To be able to interpret Propositions \ref {v1} and \ref {v2} numerically we now
want to compute the weights of the cells of codimension-1 cycles in $ \R^{2n} $
that are of the form $ \ev_* (Z) $ for a cycle $Z$ in $ \M_{0,n}^{\tn{lab}}
(\R^2,\Delta)$. For this we first need to recall some well-known linear algebra
results on elementary divisors.

\begin{lemdef}[Theorem on elementary divisors] \label{ele}
  Let $M$ be a finitely generated free module over a principal ideal domain
  $R$, and $N \subset M$ a submodule of $M$. Then there exists a basis $ \lk
  u_1,\ldots,u_m \rk$ of $M$, a basis $\lk v_1,\ldots,v_n \rk$ of $N$ and
  $e_1,\ldots,e_n \in R \setminus \lk 0\rk$ such that $v_i=u_i e_i$ for $
  i=1,\dots,n $ and $e_{i+1} \equiv 0 \mod e_i$ for $ i=1,\dots,n-1 $. The
  $e_i$ are called the \textit{elementary divisors of} $N$ and are unique up to
  units in $R$.

  For the quotient module it follows that $M/N \cong R^{m-n} \oplus
  \bigoplus_{i=1}^n R/ R e_i.$ Hence in the case $R=\Z$ the number of elements
  of the torsion part of $ M/N $ is $ \bet{\prod_{i=1}^n e_i} $. In the
  following we will denote this number by $ D(M/N) $.
\end{lemdef}

\begin {proof}
  See for example \cite [Theorem III.7.8]{L}.
\end {proof}

In the rest of the paper we will always use this result for the ring $ R=\Z $.
We then choose the $e_i$ to be positive.

\begin{lemma} \label {lem-gcd}
  In the situation of Lemma \ref{ele} (for $ R=\Z $) the number $ D(M/N) $
  is the greatest common divisor (gcd) of the $ n \times n $
  minors of any matrix $A$ representing the $ \Z $-linear map $ N
  \hookrightarrow M $. 
\end{lemma}
We therefore denote this number by $ D(A) $.
\begin{proof}
  See Remark 3 of Chapter 12.2 on page $6$ of \cite{vdW}. From Lemma \ref
  {ele} it follows that the map $ N \hookrightarrow M $ can be represented by a
  matrix $B$ with the elementary divisors on the diagonal and all other entries
  zero. As $A$ represents the same map there exist matrices $ S \in \tn{GL}
  (m,\Z) $ and $T \in \tn{GL}(n,\Z)$ such that $A=S \, B \, T$. The $n \times
  n$ minors of $A$ are then integer linear functions of the $n \times n$ minors
  of $B$, and vice versa. As the $\gcd$ of the $n \times n$ minors of $B$
  equals the product $ D(M/N) = \prod_{i=1}^n e_i$, this means that, up to
  units in $\Z$, the $\gcd$ of the $n \times n$ minors of $A$ equals $ D(M/N) $
  as well.
\end{proof}

\begin{corollary} \label{3}
  Let $B \in \Z^{(n+1) \times n}$ be a matrix having a $(n-k)\times (n-k)$ part $B_1$ in
  the upper left corner, a block $B_2$ of dimensions $(k+1) \times k$ in the
  lower right corner, an arbitrary (non-quadratic) block in the upper right
  corner, and just zeros in the lower left corner: 
    $$ B= \left( \begin{array}{c|c}
         B_1 & * \\ \hline
         0 & B_2
       \end{array} \right). $$
  Then
    \[ D(B) = |\det (B_1)| \cdot D(B_2). \]
\end{corollary}

\begin{proof}
  To compute the $n \times n$ minors of $B$, we have to erase one row of $B$ and
  look at the determinants of these matrices. If we delete one of the first
  $n-k$ rows, the vectors in the columns of the quadratic part that remain are
  linearly dependent, hence these minors vanish. Deleting the $j$-th row with $
  n-k+1 \le j \le n+1 $, we obtain a matrix that contains the block $B_1$ in
  the upper left corner, a quadratic block $C_j$ of dimensions $ k \times k $
  in the lower right corner and 0 in the lower left part. Hence the determinant
  of such a matrix equals the product of $\det(B_1)$ and $\det(C_j)$. So by
  Lemma \ref {lem-gcd} we get
  \begin{align*}
    D(B)
      &= \gcd \left\{ \det(B_1)\cdot \det(C_j) :\; j = n-k+1,\ldots,n+1
         \right\} \\
      &= |\det(B_1)| \cdot \gcd \left\{ \det(C_j) :\; j=n-k+1,\ldots,n+1
         \right\} \\
      &= |\det(B_1)| \cdot D(B_2),
  \end{align*}
  using the property $ \gcd(ma, mb) = m \, \gcd(a,b) $ for $ m \in \N $ and $
  a,b \in \Z $.
\end{proof}

We will now apply these results to obtain formulas for the weights of
push-forwards of codimension-1 cycles along the evaluation map. For this we
first have to classify those cycles.

\begin {remark}[Codimension-1 types in $ \M_{0,n}^{\tn{lab}}(\R^2,\Delta)
    $] \label {rem-types}
  In the following, a connected component of $ \Gamma \backslash
  \bigcup\limits_{j=1}^n \overline{x_j}$ will be called a \textit {region} of
  the curve. Consider a cell of a codimension-1 cycle in $ \M_{0,n}^{\tn{lab}}
  (\R^2,\Delta) $ on which the evaluation map is injective. It corresponds to a
  combinatorial type of curves having exactly one 4-valent vertex, with all
  other vertices being 3-valent.

  If this 4-valent vertex has an adjacent marking, an argument as in the proof
  of the second part of Proposition \ref {v1} shows that there is exactly one
  region that is bounded (by markings), whereas the others contain
  exactly one end. We will call this type (A); in the picture below the bounded region is
  drawn with dotted lines.

  If the 4-valent vertex has no adjacent marking it lies in a unique region.
  The same argument as above then shows that
  \begin {itemize}
  \item either all regions have exactly one end (type (B) below, with the
    region containing the 4-valent vertex drawn with dotted lines); or
  \item the region with the 4-valent vertex has exactly two ends, there is one
    other bounded region, and all other regions have exactly one end (type (C)
    below, where the bounded region is drawn with dotted lines).
  \end {itemize}
  \pict {codim1}
\end {remark}

The weights of the images of those cycles under the evaluation map can be
computed by the following formula.

\begin{lemdef} \label{index}
  Let $Z$ be a cycle of dimension $ 2n-1 $ in $ \M_{0,n}^{\tn{lab}}
  (\R^2,\Delta) $ and $\sigma' \in \ev_*(Z) \subset \R^{2n}$ a cone of the same
  dimension, i.e.\ of codimension 1 in $ \R^{2n} $. Then, with the convention
  and notations from Definition \ref {inter2} (c), the weight of $ \sigma' $ in
  the cycle $ \ev_* (Z) $ is
    $$ w_{\ev_*(Z)}(\sigma')
       = \sum_{\substack{\sigma \in Z \\ \ev(\sigma)=\sigma'}}
         w_Z(\sigma) \cdot \mult(\sigma) $$
  where the multiplicity of a cone $ \sigma $ of dimension $ 2n-1 $ in $
  \M_{0,n}^{\tn{lab}}(\R^2,\Delta) $ is defined as
    \[ \mult (\sigma) := D(\Z^{2n}/\ev(\Lambda_\sigma)). \]
  If $C$ is any curve corresponding to a point in $ \sigma $, we will write the
  multiplicity $ \mult (\sigma) $ also as $ \mult (C) $.
\end{lemdef}

\begin{proof}
  By Definition \ref {inter2} (c) we have to sum over the numbers $ w_Z(\sigma)
  \cdot \bet{\Lambda_{\sigma'}/\ev(\Lambda_\sigma)} $ for all $ \sigma
  \in Z $ with $ \ev(\sigma)=\sigma' $. But as $ \Lambda_{\sigma'} $ is by
  definition a saturated lattice in $ \Z^{2n} \subset \R^{2n} $, we have $
  \Z^{2n} = \Lambda_{\sigma'} \oplus \Z $, and thus the torsion parts of
  $ \Lambda_{\sigma'}/\ev(\Lambda_\sigma) $ and $ \Z^{2n} / \ev(\Lambda_\sigma)
  $ agree.
\end{proof}

\begin {remark} \label {length}
  By Lemma \ref {ele}, the number $ \mult (\sigma) = D(\Z^{2n}/\ev
  (\Lambda_\sigma)) $ in Lemma \ref {index} can be computed as the $ D(A) $ of
  any $ (2n)\times(2n-1) $ matrix $A$ representing the $ \Z $-linear map $ \ev:
  \Lambda_\sigma \to \Z^{2n} $. By \cite[Remark 5.2]{GKM} and
  \cite[Example 3.3]{GM2} one possibility to set up this matrix is to use the
  lengths of all bounded edges and the position in $ \R^2 $ of a root vertex as
  coordinates for $ \Lambda_\sigma $.
\end {remark}

We will now compute explicitly the multiplicities in Lemma \ref {index} for the
cases of Remark \ref {rem-types} and proceed in several steps.

\begin {lemma}[Splitting off vertex multiplicities] \label {lem-reduce}
  Let $C$ be a curve in a codimension-1 cone of $
  \M_{0,n}^{\tn{lab}}(\R^2,\Delta) $ as in Remark \ref {rem-types}. Assume that
  there is an edge of $C$ such that splitting this edge yields two parts $ C_1
  $ and $ C_2 $ of $C$, where $C_2 $ contains only regions with exactly one end and having
  only 3-valent vertices.
  Then
    \[ \mult (C) = \mult (C_1) \cdot \prod_{V \in C_2} \mult (V), \]
  where the product runs over all vertices in $ C_2 $ with no adjacent marking,
  and the multiplicity $ \mult (V) $ of such a vertex $V$ is defined as usual
  as the absolute value of the determinant of two of the adjacent direction
  vectors \cite [Definition 3.5]{GM2}. The following picture shows an example.

  \pict {vertex}
\end {lemma}

\begin {proof}
  Let $k$ be the number of unbounded ends of $ C_2 $. Then $ C_2 $ has $ k-1 $
  markings and $ 2k-2 $ bounded edges (including the split edge). Choosing the
  root vertex to be in $ C_1 $ (see Remark \ref {length}), only the $ 2k-2 $
  coordinates of the $ k-1 $ markings in $ C_2 $ depend on the $ 2k-2 $ lengths
  of the bounded edges in $ C_2 $. Hence the matrix for $ \ev $ as in Remark
  \ref {length} has the form as in Corollary \ref {3}, with $ B_1 $ the $
  (2k-2)\times(2k-2) $ block consisting of these coordinates and lengths. As
  the absolute value of the determinant of $ B_1 $ equals the product of the
  vertex multiplicities in $ C_2 $ by \cite [Proposition 3.8]{GM2} and $ B_2 $
  is precisely the matrix for the evaluation map on $ C_1 $, the claim follows
  from Corollary \ref {3}.
\end {proof}

\begin {lemma}[Multiplicity of a bounded region] \label {lem-bounded}
  Let $C$ be a $3$-valent curve that has exactly one bounded region, with all
  other regions being single ends (such a curve occurs for instance as a part of the
  types (A) and (C) in Remark \ref {rem-types}). Then the multiplicity of $C$
  is
    \[ \mult (C) = \gcd \{ w(E) :\; \text {$E$ end in $C$} \}
         \cdot \prod_{V \in C} \mult (V), \]
  where the weight $ w(E) $ of the edge $E$ is the gcd of the two coordinates
  of the direction vector of $E$, and the product is taken over all vertices of
  $C$ with no adjacent marking. The following picture shows an example.

  \pict {bounded}
\end {lemma}

\begin {proof}
  We set up the matrix $A$ for the evaluation map as in Remark \ref {length},
  with the root vertex within the bounded region. To compute the minors of $A$
  as required by Lemma \ref {lem-gcd} we have to erase one of its rows. Note
  that the rows of $A$ correspond to the coordinates of the markings in $ \R^2
  $. So let us assume that we erase the row for the $i$-th coordinate of the
  marking $ x_j $ for some $ i=1,2 $ and $ j=1,\dots,n $. Note that the length
  of the bounded edge $ E_j $ adjacent to $ x_j $ is needed only for the
  coordinates of $ x_j $ in $ \R^2 $, and so in the remaining matrix the column
  corresponding to $ E_j $ has at most one non-zero entry, namely for the other
  coordinate of $ x_j $ in $ \R^2 $. Laplace expansion of the determinant
  w.r.t.\ the $ E_j $ column thus simply gives the product of this coordinate
  of $ x_j $ in $ \R^2 $ and the determinant of the evaluation matrix for the
  curve where the marking $ x_j $ is deleted (and thus $ E_j $ becomes an
  unbounded end). But this determinant just equals the product of all vertex
  multiplicities by \cite [Proposition 3.8]{GM2}.

  Altogether we see that $ D(A) $ is the product of all vertex multiplicities
  times the gcd of both coordinates of all markings in $ \R^2 $, as we have
  claimed.
\end {proof}

\begin {corollary}[Multiplicity of the types (A) and (C)] \label {cor-ac}
  Let $C$ be a curve in codimension 1 as in Remark \ref {rem-types}. If $C$ is
  of type (A) with bounded region $ C_b $ then its multiplicity is
    \[ \mult (C) = \gcd \{ w(E):\; \text {$E$ edge in $ C_b $ with adjacent
       marking} \} \cdot \prod_{V \in C} \mult(V). \]
  If it is of type (C) with bounded region $ C_b $ then its multiplicity is
    \[ \mult (C) = \gcd \{ w(E):\; \text {$E$ edge in $ C_b $ with adjacent
       marking} \} \cdot |\det (v,v')| \cdot \prod_{V \in C} \mult(V) \]
  where $v$ and $v'$ are the directions of the two fixed adjacent edges at the
  4-valent vertex (i.e.\ the ones that do not connect to an end within their
  region when coming from the 4-valent vertex). In both formulas, the product
  is taken over all 3-valent vertices without adjacent marking in $C$.
\end {corollary}

\begin {proof}
  Let $V$ be the 4-valent vertex of $C$. If $C$ is of type (A) we can first use
  Lemma \ref {lem-reduce} to split off all vertices behind the two unmarked
  edges adjacent to $V$ that do not lead to the bounded region. This way we get
  the multiplicities of all split-off vertices as a factor, and are left with
  a curve where two of the unmarked edges adjacent to $V$ are solitary ends (as
  it is already the case in the example picture in Remark \ref {rem-types}).
  Now the evaluation matrix of this curve is precisely the same as for the
  curve where these two ends with direction vectors $ v_1 $ and $ v_2 $ are
  replaced by one end with direction $ v_1+v_2 $. Now in the remaining curve we
  can continue to split off all vertices that lie outside of the (closure of
  the) bounded region. This way we are left with a curve whose multiplicity has
  been computed in Lemma \ref {lem-bounded}. Altogether, we get the result
  stated in the corollary.

  If $C$ is of type (C) the procedure is very similar. We first split off all
  vertices behind the two free edges adjacent to $V$ and replace the resulting
  two solitary ends at $V$ by one. This makes the 4-valent vertex $V$ into a
  new 3-valent one for which two adjacent direction vectors are $v$ and $v'$.
  As above, we continue to split off all vertices that are outside of the
  bounded region (one of which will be the new one with multiplicity $ |\det
  (v,v')| $), and use Lemma \ref {lem-bounded} to obtain the result.
\end {proof}

\begin {lemma}[Multiplicity of a region with 4-valent vertex and one end]
    \label {lem-4valent}
  Let $C$ be a curve in codimension 1 as in Remark \ref {rem-types} that has a
  region with a 4-valent vertex and one end directly adjacent to it, all other
  vertices being 3-valent and all other regions being single ends. Then the
  multiplicity of $C$ is
    \[ \mult (C) = \gcd \{ w(E) \cdot |\det (v^1_E,v^2_E)| :\;
         \text {$E$ end with marking in $C$} \}
         \cdot \prod_{V \in C} \mult (V), \]
  where the product is taken over all 3-valent vertices of $C$ with no adjacent
  marking, and $ v^1_E $ and $ v^2_E $ denote the direction vectors of the two
  fixed edges adjacent to the 4-valent vertex that do not connect to $E$. The
  following picture shows an example.

  \pict {4valent}
\end {lemma}

\begin {proof}
  The proof is very similar to that of Lemma \ref {lem-bounded}. Let $V'$ be
  the $4$-valent vertex of $C$ and $E$ its unique free end. We set up the
  matrix $A$ of the evaluation map using $V'$ as the root vertex. To compute a
  maximal minor of $A$ we delete the row corresponding to the $i$-th coordinate
  of the marking $ x_j $. Performing a Laplace expansion of the minor w.r.t.\
  the column corresponding to the length of the bounded edge adjacent to $ x_j
  $ we obtain the other coordinate of $ x_j $ in $ \R^2 $ times the determinant
  of the evaluation matrix corresponding to the curve where the marking $ x_j $
  has been deleted. In this new curve we can use the technique of Lemma \ref
  {lem-reduce} to split off all vertices behind the one that lead to $ x_j $.
  We can then replace the two resulting solitary ends at $V'$ ($E$ and the one
  just created by splitting off vertices) at $V'$ by one, leading to a new
  3-valent vertex with multiplicity $ |\det (v^1_E,v^2_E)| $. The resulting
  determinant gives the product of all vertex multiplicities by 
  \cite[Proposition 3.8]{GM2}. Taking the gcd of these expressions for all rows of
  $A$ yields the desired result.
\end {proof}

\begin {corollary}[Multiplicity of the type (B)] \label {cor-b}
  Let $C$ be a curve in codimension 1 as in Remark \ref {rem-types}. If $C$ is
  of type (B), and $C'$ denotes the region with the 4-valent vertex, then its
  multiplicity is
    \[ \mult (C) = \gcd \{ w(E) \cdot |\det (v^1_E,v^2_E)| \} \cdot
       \prod_{V \in C} \mult(V) \]
  where
  \begin {itemize}
  \item the product is taken over all 3-valent vertices without adjacent
    marking in $C$;
  \item the $ \gcd $ is taken over all edges in $C'$ that are adjacent to a
    marking and lie behind one of the three fixed edges adjacent to the
    4-valent vertex; and
  \item $ v^1_E $ and $ v^2_E $ denote as in Lemma \ref {lem-4valent} the
    directions of the two fixed edges adjacent to the 4-valent vertex that do
    not connect to $E$.
  \end {itemize}
\end {corollary}

\begin {proof}
  The proof is similar to that of Corollary \ref {cor-ac}: first we can split
  off all vertices outside of $C'$ and behind the one free end of the 4-valent
  vertex, and then we use Lemma \ref {lem-4valent} for the resulting curve.
\end {proof}

Summarizing, we can now rephrase Lemma \ref{index} as follows.

\begin{corollary} \label{resum}
  Let $Z$ be a cycle of dimension $ 2n-1 $ in $ \M_{0,n}^{\tn{lab}}
  (\R^2,\Delta) $ and $\sigma' \in \ev_*(Z) \subset \R^{2n}$ a cone of the same
  dimension, i.e.\ of codimension 1 in $ \R^{2n} $. Then, with the convention
  and notations from Definition \ref {inter2} (c), the weight of $ \sigma' $ in
  the cycle $ \ev_* (Z) $ is
    $$ w_{\ev_*(Z)}(\sigma')
       = \sum_{\substack{\sigma \in Z \\ \ev(\sigma)=\sigma'}}
         w_Z(\sigma) \cdot \mult(\sigma) $$
  where the multiplicity $ \mult(\sigma) $ of a cone $ \sigma $ of dimension $
  2n-1 $ in $ \M_{0,n}^{\tn{lab}}(\R^2,\Delta) $ is given by Corollaries
  \ref{cor-ac} and \ref{cor-b} depending on their type as in Remark
  \ref{rem-types}.

  In particular, this gives an explicit formula for the cycles in $ \R^{2n} $
  of points in special position (v1) and (v2) as in Propositions \ref {v1} and
  \ref {v2}.
\end{corollary}

