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\section{Saddle connection Siegel-Veech constants}
\label{sec:VRandSC}
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In this section we deduce from the volume recursion and its refinement for
spin and hyperelliptic components a proof of Theorem~\ref{intro:SVproduct}.
Almost all we need here has been proven already in \cite{emz}. We
start with two more auxiliary statements.
\par
\begin{Prop} \label{prop:fullsetSC}
The full set of saddle connection configurations for the strata
given in Proposition~\ref{prop:emzsummary} is in bijection with
(possibly unstable) 
backbone graphs and a cyclic ordering of its vertices at level zero. 
The subset of relevant configurations is in bijection
with stable backbone graphs. 
\end{Prop}
\par
Although not needed in the sequel, we relate for the convenience of the
reader our notion of twists and the angle information that \cite{emz} recorded. Let $\{\gamma_i\}_{i=1}^k$ be the arcs of
a configuration realized by~$(X,\omega)$ labeled cyclically
and let $a_i'$ and $a_i''$ be the angles between $\gamma_i$ and $\gamma_{i+1}$
at $z_1$ and $z_2$ respectively. If an edge~$e$ is separated by the loop
formed by $\gamma_i$ and $\gamma_{i+1}$, then the twist is
\be\label{eq:angle}
\bfp(e) \= \tfrac{1}{2\pi}(a_i' + a_i'') - 1\,. 
\ee
\par
The above proposition can be seen as follows. Given a collection of~$k$ homologous short saddle
connections there is a sphere (with $z_1$ as its south pole and $z_2$
as its north pole, see~\cite[Figure~5]{emz}) supporting the~$k$
saddle connections. This sphere is the source of the backbone of the graph. The components
at level zero are bounded by the arcs $\gamma_i$ and $\gamma_{i+1}$.
The converse is obvious, given that all the edges of a (stable) backbone graph are separating by definition.
Finally formula~\eqref{eq:angle} is just a restatement of the Gauss-Bonnet theorem.
\par
Recall that a backbone graph (being of compact type) is compatible
with a unique twist $\bfp(\cdot)$. If the vertices at level zero are
labeled as $1,\ldots, k$ as usual and if $(h_j, i(h_j))$ is the edge connecting
the $j$-th vertex to level~$-1$, we write  $p_j = |\bfp(h_j)|$ as we
did in Section~\ref{sec:RelMZInt}.
\par
\begin{Prop} \label{prop:boundaryvol}
For each configuration~$\cCC$ corresponding to a backbone graph~$\Gamma$,
the volume of the subspace $\omoduli[g,n]^\ve(\mu, \cCC)$ satisfies
\bas
& \phantom{\=}\,\, \sum_{k \geq 1} \sum_{\bfg, \bfmu} \pi \ve^2
\frac{h_{\PP^1}((m_1,m_2),\bfp)}
{k! \, (2g-3+n)!}
\cdot \prod_{i=1}^k
{(2g_i-1+n_i)!}\,{p_i}\,\vol(\omoduli[g_i, n_i+1](\mu_i,p_i-1)) \\
&\= 2^{1-k} \vol(\omoduli[g,n]^\ve(\mu, \cCC)) \,+\, o(\ve^2)\,
\eas
as $\ve \to 0$, where the summation conventions and $\bfp$ are
as in Theorem~\ref{intro:VolRec} and where $n_i = n(\mu_i)$. 
\end{Prop}
\par
\begin{proof} This is mainly contained in~\cite[Corollary~7.2, Formulas~8.1
and~8.2]{emz}, stating that the volume of the locus with an $\ve$-short
configuration is $\pi\ve^2$ times the volume of the corresponding boundary.
We now explain the combinatorial factors that appear. First, the factor of
two and the factorials result from the passage of the boundary volume
element in the ambient stratum to the product of the volume elements
of the components at the boundary, as explained in detail
in \cite[Section~6]{emz}. The $1/k!$ stems from labelling the level zero
vertices. Second, we need to count the ways to obtain a surface in
$\omoduli[g,n]^\ve(\mu, \cCC)$ by gluing a collection of surfaces
$(X_i,\omega_i)$ in $\omoduli[g_i,n_i+1](\mu_i,p_i-1)$.
\par
Suppose we are given a branched cover $b\colon \PP^1 \to \PP^1$ that has
ramification profile $\Hmu= ((m_1+1), (m_2+1),(p_1,\ldots,p_k))$ over the
points $0,1$ and $\infty$. We provide the domain with the differential
$\omega_{-1} = b^* dz$. Since this differential has no
residues we can glue $t \cdot \omega_{-1}$ with the surfaces $(X_i,\omega_i)$
by cutting the pole of order $p_i+1$ and gluing it to an
annular neighborhood of the zero of order $p_i-1$ of $\omega_i$.
For $t \leq \ve$ this provides a surface in $\omoduli[g,n]^\ve(\mu, \cCC)$,
see e.g.\ \cite[Section~4]{strata} for details of the construction.
The plumbing construction also depends on the choice of a $p_i$-th root of
unity at each pole (from the choice of a horizontal slit at a zero of order
$p_i-1$).  In total there are $h_{\PP^1}((m_1,m_2),\bfp)\cdot \prod_{i=1}^k p_i$
possibilities involved in the construction, thus justifying the remaining
combinatorial factors in the formula.
\par
We claim that this construction provides a collection of maps to
$\omoduli[g,n]^\ve(\mu, \cCC)$ that are almost everywhere injections
if none of the
surfaces  $(X_i,\omega_i)$ has a period of length smaller than~$\ve$.
In fact, if two such plumbed surfaces are isomorphic, this isomorphism
restricts to an automorphism of $(\PP^1, \omega_{-1})$ (see
\cite[Lemma~8.1]{emz} for more details) and this happens only on
a measure zero set. The locus where one of the $(X_i,\omega_i)$ has a short
period is subsumed in the $o(\ve^2)$ (\cite[Lemma~7.1]{emz}). Conversely,
for each surface~$(X,\omega)$
in $\omoduli[g,n]^\ve(\mu, \cCC)$ we can cut a ribbon graph around
the configuration $\cCC$. The restriction of $\omega$ has no periods
since the boundary curves are homologous by definition of~$\cCC$.
It can thus be integrated and completed to a map $b\colon \PP^1 \to \PP^1$
with ramification profile as above. 
\end{proof}
\par
\begin{proof}[Proof of Theorem~\ref{intro:SVproduct}]
We first focus on the case that $\omoduli[{g,n}](\mu)$ is connected.
A decomposition $g = \sum_{i=1}^k g_i$ and $(m_3,\ldots, m_n)= \mu_1\sqcup
\cdots \sqcup  \mu_k$ (as in equation~\eqref{eq:volintro} we proved)
determines uniquely a configuration and the converse is true up to
the labeling of the~$k$ vertices at level zero by Proposition~\ref{prop:fullsetSC}.
The configuration is relevant if and only if the volumes on the right-hand
side of~\eqref{eq:volintro} are non-zero. Since the saddle connection
Siegel-Veech constant
is the sum of ratios of the boundary volumes over the total volume
(by Proposition~\eqref{prop:SVviaboundary}), comparing the formula in Proposition~\ref{prop:boundaryvol} with equation~\eqref{eq:volintro} thus implies
Theorem~\ref{intro:SVproduct}. More precisely, note that the rescaled volume $v(\mu)$ defined
in~\eqref{eq:vol-rescale} involves a product of all $(m_i+1)$, while on the
right-hand side $(m_1+1)(m_2+1)$ is missing and this factor gives the right
hand side of the desired formula in Theorem~\ref{intro:SVproduct}. The factor
of~$\pi$ in Proposition~\ref{prop:boundaryvol} cancels with the one in
Proposition~\ref{prop:fullsetSC}.
\par
For disconnected strata with components parameterized by
$S \subseteq \{\odd,\even,\hyp\}$ the same proof gives the averaged version that 
\bes
\frac{1}{v(\mu)} \, \sum_{\bullet \in S} v(\mu)^\bullet \,c^\ho_{1\lra 2}(\mu)^\bullet
\= (m_1 +1)(m_2 + 1)
\ees
by rewriting equation~\eqref{eq:SVviaboundary} as a sum over the
connected components. It remains to prove the statement for the
odd spin subspaces (that may be disconnected for some
strata in our notation, in which case there is an extra hyperelliptic component with the same spin parity) as well as for
the hyperelliptic components. Then one obtains the desired result for each connected component
by taking suitable differences.
\par
For the odd spin subspaces we now focus on Theorem~\ref{thm:refinedVR} instead
of~\eqref{eq:volintro} and note that each decomposition of~$g$ and~$\mu$
still determines uniquely a configuration. Moreover, since the Arf-invariant
on stable curves of compact type is additive (see \cite[Proposition~4.6]{chenPB}
and \cite[Lemma~10.1]{emz}) and since by definition the spin assignment is additive
on the vertices, the configurations appearing on the right-hand side
in Theorem~\ref{thm:refinedVR} are precisely the configurations that contribute
(in the sense of Proposition~\ref{prop:SVviaboundary}) to the
Siegel-Veech constant of the odd spin subspaces. We thus obtain the analogous 
statement of Theorem~\ref{intro:SVproduct} for the odd spin subspaces  
 by comparing Proposition~\ref{prop:boundaryvol} to Theorem~\ref{thm:refinedVR}.
\par
For the hyperelliptic components we use Proposition~\ref{prop:HRvolrec}
and note that the summands on the right-hand side there correspond
exactly to the configurations on  hyperelliptic components, as
explained in \cite[Proposition~4.3]{chenPB} and \cite[Lemma~10.3]{emz}.
\end{proof}
\par

