carries the structure of a smooth projective variety
\cite{B,FP}.

In both the tropical and the algebraic setup one can then count such curves
through given points, where the number of these points has been chosen so that
one expects a finite answer. We are interested in the description of the set of
points in special position for these counting problems. Remember that in the
classical situation, when the ambient space for the rational stable maps is
$X=\PP^2$, it is known that the points in special position --- in the sense
that there are infinitely many curves passing through them --- form a
subvariety of $(\PP^2)^n$ (see \cite [Corollaire 13.1.5]{EGA} applied to the
(product) evaluation map). In the tropical context a more striking result
holds, namely that these points form a tropical subfan of $(\R^2)^n$ of
codimension one, which we can describe as a push-forward of some divisor in
$\M_{0,n}$. To be more precise, we deal here with two notions of ``points in
special position'' which both as sets arise as a divisor pushed forward by the
evaluation map. We restrict ourselves to the case where the ambient space is
$\R^2$ as our arguments just hold there.

But let us start by recalling the necessary tropical vocabulary. More details
about the following can be found in \cite [section 4]{GKM}.

\begin {definition}[Rational $n$-marked (labeled) parametrized tropical curve
    of degree $\Delta$ in $\R^2$]
   We denote a tuple $(v_1,\ldots,v_m)\in (\Z^2 \setminus \lk 0 \rk)^m$ by
  $\Delta$.

  A \textit {rational $n$-marked (labeled) parametrized tropical curve of
  degree $\Delta$ in $\R^2$} is a pair $(C, h)$ consisting of a rational
  $(n+m)$-marked abstract tropical curve $C=\!(\Gamma; x_1, \ldots, x_{n+m})$
  and a continuous map $h:\Gamma \to \R^2$ such that the following holds.
  \begin {itemize}
  \item The map $h$ is integer affine linear on each edge $E$ of $ \Gamma $,
    i.e.\ of the form $ h(t)=a+vt $ for some $ a \in \R^2 $ and $ v \in \Z^2 $.
    If we start parametrizing $E$ at the vertex $ V \in \partial E $ we call
    $v$ the \textit {direction} $ v(E,V) $ of $E$ with respect to $V$.
  \item At each vertex $V$ the \textit {balancing condition}
      \[ \qquad \quad \sum_{E: V \in \partial E} v(E,V) = 0 \]
    holds.
  \item The direction of $ x_i $ is $0$ for all $ i = 1, \ldots,n$ (contracted
    ends).
  \item The direction of $ x_i $ is $ v_{i-n} $ for all $ i=n+1,\ldots,n+m $
    (non-contracted ends).
  \end {itemize}
  Let $V$ be a 3-valent vertex which is not adjacent to a contracted end. Let
  $w_1,w_2,w_3$ be the direction vectors of its adjacent edges. Then we define
  the \textit {multiplicity} $ \mult(V) $ of $V$ to be $ \bet{\det(w_1|w_2)} =
  \bet{\det(w_1|w_3)} = \bet{\det(w_2|w_3)} $.

  In our pictures we will usually only draw the image curve $ h(\Gamma) $
  together with the points $ h(x_1),\dots,h(x_n) $. This image then has $m$
  (labeled) unbounded edges whose directions are contained in $\Delta$. We can
  think of the \textit {degree} $\Delta$ as the tropical equivalent of the
  homology class of an algebraic stable map. Note that the stability condition
  translates to the requirement that $\Gamma$ has at least $3$-valent vertices.

  In the following we will sometimes abbreviate the term ``rational $n$-marked
  (labeled) parametrized tropical curve of degree $\Delta$ in $\R^2$'' by
  ``parametrized tropical curve''. 
\end {definition}

\begin{construction} \label{mlab}
  Two parametrized tropical curves $(C,h)$ and
  $(\tilde{C},\tilde{h})$ are
  \textit{isomorphic} if there is an isomorphism $\varphi$ between the underlying abstract
  tropical curves $C$ and $\tilde{C}$ satisfying $\tilde{h} \circ \varphi =h$. The
parameter space of  rational $n$-marked (labeled) parametrized  tropical
  curves of given degree $\Delta$ in $\R^2$ modulo isomorphisms is denoted by
  $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$. Observe that the directions of the
  bounded edges in a parametrized tropical curve are not fixed by $\Delta$. But
  when fixing a combinatorial type (see Remark \ref{combtypes}) in $\M_{0,n+m}$ for $
m=|\Delta| $ there is
  a unique choice for the directions of the bounded edges for a parametrized
  tropical curve of degree $\Delta$ in $\R^2$ such that the underlying graph
  $\Gamma$ is of this combinatorial type. Hence there is a bijection between
  combinatorial types of the moduli spaces $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$
  and $\M_{0,n+m}$ \cite[Lemma 4.6]{GKM}. Note that the number of combinatorial
  types in $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$ is finite too for this reason.

  Coordinates in $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$ are given by the position
  of a \textit{root vertex} in $\R^2$ of the para\-metrized tropical curve and
  the lengths of the bounded edges (see \cite[Proposition 2.11]{GM2}).
Hence,
  given a parametrized tropical curve in $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$ we
  can send it to the image under the map $v$ of its underlying graph $\Gamma$
  (see Construction \ref{moduli}). So we get an isomorphism of polyhedral
  complexes \cite[Proposition 4.7]{GKM}
    $$ \M_{0,n}^{\tn{lab}}(\R^2,\Delta) \stackrel{\cong}{\rightarrow}
       v(\M_{0,n+m}) \times \R^2. $$
  In particular, we can consider $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$ as a
  tropical fan of dimension $(n+m-3)+2$.
\end{construction}

\begin{definition}[More intersection theory and tropical morphisms]
    \label{inter2}
  We extend the definitions of Definition \ref{intersection}.
  \begin {itemize}
  \item[(a)]
    Let $X$ and $Y$ be two $k$-cycles. After possibly adequately refining
    $X$ and $Y$ we can construct a $k$-cycle on $X \cup Y$, called the
    \textit{sum of the cycles $X$ and $Y$}, which is denoted by $X + Y$
    \cite[Construction 2.13]{AR}.
  \item[(b)]
    A \textit {morphism} $f: X \to Y$ \textit{of cycles $X$ and $Y$
    (resp.\ of fans $X$ and $Y$)} is a $\Z$-linear map, i.e.\ a map $\tilde{f}:
    \bet{X} \to \bet{Y}$ induced by a $\Z$-linear map between the underlying
    lattices $\Lambda_X$ and $\Lambda_Y$. 
  \item[(c)]
    Let $f: X \to Y$ be a morphism of a cycle $X$ to an $m$-cycle $Y$, $Z$ an
    $n$-cycle in $X$ where $n \leq m$. Using an appropriate refinement of $X$
    we can assume that the image of each cone in $X$ is a cone of $Y$. We
    define the \textit {push-forward of $Z$ along $f$} by $ f_*(Z) =
    \lk f(\sigma)|~\sigma \in Z\rk$. This polyhedral complex $f_*(Z)$ becomes
    an $n$-cycle in $Y$ by giving the weights
      $$ \qquad \quad
         w_{f_*(Z)} (\sigma') =
         \sum_{\substack{\sigma \in Z \\ f(\sigma)=\sigma'}}
           w_Z(\sigma) \cdot \bet{\Lambda_{\sigma'}/f(\Lambda_\sigma)} $$
    to the $n$-dimensional cones $\sigma' \in Y$ in the image of $Z$ under $f$
    \cite[Proposition 4.6]{AR}, where $\Lambda_\sigma$ is the sublattice of
    $\Lambda_X$ generated by $\sigma$.
  \item[(d)]
    The tropical Psi-classes of Definition \ref {psi} can be written as the
    divisors associated to certain rational functions \cite [Chapter 3]{KM}.
    As such, we can intersect several Psi-classes by consecutively intersecting
    with these rational functions. The product $ \psi_1^{k_1} \cdot \; \cdots
    \; \cdot \psi_n^{k_n} $ is then a cycle whose support consists of all
    curves such that a vertex with the markings $ i_1,\dots,i_m $ has valence
    at least $ k_{i_1}+\cdots+k_{i_m}+3 $ \cite [Chapter 4]{KM}.
  \end {itemize}
\end{definition}

\begin{definition}[Evaluation maps] \label {def-ev}
  For all $i=1,\ldots,n$ define the \textit{$i$-th evaluation map} by
    $$ \ev_i: \M_{0,n}^{\tn{lab}}(\R^2, \Delta) \to \R^2,~(C,h)
       \mapsto h(x_i). $$
  These maps are well-defined and are morphisms of fans
  \cite[Proposition 4.8]{GKM} using Construction \ref {mlab}. The \textit
  {evaluation map}
    $$ \ev:=\prod_{i=1}^n \ev_i: \M_{0,n}^{\tn{lab}}(\R^2,\Delta) \to
       \R^{2n} $$
  is a morphism of fans, too.
\end{definition}

\begin{definition} [Two versions of points in special position]
    \label {def-special}
  Fix $n>0$ and a degree $\Delta$ such that $ n=|\Delta|-1 $ (i.e.\ so that
  the source and target of $\ev$ have the same dimension and we expect a finite
  number of curves of degree $ \Delta $ through $n$ given points). A collection
  $Q=(Q_1,\ldots,Q_n)$ of $n$ points in $\R^2$ is said to be
  \begin {itemize}
  \item \textit {in special position (v1)} if $\ev^{-1}(Q)$ is infinite;
  \item \textit {in special position (v2)} if $\ev^{-1}(Q)$ is infinite or
    intersects the codimension-1 skeleton of $\M_{0,n}^{\tn{lab}}
    (\R^2,\Delta)$.
  \end {itemize}
  Otherwise we say that $Q$ is \textit {in general position} (for (v1) or
  (v2)).
  As $\ev$ is linear on each cone of $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$, note
  that $\ev^{-1}(Q)$ being infinite is equivalent to saying that the map $\ev$
  is not injective on (at least) one cone of $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$
  that intersects $ \ev^{-1}(Q) $.
\end{definition}

\begin{remark}
  Version (v2) is typically used for enumerative purposes: fix a point configuration
  $Q$ of points in general position (v2) and consider curves $C$ in $\ev^{-1}(Q)$, which
  contain only $3$-valent vertices. We assign a multiplicity $\mult(C)$ to each
  curve $C$ (see for example \cite{GKM}). It can then be shown that 
  $\sum_{C \in \ev^{-1}(Q)} \mult(C)$ is independent of the choice of $Q$; it
  is an invariant.
\end{remark}
 
\begin{remark} [Welschinger invariants as intersection products]
  Inspired by \cite{MR}, one could hope to write real enumerative invariants,
  for example Welschinger invariants, as intersection products of adequate
  cycles in $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$. Contrarily to the
  classical situation, the part of $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$
  corresponding to real $n$-marked rational projective curves has
  the same dimension as $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$. Hence,
  in order to impose some incidence conditions to these curves
  (like point, line or Psi-conditions) we would have
  to intersect a cycle of codimension $0$ in $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$
  with pull-backs of rational functions cutting out points, lines,
  or Psi-classes. For real curves, this cycle should be different
  from $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$, which corresponds to
  the case of complex curves. But Proposition \ref{mir} states that
  such a cycle cannot exist.
\end{remark}

We now want to derive formulas for the locus in $ \R^{2n} $ of points in
special position, for both versions (v1) and (v2).

\begin{remark}[Strings] \label {string}
  A parametrized tropical curve has a \textit{string} if the  underlying graph $\Gamma$
  contains a subgraph homeomorphic to $\R$ which does not  intersect the closures
  $\overline{x_i}$ of the markings $x_1,\ldots,x_n$. By \cite [Remark 3.7]{GM2} curves
  lying in codimension $0$ of $ \M_{0,n}^{\tn{lab}}(\R^2,\Delta) $ passing through points
  in special position have at least one string. Such a curve can have several strings
  which are not necessarily disjoint. 
\end{remark}

\begin {definition}[Free and fixed edges] \label{def-free-fixed}
  Let $(C,h)$ with $ C=(\Gamma; x_1,\ldots,x_n) $ be a pa\-ram\-e\-trized
  tropical curve in $\R^2$. Let $V$ be a vertex of $C$ and $E$ an adjacent
  edge. Then $E$ is called a \textit {free edge} at $V$ if it can be connected
  in $ \Gamma \setminus (V \cup \bigcup_{i=1}^n \overline{x_i})$ to an unmarked
  end. Otherwise we call $E$ a \textit {fixed edge} at $V$.
\end {definition}

\begin{proposition}[Points in special position (v1)] \label{v1}
  Let $n=\bet{\Delta}-1$ and assume $n>1$. Then the set of points in special
  position (v1) for curves in $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$ equals
  the support of the cycle $\ev_*(\psi_1 +\dots + \psi_n)$.
\end{proposition}

\begin{proof}
  We have to show two inclusions.

  In order to prove that the set of points in special position is contained in
  the support of $ \ev_*(\psi_1+\dots + \psi_n) $ we consider curves lying in
  codimension $0$ of $ \M_{0,n}^{\tn{lab}}(\R^2,\Delta) $ passing through
  points in special position. By Remark \ref{string} these curves have at least
  one string. The idea of the proof is that moving such a string yields a curve
  passing through the same point configuration, but lying in a codimension-one
  cone of $ \M_{0,n}^{\tn{lab}}(\R^2,\Delta) $ and having one $4$-valent vertex
  where one of the adjacent edges is a marking $x_1,\ldots,x_n$. (Remember that
  a marking $x_i$ is an edge of the underlying abstract tropical curve that is
  mapped to a point in the parametrized curve, it is a contracted end.
  So a marking $x_i$ adjacent to a vertex is depicted by a point on the vertex
  below.) This cone is often not unique as we might move the string in
  different directions. Moreover, the resulting curve does not necessarily lie
  in the boundary of the original cone: it might happen that the string first
  runs into a $4$-valent vertex such that no marking $x_i$ is adjacent to it.
  When resolving this vertex a new string appears which can be moved again. The
  following sequence of pictures shows the idea of these movements; there is
  one string which is drawn in dashed lines. In the second picture there is a
  4-valent vertex without adjacent marking. The final curve has a marking at a
  4-valent vertex $ x_i $, so it lies in the support of $ \psi_i $, which means
  that our set of points in special position lies in the support of $ \ev_*
  \psi_i $ and thus also of $ \ev_* (\psi_1+\cdots+\psi_n) $.

  \pict {bew}

  To make this argument rigorous we have to give an algorithm how to move a
  string so that it runs into a marking. For this let us first consider a curve
  with a 4-valent vertex on a string without adjacent marking (as in the second
  picture above). The following picture shows the types of 4-valent vertices
  without marking, where the types distinguish which of the adjacent edges are
  parallel.

  \pict {fall}

  Note that at most two of the adjacent edges are fixed by the point conditions
  since at least two of them lie on a string. If none of the adjacent edges is fixed,
  the $4$-valent vertex arises from the string movement in curves in codimension $0$
  having at least two strings joining in codimension $1$ at the 4-valent vertex. In
  this case, it is possible to move one of the strings differently in order to obtain a
  4-valent vertex with at least one adjacent fixed edge. Let us assume this in the
  following. Then, considering all possibilities which of the
  edges can be fixed in each of the above types, one can see that in each
  case there is a resolution of the 4-valent vertex such that at least one of
  the fixed adjacent edges becomes shorter. For instance, for the first 4-valent vertex
  from the left in the picture above the table below lists the resolution(s) (A), (B) or
  (C), where at least one of the fixed edges a, b, c or d gets shorter,
  depending on which of the adjacent edges are fixed. Note that, in the case of one fixed
  edge, this edge becomes shorter in each of the resolutions.

  \pict{fall1}
  \begin{center}
    \begin{tabular}{|c||c@{\;\;}|c@{\;\;}|c@{\;\;}|c@{\;\;}|c@{\;\;}|c|} \hline
      edges fixed & a, b & a, c & a, d & b, c & b, d & c, d \\ \hline \hline
      resolutions &(A), (C)&(B)&(A), (B)&(A), (B)&(B), (C)&(A), (C)\\ \hline
    \end{tabular}
  \end{center}


  The algorithm now works as follows. Start with a triple $ (C,S,E) $ where $C$
  is the original curve, $S$ a string on $C$, and $E$ a fixed edge adjacent to
  $S$ (such a choice is possible since there is at least one marking). Denote
  by $ M=M(C,S,E) $ the maximum distance in $C$ from $S$ to a marking behind
  $E$ which can be reached from $S$ without passing other markings. The
  following picture on the left shows such a choice; $S$ is again drawn with
  dashed lines, and $M$ is the length of the two line segments drawn in bold.

  \pict {string}

  We now claim that we can always change the curve by moving the string (as
  indicated by the arrows) so that $M$ decreases --- until either $M$ becomes
  zero and thus the string runs into a marking ($x_1$ in the picture above), or
  the string runs into another marking elsewhere earlier (maybe $x_6$ in the
  picture above). The possibility of such a movement is obvious as long as the
  length of $E$ is positive. If $E$ shrinks to a point in the movement (as in
  the picture above on the right) we have a 4-valent vertex at the string with
  at least one fixed adjacent edge, and by our above argument we know that we
  can always continue to move the string so that at least one of the fixed
  adjacent edges becomes shorter. Choosing this edge to be $E$ we can thus
  continue to decrease $M$ (note that by changing $E$ the set of first
  markings behind $E$ is replaced by a smaller one, so this step cannot make
  the maximum $M$ of their distances to $S$ bigger). This completes the
  argument and yields the first inclusion of the proposition.

  For the other direction, we have to show that $\ev_* (\psi_1 + \dots +
  \psi_n)$ contains no points in general position. As $\sum_{i=1}^n \psi_i$ is
  a divisor, we just have to consider curves lying in codimension one of $
  \M_{0,n}^{\tn{lab}}(\R^2,\Delta) $. So consider such a curve $ C =
  (\Gamma;x_1,\dots,x_n) $ in $ \psi_i $, i.e.\ a curve with one $4$-valent
  vertex $V$ with an adjacent marking $x_i$ and only 3-valent vertices
  otherwise. We have to prove that we can deform $C$ to a codimension-0 curve
  that still satisfies the same point conditions.

  This is obvious if $C$ contains a string, so let us assume that this is not
  the case. Note that removing $ \overline {x_i} $ from $ \Gamma $ separates
  $\Gamma$ into $3$ parts, whereas removing each of the other $ n-1 =
  |\Delta|-2 $ causes one more separation. So $ \Gamma \backslash
  \bigcup_{j=1}^n \overline{x_j} $ consists of $ |\Delta|+1 $ connected
  components. As none of these components can have more than one end (otherwise
  we would have a string) we conclude that there is precisely one bounded
  component with no end, whereas all other $|\Delta|$ components contain
  exactly one end. This means that at $V$ (which has $ \psi_i $ and three more
  edges adjacent to it) at least two of the unmarked adjacent edges must be
  connected in $ \Gamma \backslash \bigcup_{j=1}^n \overline{x_j} $ to an
  unbounded edge. We can then resolve $V$ so that these two edges $ E_1 $ and $
  E_2 $ remain together but separate from $ x_i $, forming a string and thus a
  movement of the curve with the positions of the markings fixed.

  \pict {move}

  Hence the points are by definition in special position.
\end{proof}

\begin{corollary} \label {cor-v1}
  In the same situation as above, the set of points in special position
  (v1) for curves in $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$ can be given the
  structure of a tropical subfan in $\R^{2n}$ of codimension one.
\end{corollary}

\begin{proof}
  Observe that $\ev$ is a morphism of fans of dimension $2n$. The claim then
  follows directly from Proposition \ref{v1} and Definition \ref{inter2} (a)
  and (c).
\end{proof}

\begin{remark}[Comparison to the classical situation]
  As stated in the introduction of this chapter, the set of $n$ points in
  special position (v1) in the corresponding classical situation is a
  subvariety of $(\PP^2)^n$. But in contrast to Corollary \ref {cor-v1} it is
  not necessarily a subvariety of codimension one: consider for instance conics
  in $\PP^2$ through 5 points. There are infinitely many conics through these
  points if and only if two of them coincide or four of them lie on a line ---
  and this forms a subvariety of codimension $2$ in $ (\PP^2)^5 $.

  The reason for the bigger dimension on the tropical side is that there are
  infinitely many liftings in the sense of \cite{JMM} to $\PP^2$ of the points
  in $\R^2$. When tropicalizing, the algebraic curves through each such
  configuration in $(\PP^2)^n$ give rise to tropical curves passing through the
  given configuration of points in $(\R^2)^n$. As a consequence, the number of
  such tropical curves through the given points can be infinite although the
  number of algebraic curves through any lifting of them is not, i.e.\ the
  point configuration in $ (\R^2)^n $ can be in special position (v1) although
  their liftings are not in special position classically.
  
  On the other hand, returning to the case of conics, the locus of points where
  we find reducible curves through them (which roughly corresponds to (v2)) is
  the image of the locus of reducible curves in $ \overline\M_{0,5} (\PP^2,2
  \ell)$ under the evaluation map, where $\ell$ is the homology class of a
  line. Here, both this locus in $ \overline\M_{0,5} (\PP^2,2 \ell)$ and its
  image in $ (\PP^2)^5 $ has codimension 1, the latter being the space of all
  points where three of them lie on a line.
\end{remark}

\begin {remark}[Generalization to curves with Psi-class conditions]
    \label {rem-v1}
  One can generalize the result of Proposition \ref {v1} as follows to the case
  of counting curves satisfying Psi-class conditions (i.e\ to tropical
  descendant Gromov-Witten invariants) as in \cite {MR}: fix $ n>0 $, a degree
  $ \Delta $, and $ k_1,\dots,k_n \ge 0 $ such that $ n =
  |\Delta|-1-k_1-\cdots-k_n $. If we then modify Definitions \ref {def-ev} and
  \ref {def-special} so that they use the moduli space $ \psi_1^{k_1} \cdot \;
  \cdots \; \cdot \psi_n^{k_n} \cdot \M_{0,n}^{\tn{lab}}(\R^2,\Delta) $ instead
  of $ \M_{0,n}^{\tn{lab}}(\R^2,\Delta) $ we count curves through given points
  in $ \R^2 $ with additional Psi-class conditions, i.e.\ such that the valence
  of the vertex with attached marking $ x_i $ is (at least) $ k_i + 3 $ for all
  $i$ \cite [Theorem 4.1]{KM}. The proof of Proposition \ref {v1} can then
  easily be adapted to show that the set of points in special position (v1)
  equals the support of the cycle $ \ev_* (\psi_1^{k_1} \cdot \; \cdots \;
  \cdot \psi_n^{k_n} \cdot (\psi_1 + \cdots + \psi_n)) $. In fact, the first
  direction in the proof of the proposition remains unchanged since it is still
  true that curves passing through points in special position contain a string.
  In the second direction the curves in question will still contain exactly one
  bounded region in $ \Gamma \backslash \bigcup_{j=1}^n \overline{x_j} $; it
  follows that the required movement of the curve is still possible, now
  resolving a $ (k_i+4) $-valent vertex to a $ (k_i+3) $-valent (containing the
  marking) and a 3-valent vertex.
\end {remark}

\begin {proposition}[Points in special position (v2)] \label{v2}
  Fix $\Delta$ and let $n$ equal $\bet{\Delta}-1$. Then the set
  of points in special position (v2) for curves in $ \M_{0,n}^{\tn{lab}}
  (\R^2,\Delta) $ equals the support of the push-forward $ \ev_*
  (\M_{0,n}^{\tn{lab}}(\R^2,\Delta)_{\tn{sk}}^{(2n-1)}) $ of the
  codimension-one skeleton of $ \M_{0,n}^{\tn{lab}} (\R^2,\Delta) $ (note that
  this codimension-one skeleton is a cycle by Proposition \ref {comod} and
  Construction \ref {mlab}).
\end {proposition}

\begin{proof}
  By definition the support of the cycle $ \ev_* (\M_{0,n}^{\tn{lab}}
  (\R^2,\Delta)_{\tn{sk}}^{(2n-1)}) $ contains only points in special position.
  In the other direction, if the points are in special position we can use the
  same argument as in the proof of Proposition \ref {v1} to show that they lie
  in the push-forward of the codimension-one skeleton of $ \M_{0,n}^{\tn{lab}}
  (\R^2,\Delta) $.
\end{proof}

\begin{corollary} \label {cor-v2}
  In the same situation as above, the set of points in special position (v2)
  for curves in $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$ can be given the structure
  of a tropical subfan in $ \R^{2n} $ of codimension one.
\end{corollary}

\begin{proof}
  Observe that $\ev$ is a morphism of tropical fans of dimension $2n$. The
  claim thus follows directly from Proposition \ref {v2} and Definition \ref
  {inter2} (c).
\end{proof}

\begin{remark} [Reducible cycles]\label{rem-irr}
  Using the characterization of Proposition \ref{v1}, the set of points in special
  position (v1) for curves in $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$ cannot be an irreducible
  cycle of $\R^{2n}$ as it is the push-forward of the reducible divisor $\psi_1 + \cdots
  + \psi_n$. Likewise, the set of points in special position (v2) for curves in
  $\M_{0,n}^{\tn{lab}}(\R^2,\Delta)$ is not an irreducible cycle as Example
  \ref{petersen} shows.
\end{remark}


\begin{remark} [Psi-condition] \label {rem-v2}
  As in Remark \ref {rem-v1} we can easily generalize Proposition \ref {v2} to
  the case of curves satisfying a Psi-condition $ \psi_i $ in addition to
  incidence conditions with points; the result is then that the set of points
  in special position is the push-forward by $ \ev $ of the codimension-one
  skeleton of $ \psi_i $ (which is a cycle by Proposition \ref {copsi}).
  However, for more than one Psi-condition there is no similar statement as the
  codimension-one skeleton of a product of Psi-classes does not have a
  canonical choice of weights.
\end {remark}

\begin{example} \label{petersen}
  Consider $\M_{0,2}^{\tn{lab}}(\R^2,\Delta)$ with $\Delta=\lk e_1
  +e_2,-e_1,-e_2\rk$ where $e_i$ is the $i$-th standard basis vector of $\R^2$
  (i.e.\ curves of degree one in the notation of \cite[Definition 4.1]{GKM}).
  Then $ \M_{0,2}^{\tn{lab}}(\R^2,\Delta) = \M_{0,5} \times \R^2 $ by
  Construction \ref {mlab}. The space $\M_{0,5}$ can be represented by the
  Petersen graph, depicted below twice. The \textit{Petersen graph} is a
  nonplanar, connected graph with $10$ vertices denoted here $ij$ with $i,j \in
  \{1,\ldots,5\} $ and $ i<j $, which are connected by $15$ edges s.t.\ a
  vertex $ij$ is only linked to the three vertices $kl$ with $k,l \in
  \{1,\ldots,5\}\setminus \{i,j\}$. In this graph, the two-dimensional cones of
  $ \M_{0,5} $ appear as edges, and the one-dimensional cones as vertices. The
  vertex $ ij $ corresponds to the ray of $ \M_{0,5} $ generated by the vector
  $v(\lk i,j \rk)$.

  $\M_{0,2}^{\tn{lab}}(\R^2,\Delta)$ is particularly interesting as in this
  case the sets of points in special position of the two versions coincide,
  more precisely they even coincide as tropical fans with the weights of
  Corollaries \ref {cor-v1} and \ref {cor-v2}. To see this, we observe that the
  codimension-one skeleton of $\M_{0,5}$ contains the cone $12$ which is not
  contained in $\psi_1+\psi_2$. Furthermore, the cones $34$, $35$, and $45$
  have each weight $2$ in the fan $\psi_1+\psi_2$ but just weight $1$ each in
  the codimension-one skeleton of $\M_{0,5}$. But these cones $12$, $34$, $35$,
  $45$ vanish when pushed forward by $ \ev $, since $ \ev $ is not injective on
  them (in fact their images consist of the configurations of two equal
  points in $ \R^2 $ and thus have codimension 2 in $ \R^2 \times \R^2 $).
  
  For more markings, the notions (v1) and (v2) will in general differ.

  \pictwo{petersen} \hfill $\circ$
\end{example}

