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\title[Virtual Canonical Bundle]
{Divisor classes and the virtual canonical bundle for genus 0 maps}

\author[de Jong]{A. J. de Jong}
\address{Department of Mathematics \\
  Columbia University \\ New York NY 10027}
\email{dejong@math.columbia.edu}

\author[Starr]{Jason Starr}
\address{Department of Mathematics \\
  Massachusetts Institute of Technology \\ Cambridge MA 02139}
\email{jstarr@math.mit.edu} 

\date{\today}
\begin{document}

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\begin{abstract}
We prove divisor class relations for families of genus 0 curves and
used them to compute the divisor class of the ``virtual'' canonical
bundle of the Kontsevich space of genus 0 maps to a smooth target.
This agrees with the canonical bundle in good cases.  
This work generalizes Pandharipande's results in the special case that
the target is projective space, \cite{Pand97}, \cite{QDiv}.  Our
method is completely different from Pandharipande's.
\end{abstract}

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\maketitle

\section{Statement of results}

\noindent
Much geometry of
a higher-dimensional complex variety $X$ is captured by the rational
curves in $X$.
For uniruled and rationally connected varieties 
the parameter spaces for rational curves in $X$ are also interesting.  
These parameter spaces are rarely compact, but
there are natural compacitifications: the Chow variety, the
Hilbert scheme and the Kontsevich moduli space.  
Of these, the most
manageable is the Kontsevich space.
Briefly $\Kgnb{0,r}(X,\beta)$
parametrizes data $(C,p_1,\dots,p_r,f)$ of a proper,
connected, at-worst-nodal, arithmetic genus $0$ curve $C$, a
collection $p_1,\dots,p_r$ of distinct, smooth points of $C$, and a
morphism $f:C\rightarrow X$ with $f_*[C] = \beta$ and satisfying a
natural stability condition.  

\mni
When 
$\Kgnb{0,r}(X,\beta)$ is irreducible and reduced and when the
dimension equals the expected, or \emph{virtual}, dimension, one can
ask more refined questions about
the geometry of $\Kgnb{0,r}(X,\beta)$.  For instance, 
what is its Kodaira dimension?
The first step in answering this and other questions is understanding the
canonical bundle.

\mni
In this article we give a formula for the
\emph{virtual canonical bundle} of $\Kgnb{0,r}(X,\beta)$.  The virtual
canonical bundle is a naturally defined
line bundle which equals 
the actual canonical bundle if
$\Kgnb{0,r}(X,\beta)$ is irreducible and reduced and its dimension
equals the virtual dimension.

\begin{thm} \label{thm-main}
\marpar{thm-main}
Assume that $e := \langle C_1(T_X), \beta \rangle \neq 0$.  For
$\Kgnb{0,0}(X,\beta)$, the virtual canonical bundle equals
$$
\frac{1}{2e} [
2e \pi_* f^* C_2(T_X) 
- (e + 1) \pi_* f^* C_1(T_X)^2 +  
$$
$$
\sum_{\{\beta',\beta''\}, \beta'+\beta'' = \beta} 
(\langle f^*C_1(T_X),\beta' \rangle \langle
f^*C_1(T_X),\beta'' \rangle -4e)
\Delta_{\beta',\beta''}].
$$
For $\Kbm{0,1}{X,\beta}$, the virtual canonical bundle equals
$$
\frac{1}{2e} [ 
2e\pi_*f^* C_2(T_X) -(e+1) \pi_* f^* C_1(T_X)^2 + 
$$
$$ 
\sum_{\{\beta',\beta''\}, \beta'+\beta''=\beta} 
(\langle f^*C_1(T_X),\beta' \rangle \langle
f^*C_1(T_X),\beta'' \rangle -4e)
\Delta_{\beta',\beta''}]
+ \psi_1.
$$
Finally, for $r\geq 2$, the virtual canonical bundle of
$\Kbm{0,r}{X,\beta}$ equals
$$
\frac{1}{2e} [
2e\pi_* f^* C_2(T_X) 
-(e + 1) \pi_* f^* C_1(T_X)^2 +
$$
$$  
\sum_{\{\beta',\beta''\}, \beta'+\beta''=\beta} 
(\langle f^*C_1(T_X),\beta' \rangle \langle
f^*C_1(T_X),\beta'' \rangle -4e)
\Delta_{\beta',\beta''}] 
+ 
$$
$$
\frac{1}{r-1} \sum_{(A,B), 1\in
  A} \#B(r-\#B)\Delta_{(A,B)}.
$$
\end{thm}

\mni
In order to prove these formulas, we need to prove some divisor class
relations for families of genus $0$ curves.  These relations are of
some independent interest.

\begin{prop} \label{prop-dcr}
\marpar{prop-dcr}
Let $\pi:\mc{C}\rightarrow M$ be a proper, flat family of connected,
at-worst-nodal, arithmetic genus $0$ curves over a quasi-projective
variety $M$ or over a Deligne-Mumford stack $M$ with quasi-projective coarse
moduli space.  Let $D$ be a $\QQ$-Cartier divisor class on $\mc{C}$.  
\begin{enumerate}
\item[(i)]
There
is an equality of $\QQ$-divisor classes on $M$
$$
\pi_*(D\cdot D) + \langle D,\beta \rangle \pi_*(D\cdot
C_1(\omega_\pi)) = \sum_{\{\beta',\beta''\}, \beta'+\beta'' = \beta} 
\langle D,\beta' \rangle \langle D,\beta'' \rangle \Delta_{\beta',\beta''}.
$$
\item[(ii)]
Also, there is an equality of $\QQ$-divisor classes on $\mc{C}$
$$
2\langle D,\beta \rangle D - \pi^*\pi_*(D\cdot D) + \langle D,\beta
\rangle^2 C_1(\omega_\pi) = 
\sum_{(\beta',\beta'')} \langle D,\beta'' \rangle^2
\widetilde{\Delta}_{\beta',\beta''}.
$$
\end{enumerate}
\end{prop}

\mni
The notation $\Delta$ indicates a boundary
divisor.  The precise meaning of each term is given in Section
~\ref{subsec-bound}.  
Roughly, $\Delta_{(A,B)}$ is the divisor in $M$ 
parametrizing reducible curves where the marked points indexed by $A$
lie in one component, and the marked points indexed by $B$ lie in the
other component. The divisor
$\Delta_{\beta',\beta''}$ in $M$ parametrizes reducible
curves with one component of class $\beta'$ and one component of class
$\beta''$.  And the divisor $\widetilde{\Delta}_{\beta',\beta''}$ 
in $\mc{C}$ is the closed subset of $\pi^{-1}(\Delta_{\beta',\beta''})$ which
is the union of all components of fibers with class $\beta'$.  The
bundle $\omega_\pi$ on $\mc{C}$ is the relative dualizing sheaf of $\pi$.  
Finally the bundle
$\psi_i$ on $M$ is the pullback of $\omega_\pi$ 
by the ``$i^\text{th}$ marked point'' section.

\mni
The \emph{virtual canonical bundle} is the determinant of the
Behrend-Fantechi perfect obstruction theory for $\Kbm{0,r}{X,\beta}$,
cf. \cite{BF}.
This is a complex $E^\bullet$ of $\OO$-modules on $\Kbm{0,r}{X,\beta}$
which is perfect of amplitude $[-1,0]$,
together with a map to the cotangent complex 
$$
\phi:E^\bullet
\rightarrow L^\bullet_{\Kbm{0,r}{X,\beta}}
$$
such that $h^0(\phi)$ is
an isomorphism and $h^{-1}(\phi)$ is surjective.  The determinant of a
perfect complex is an invertible sheaf that, roughly, is the
alternating tensor product of the determinants of the terms of the
complex, cf. \cite{detdiv}.  

\mni
If the dimension of $\Kbm{0,r}{X,\beta}$ equals the virtual dimension
$$
\langle c_1(T_X), \beta \rangle + \text{dim}(X)+r-3
$$
and if $\Kbm{0,r}{X,\beta}$ is reduced at all codimension 1 points, then the
virtual canonical bundle equals the canonical bundle.  Even when they
differ, the virtual canonical bundle can be quite useful.  For
instance, there is a
lower bound for the dimension of the space
of maps from a curve $C$ into a smooth projective variety $Y$ mapping
an effective divisor $B\subset C$ to specified points of $Y$
$$
\text{dim}_{[f]}\text{Hom}(C,Y;f|_B) \geq \langle -K_Y, f_*[C]
\rangle + \text{dim}(Y)(1-p_a(C) - \text{deg}(B))
$$
cf. \cite[Theorem II.1.2, Theorem II.1.7]{K}.  An analogous results holds for
$Y=\Kgnb{0,r}(X,\beta)$ when $K_Y$ is replaced by the virtual
canonical bundle and $\text{dim}(Y)$ is replaced by the virtual
dimension, cf. \cite{dJS9}.  

\begin{prop}\cite[Lemma 2.2]{dJS9} \label{prop-dim}
\marpar{prop-dim}
Let $C$ be a projective Cohen-Macaulay curve, let $B\subset C$ be a
divisor along which $C$ is smooth, and let $f:C\rightarrow
\Kgnb{0,r}(X,\beta)$ be a 1-morphism.  Assume that every generic point
of $C$ parametrizes a smooth, free curve in $X$.  Then for
$Y=\Kgnb{0,r}(X,\beta)$
$$
\text{dim}_{[f]}\text{Hom}(C,Y;f|_B) \geq \langle -K^{\text{virt}}_Y, f_*[C]
\rangle + \text{dim}^\text{virt}(Y)(1-p_a(C) - \text{deg}(B)).
$$
\end{prop}

\subsection{Outline of the article} \label{subsec-out}
\marpar{subsec-out}

\mni
There is a universal family of stable maps over $\Kgnb{0,r}(X,\beta)$
$$
(\pi:\mc{C}\rightarrow
\Kgnb{0,r}(X,\beta),\sigma_1:\Kgnb{0,r}(X,\beta)\rightarrow
\mc{C},\dots,\sigma_r:\Kgnb{0,r}(X,\beta), f:\mc{C}\rightarrow X).
$$
The Behrend-Fantechi obstruction theory is defined in terms of total
derived pushforwards under $\pi$ of the relative cotangent sheaf of $\pi$
and the pullback under $f$ of the cotangent bundle of $X$.  Thus
the Grothendieck-Riemann-Roch theorem gives a formula for the virtual
canonical bundle.  Unfortunately it is not a very useful formula.  For
instance, using this formula
it is difficult to determine whether
the virtual canonical bundle is NEF, ample, etc.  
But combined with Proposition ~\ref{prop-dcr},
Grothendieck-Riemann-Roch gives the formula from Theorem
~\ref{thm-main}.  The main work in this article is proving Proposition
~\ref{prop-dcr}.  

\mni
The proof reduces
to local computations for the universal family over the Artin stack of
all prestable curves of genus $0$, cf. Section~\ref{sec-local}.
Because of this, most results are stated for Artin stacks.  This leads
to one \emph{ad hoc} consruction: since there is as yet no theory of
cycle class groups for Artin stacks admitting Chern classes for all
perfect complexes of bounded amplitude, a Riemann-Roch theorem for all
perfect morphisms relatively representable by proper algebraic spaces,
and arbitrary pullbacks for all cycles coming from Chern classes, a
stand-in $Q_\pi$ is used, cf. Section~\ref{sec-Qpi}. (Also by avoiding
Riemann-Roch, this allows some relations to be proved ``integrally''
rather than ``modulo torsion'').  
% Also, although the relative Picard
% functor of the universal family of genus $0$ curves is well-known, a
% description is included in Sections ~\ref{sec-dec} and ~\ref{sec-cpct}
% for completeness.

\mni  
In the special case $X=\PP^n_k$, Pandharipande proved both
Theorem~\ref{thm-main} and Proposition ~\ref{prop-dcr} in
\cite{QDiv} and \cite{Pand97}.  
Pandharipande's work was certainly our inspiration. 
But our proofs are
completely different, yield a more general virtual canonical bundle
formula, and hold modulo torsion (and sometimes ``integrally'') rather
than modulo numerical equivalence.  

% \section{Decorated prestable curves} \label{sec-dec}
% \marpar{sec-dec}
 
% \noindent
% This section and the next give a precise description of the relative
% Picard functor for the universal proper, flat family of connected,
% at-worst-nodal, 
% arithmetic genus $0$ curves -- which is a $1$-morphism of Artin stacks
% rather than a morphism of varieties.  
% The description follows easily from the
% work of Raynaud, \cite{Raynaud} and \cite{Ner}, and is mostly just
% bookkeeping notation.
% But the precise
% description has not been written down.  And we need to
% introduce the notation anyway to prove
% the divisor class relations.  
% However, the reader who is familiar with
% divisor classes on Kontsevich moduli spaces 
% may safely skip to Section
% ~\ref{sec-Qpi}.  

% There exists an Artin stack parametrizing prestable curves whose dual
% graph is a given modular graph.  This section describes a variant
% Artin stack obtained by ``decorating'' the modular graph.  Although
% the variation is simple, it arises often enough to warrant a few
% words.  This variant is used in the next section to describe the
% closure of the identity section in the relative Picard functor for the
% universal family of prestable curves of compact type.
% The reference for this section is ~\cite{BM}.

% \mni
% A \emph{modular graph} is a (not necessarily connected) graph $\sigma$
% -- edges are undirected and \emph{tails} or half-edges are allowed --
% together with a \emph{genus function} $g:\text{Vertex}(\sigma)
% \rightarrow \ZZ_{\geq 0}$.  There are 2 collections of morphisms
% between modular graphs: \emph{contractions} are surjective on
% vertices, roughly contracting subgraphs of the domain to vertices of
% the target, and \emph{graph inclusions} are inclusions of
% subgraphs. (In ~\cite{BM}, the \emph{combinatorial morphisms} are
% obtained from graph inclusions by adjoining formal inverses of certain
% ``stabilizing'' contractions; stability is not an issue here, so graph
% inclusions are more appropriate).  Also, for every diagram,
% $$
% \begin{CD}
% & & \sigma_3 \\ 
% & & @VV a V \\
% \sigma_2 @> \phi >> \sigma_1
% \end{CD}
% $$
% of a contraction $\phi$ and a graph inclusion $a$, there is a
% \emph{pullback diagram},
% $$
% \begin{CD}
% \sigma_4 @> a^*\phi >> \sigma_3 \\
% @V \phi^* a VV @VV a V \\
% \sigma_2 @> \phi >> \sigma_1
% \end{CD}
% $$
% of a contraction $a^*\phi$ and a graph inclusion $\phi^*a$ such that
% the maps on vertices, $a\circ a^*\phi, \phi \circ
% \phi^*a:\text{Vertex}(\sigma_4) \rightarrow \text{Vertex}(\sigma_1)$
% are equal.  The diagram is unique up to a unique isomorphism (both as
% a contraction and a graph inclusion) of $\sigma_4$.  The category of
% modular graphs is denoted $\mf{G}$.

% \mni
% To each prestable curve there is an associated modular graph, and to
% each modular graph $\sigma$ there is an Artin stack $\mf{M}(\sigma)$
% parametrizing prestable curves along with a contraction of the
% associated modular graph to $\sigma$.  This defines a lax 2-functor
% from $\mf{G}$ to the 2-category of Artin stacks, covariant for
% contractions, contravariant for graph inclusions, and such that for
% every pullback diagram there is a 2-equivalence $\mf{M}(a)\circ
% \mf{M}(\phi) \Rightarrow \mf{M}(a^*\phi) \circ \mf{M}(\phi^*a)$.

% \begin{defn} \label{defn-decg}
% \marpar{defn-decg}
% A \emph{category of decorated modular graphs} is a category $\mf{H}$
% with 2 sets of morphisms -- $\mf{H}$-contractions and $\mf{H}$-graph
% inclusions -- together with a functor $p:\mf{H} \rightarrow \mf{G}$
% compatible with both contractions and graph inclusions satisfying the
% following axioms.
% \begin{enumerate}
% \item[(i)]
% For every $\mf{H}$-contraction $\phi:\tau_2 \rightarrow \tau_1$ and
% $\mf{H}$-graph inclusion $a:\tau_3 \rightarrow \tau_1$, there exists
% an object $\tau_4$, an $\mf{H}$-contraction $a^*\phi:\tau_4
% \rightarrow \tau_3$ and an $\mf{H}$-graph inclusion $\phi^*a:\tau_4
% \rightarrow \tau_2$ mapping under $p$ to a pullback diagram.  Moreover
% this is unique up to unique isomorphism of $\tau_4$. 
% \item[(ii)]
% For every object $\tau$ in $\mf{H}$ and every contraction $\phi:p(\tau)
% \rightarrow \sigma$ in $\mf{G}$, there is a $\mf{H}$-contraction $\psi$
% such that $p(\psi)=\phi$.  And $\psi$ is unique up to unique isomormphism.
% \end{enumerate}
% \end{defn}

% \begin{constr} \label{constr-semigp}
% \marpar{constr-semigp}
% Let $A$ be an Abelian semigroup.  Define $\mf{G}_A$ to be the category
% whose objects are pairs $(\sigma,\alpha)$ of a modular graph $\sigma$
% together with a function $\alpha:\text{Vertex}(\sigma) \rightarrow A$,
% where $\mf{G}_A$-contractions, $\phi:(\sigma_1,\alpha_1) \rightarrow
% (\sigma_2,\alpha_2)$, are contractions $\phi:\sigma_1 \rightarrow
% \sigma_2$ such that $\alpha_2(v) = \sum_{w\in \phi^{-1}(v)}
% \alpha_1(w)$ for every $v\in \text{Vertex}(\sigma_2)$, and where
% $\mf{G}_A$-graph inclusions, $a:(\sigma_1,\alpha_1) \rightarrow
% (\sigma_2,\alpha_2)$, are graph inclusions $a:\sigma_1 \rightarrow
% \sigma_2$ such that $\alpha_2(a(v)) = \alpha_1(v)$ for every vertex
% $v\in \text{Vertex}(\sigma_1)$.  Define $p:\mf{G}_A \rightarrow
% \mf{G}$ to be the obvious forgetful functor.  This is a category of
% decorated modular graphs; the only one used in the rest of this paper.
% \end{constr}

% \mni
% The aim of this section is to construct for every object $\tau$ of
% $\mf{H}$ an Artin stack $\mf{M}_{\mf{H}}(\tau)$ parametrizing
% prestable curves along with a lifting of the associated modular graph
% to an object of $\mf{H}$ contracting to $\tau$.  The association $\tau
% \mapsto \mf{M}_{\mf{H}}(\tau)$ should define a lax 2-functor from
% $\mf{H}$ to the 2-category of Artin stacks, covariant for
% $\mf{H}$-contractions, contravariant for $\mf{H}$-graph inclusions,
% and such that for every pullback diagram there is an associated
% 2-equivalence.

% \begin{defn} \label{defn-sat}
% \marpar{defn-sat}
% Let $\mf{H}$ be a category of decorated modular graphs, considered as
% a usual category whose morphisms are $\mf{H}$-contractions.  A
% subcategory $\mf{H}'$ is \emph{saturated} if $\mf{H}'$ contains every
% $\mf{H}$-contraction whose domain is in $\mf{H}'$.  A subcategory
% $\mf{H}'$ is \emph{$p$-embedding} if the functor of categories with
% contractions as morphisms, $p:\mf{H}' \rightarrow \mf{G}$, is an
% equivalence to a (necessarily full) subcategory of $\mf{G}$.

% \noindent
% Let $\tau$ be an object of $\mf{H}$ and denote by $\mf{H}_\tau$ the
% category whose objects are contractions $\phi:\tau_\phi \rightarrow
% \tau$ and whose morphisms are commutative diagrams of contractions.  A
% subcategory $\mf{H}'$ of $\mf{H}_\tau$ is \emph{saturated} if
% $\mf{H}'$ contains every morphism in $\mf{H}_\tau$ whose domain is in
% $\mf{H}'$.  A subcategory $\mf{H}'$ of $\mf{H}_\tau$ is
% \emph{$p$-embedding} if the functor $p:\mf{H}' \rightarrow
% \mf{G}_{p(\tau)}$ is an equivalence to a (necessarily full)
% subcategory of $\mf{G}_{p(\tau)}$.  Denote by
% $\text{Sat}(\mf{H}_\tau)$ the set of saturated, $p$-embedding
% subcategories of $\mf{H}_\tau$ directed by reverse inclusion of
% subcategories.
% \end{defn}

% \mni
% Let $\tau$ be an object of $\mf{H}$ and let $\mf{H}'$ be a saturated,
% $p$-embedding subcategory of $\mf{H}_\tau$.  Define
% $U_{\mf{H}'}(\tau)$ to be the open substack of $\mf{M}(p(\tau))$ whose
% complement is the union of the images of all 1-morphisms
% $\mf{M}(\phi):\mf{M}(\sigma) \rightarrow \mf{M}(p(\tau))$ such that
% $\phi$ is not in the image of $p:\mf{H}' \rightarrow
% \mf{G}_{p(\tau)}$.  It is straightforward that $U_{\mf{H}'}(\tau)$ is
% open: the intersection with every quasi-compact open substack of
% $\mf{M}(p(\tau))$ is open, and $U_{\mf{H}'}(\tau)$ is the union of
% these open sets.

% \mni
% Let $\mf{H}' \subset \mf{H}''$ be saturated, $p$-embedding
% subcategories of $\mf{H}_\tau$.  Then $U_{\mf{H}'}(\tau) \subset
% U_{\mf{H}''}(\tau)$ as subsets of $\mf{M}(p(\tau))$.  Therefore
% $\mf{H}' \mapsto U_{\mf{H}'}(\tau)$ is a directed system of open
% immersions of Artin stacks indexed by $\text{Sat}(\mf{H}_\tau)$.
% Because this is a directed system of open immersions, the direct limit
% is an Artin stack.

% \begin{notat} \label{notat-MHtau}
% \marpar{notat-MHtau}
% Denote by $\mf{M}_{\mf{H}}(\tau)$ the direct limit of the directed
% system $\mf{H}' \mapsto U_{\mf{H}'}(\tau)$.  Denote by
% $\mf{M}_p(\tau): \mf{M}_{\mf{H}}(\tau) \rightarrow \mf{M}(p(\tau))$
% the natural 1-morphism.  If $\mf{H}=\mf{G}_A$, also denote
% $\mf{M}_{\mf{H}}(\tau)$ by $\mf{M}_A(\tau)$.
% \end{notat}

% \mni
% The ``points'' of $\mf{M}_{\mf{H}}(\tau)$ have a simple description.

% \begin{defn} \label{defn-strict}
% \marpar{defn-strict}
% For every modular graph $\sigma$, define
% $\mf{M}^{\text{strict}}(\sigma)$ to be the open substack of
% $\mf{M}(\sigma)$ that is the complement of the images of all
% $\mf{M}(\phi)$ where $\phi:\sigma' \rightarrow \sigma$ is a
% non-invertible contraction.
% \end{defn}

% \begin{lem} \label{lem-pts}
% \marpar{lem-pts}
% Let $\tau$ be an object of $\mf{H}$ and let $\phi:\sigma \rightarrow
% p(\tau)$ be a contraction.  The 2-fibered product
% $\mf{M}^{\text{strict}}(\sigma) \times_{\mf{M}(p(\tau))}
% \mf{M}_{\mf{H}}(\tau)$ is equivalent to a disjoint union of copies of
% $\mf{M}^{\text{strict}}(\sigma)$ indexed by equivalence classes of
% contractions $\psi$ in $\mf{H}_\tau$ such that $p(\psi)=\phi$.
% \end{lem}

% \begin{proof}
% Let $(\eta,\zeta,\theta)$ be an object of the 2-fibered product, i.e.,
% a triple of an object of $\mf{M}^{\text{strict}}(\sigma)$, an object
% of $\mf{M}_{\mf{H}}(\tau)$ and an equivalence
% $\theta:\mf{M}(\phi)(\eta) \rightarrow \mf{M}_{p}(\tau)(\zeta)$.
% There is a saturated, $p$-embedding subcategory $\mf{H}'$ such that
% $\mf{M}_p(\tau)(\zeta)$ is in $U_{\mf{H}'}(\tau)$.  Because this is in
% the image of $\mf{M}(\phi)$, there is a contraction $\psi:\tau_\psi
% \rightarrow \tau$ in $\mf{H}'$ such that $\phi=p(\psi)$.  Because
% $\mf{H}'$ is $p$-embedding, $\psi$ is unique up to unique isomorphism.
% By the nature of the direct limit, $\psi$ is independent of the choice
% of $\mf{H}'$.

% \mni
% Conversely, given an object $\eta$ of $\mf{M}^{\text{strict}}(\sigma)$
% and a contraction $\psi:\tau_\psi \rightarrow \tau$ in $\mf{H}_\tau$
% such that $p(\psi) = \phi$, define $\mf{H}'$ to be the subcategory of
% $\mf{H}_\tau$ consisting of all contractions through which $\psi$
% factors.  By Definition~\ref{defn-decg}(ii), this is a saturated,
% $p$-embedding subcategory.  And $\mf{M}(\phi)(\eta)$ is in
% $U_{\mf{H}}(\tau)$.  The image in the direct limit is an object
% $\zeta$, and there is a canonical isomorphism $\theta:
% \mf{M}(\phi)(\eta) \rightarrow \mf{M}_p(\tau)(\zeta)$.  Thus
% $(\eta,\zeta,\theta)$ is an object of the 2-fibered product.

% \mni
% It is left to the reader to verify these operations give an
% equivalence of stacks.
% \end{proof}

% \mni
% \textbf{Note:}  
% The functorialities are only sketched.  Given an $\mf{H}$-contraction
% $\phi:\tau_1 \rightarrow \tau_2$, Definition~\ref{defn-decg}(ii) gives
% a map of directed sets $\text{Sat}(\phi):\text{Sat}(\mf{H}_{\tau_1})
% \rightarrow \text{Sat}(\mf{H}_{\tau_1})$, and composition with
% $\mf{M}(p(\phi))$ gives a compatible family of 1-morphisms of directed
% systems.  This defines the 1-morphism $\mf{M}_{\mf{H}}(\phi)$.  Given
% an $\mf{H}$-graph inclusion $a:\tau_1 \rightarrow \tau_2$, existence
% of pullback diagrams, Definition~\ref{defn-decg}(i), gives a map of
% directed sets $\text{Sat}(a):\text{Sat}(\mf{H}_{\tau_2}) \rightarrow
% \text{Sat}(\mf{H}_{\tau_1})$, and composition with $\mf{M}(p(a))$
% gives a compatible family of 1-morphisms of directed systems.  This
% defines the 1-morphism $\mf{M}_{\mf{H}}(a)$.  The rest of the
% compatibilities are straightforward.

% \section{The universal relative Picard for curves of compact type}
% \label{sec-cpct}
% \marpar{sec-cpct}

% \noindent
% The results in this section are well-known, and easily follow from
% work of Raynaud,
% ~\cite{Raynaud} and ~\cite{Ner}.  It is useful in the rest of the
% paper to gather the results here.

% \begin{notat} \label{notat-H}
% \marpar{notat-H}
% Denote by $\mf{H} \subset \mf{G}_\ZZ$ the full subcategory of objects
% $(\sigma,\alpha)$ such that $\sigma$ is a forest of trees, i.e., the
% graph has no cycles.  For each triple of integers $g,n \geq 0$ and
% $e$, denote by $\tau_{g,n}(e)$ the object of $\mf{H}$ consisting of a
% tree $\sigma_{g,n}$ with a single vertex of genus $g$ and $n$ flags,
% such that $\alpha(v)=n$.
% \end{notat}

% \mni
% Denote by $\pi:\mc{C} \rightarrow \mf{M}_{\mf{H}}(\tau_{g,0}(0))$ the
% pullback from $\mf{M}(\sigma_{g,0})$ of the universal curve.  For each
% 4-tuple $A=((g',g''),(e',e''))$ of integers $g',g''\geq 0$,
% $g'+g''=g$, and integers $e',e''$, $e'+e''=0$, denote by $\tau_A$ the
% tree with vertices $v',v''$ such that $g(v')=g', \alpha(v')=e'$ and
% $g(v'')=g'', \alpha(v'')=e''$.  Denote by $\phi:\tau_A \rightarrow
% \tau_{g,0}(0)$ the canonical contraction.  The 2-fibered product
% $\mf{M}_{\mf{H}}(\tau_A) \times_{\mf{M}_{\mf{H}}(\tau_{g,0}(0))}
% \mc{C}$ has 2 irreducible components $\mc{C}', \mc{C}''$ corresponding
% to the vertices $v', v''$.  There is a unique effective Cartier
% divisor $\mc{D} \subset \mc{C}$ such that for every
% $A=((g',g''),(e',e''))$,
% $$
% \mf{M}^{\text{strict}}_{\mf{H}}(\tau_A)
% \times_{\mf{M}_{\mf{H}}(\tau_{g,0}(0))} \mc{D}
% $$ 
% is empty if $e'=e''=0$ and is $e'\mc{C}''$ if
% $e'>0$.  

% \mni
% Let $U\subset \mf{M}(\sigma_{g,0})$ denote the open substack that is
% the image of $\mf{M}_p(\tau_{g,0}(0))$, i.e., $U$ is the Artin stack
% of $n$-pointed, genus $g$ curves of \emph{compact type}.  The
% 1-morphism $\pi:\mc{C}_U \rightarrow U$ is cohomologically flat, so by
% \cite[Thm. 7.3]{Artin} the relative Picard functor of the universal
% curve over $U$ is a 1-morphism $\text{pr}:\text{Pic}_{\mc{C}_U/U}
% \rightarrow U$ relatively representable by \emph{non-separated}
% algebraic spaces.  The closure $E_{\mc{C}_U/U}$ of the identity
% section gives a closed substack of $\text{Pic}_{\mc{C}_U/U}$ which is
% relatively representable over $U$ by non-separated group algebraic
% spaces, ~\cite[Prop. 5.2]{Raynaud}.  The quotient $Q_{\mc{C}_U/U}$ of
% $\text{Pic}_{\mc{C}_U/U}$ by $E_{\mc{C}_U/U}$ is a stack that is
% relatively representable over $U$ by a countable disjoint union of
% smooth, proper group algebraic spaces, \cite[Thm. 4.1.1]{Raynaud}
% (properness requires a bit more, see \cite[Ex. 8, p. 246]{Ner}).  The
% next lemma describes $E_{\mc{C}_U/U}$.

% \mni
% The invertible sheaf $\OO_{\mc{C}}(\mc{D})$ defines a
% 1-morphism $f:\mf{M}_{\mf{H}}(\tau_{g,0}(0)) \rightarrow
% \text{Pic}_{\mc{C}_U/U}$, and there is a natural 2-equivalence of
% $\text{pr}\circ f$ with $\mf{M}_{\mf{H}}(\tau_{g,0}(0))$.

% \begin{lem} \label{lem-subgp} 
% \marpar{lem-subgp}
% The 1-morphism $f$ defines an equivalence to $E_{\mc{C}_U/U}$, the
% closure of the 
% identity section of $\text{Pic}_{\mc{C}_U/U}$.  
% Denoting by
% $Q^0_{\mc{C}_U/U}$ the identity component of the quotient and by
% $\text{Pic}^0_{\mc{C}_U/U}$ the preimage, there are 1-morphisms
% $$
% \text{Pic}^0_{\mc{C}_U/U} \rightleftarrows
% \mf{M}_{\mf{H}}(\tau_{g,0}) \times
% Q^0_{\mc{C}_U/U}
% $$ 
% giving an equivalence of stacks
% over $U$, and splitting the extension of group algebraic spaces over
% $U$.
% \end{lem}  

% \begin{proof}
% It is easy to see $f$ is an equivalence to its image which is a
% subgroup of $E_{\mc{C}_U/U}$.  To prove the image of $f$ is all of
% $E_{\mc{C}_U/U}$, by the valuative criterion of closedness it suffices
% to check equality of pullbacks for every map of a DVR to
% $\mf{M}_{g,0}$ sending the generic point to
% $\mf{M}_{g,0}^{\text{strict}}$. By \cite[Prop. 6.1.3]{Raynaud}, the
% sections of $E_{\mc{C}_U/U}$ over a DVR are just the quotient of
% the free Abelian group on the irreducible components of the closed
% fiber by the subgroup generated by the entire fiber.  By
% Lemma~\ref{lem-pts} the same is true for the pullback of
% $\mf{M}_{\mf{H}}(\tau_{g,0}(0))$, and it is clear the map between them
% is an isomorphism.

% \mni
% The splitting of $\text{Pic}^0_{\mc{C}_U/U} \rightarrow
% Q^0_{\mc{C}_U/U}$ is given by the subfunctor of
% $\text{Pic}^0_{\mc{C}_U/U}$ of invertible sheaves whose degree on
% every irreducible component of every fiber is $0$, denoted by $P^0$ in
% \cite{Raynaud}.
% \end{proof}

% \mni
% In the special case that $g=0$, more is true.  First of all
% $\mf{M}_{\ZZ}(\tau) = \mf{M}_{\mf{H}}(\tau)$ for every $\tau$ of genus
% $0$.  Secondly  $U=\mf{M}_{0,n}$.  

% \begin{cor}[Raynaud, Prop. 9.3.1, ~\cite{Raynaud}] \label{cor-subgp}
% \marpar{cor-subgp}
% For $g=0$, $\text{Pic}^0_{\mc{C}/\mf{M}_{0,0}}$ is equivalent
% to $\mf{M}_{\ZZ}(\tau_{0,0}(0))$.
% \end{cor}

% \mni
% Moreover, the union $\cup_{e\in \ZZ} \mf{M}_{\ZZ}(\tau_{0,0}(e))$ is a
% group algebraic space over $\mf{M}_{0,0}$ containing
% $\mf{M}_{\ZZ}(\tau_{0,0}(0))$ as a subgroup algebraic space over
% $\mf{M}_{0,0}$.  Essentially, given a contraction
% $\phi:\sigma\rightarrow \sigma_{0,0}$ and given liftings
% $\psi_i:(\sigma,\alpha_i) \rightarrow \tau_{0,0}(e_i)$ for $i=1,2$,
% addition is determined by $\psi_1+\psi_2=\psi:(\sigma,
% \alpha_1+\alpha_2) \rightarrow \tau_{0,0}(e_1+e_2)$.  The \emph{total
% degree map} gives an isomorphism of $\text{Pic}/\text{Pic}^0$ with
% $\ZZ$.  The following result is easy.

% \begin{lem} \label{lem-g0}
% \marpar{lem-g0}
% For $g=0$ and for every $e$ there is an equivalence of stacks over
% $\mf{M}_{0,0}$, $\mf{M}_{\ZZ}(\tau_{0,0}(e)) \rightleftarrows
% \text{Pic}^e_{\mc{C}/\mf{M}_{0,0}}$ such that the equivalence
% $\cup_{e\in \ZZ} \mf{M}_{\ZZ}(\tau_{0,0}(e)) \rightleftarrows
% \text{Pic}_{\mc{C}/\mf{M}_{0,0}}$ is an equivalence of group algebraic
% spaces over $\mf{M}_{0,0}$ and is compatible with the equivalence in
% Corollary~\ref{cor-subgp}.
% \end{lem}

\section{Notation for moduli spaces and boundary divisor classes}
\label{subsec-bound}
\marpar{subsec-bound}

\mni
Denote by $\mf{M}_{0,0}$ the category whose objects are proper, flat
families $\pi:C\rightarrow M$ of connected, at-worst-nodal, arithmetic
genus $0$ curves, and whose morphisms are Cartesian diagrams of such
families.  This category is a smooth Artin stack over $\SP \ZZ$ (with
the flat topology),
cf. ~\cite{B}.  The subcategory $\Delta$ parametrizing families with
reducible fibers is a closed substack.  With respect to the smooth
topology, the pair
$(\mf{M}_{0,0},\Delta)$ is locally isomorphic to an irreducible  
simple normal crossings divisor in a smooth variety.  Denote by
$\Delta'$ the singular locus in $\Delta$.

\mni
Let $U_1$ be $\mf{M}_{0,0}-\Delta'$ and let $U_2$ be
$\mf{M}_{0,0}-\Delta$.  There are smooth atlases for each of these
stacks as follows.  For $U_2$, the family $\pi:\PP^1_\ZZ \rightarrow \SP
\ZZ$ defines a $1$-morphism $\zeta_2: \SP \ZZ \rightarrow U_2$.  This
is a smooth surjective morphism.  The $2$-fibered product
$$
\SP \ZZ \times_{\zeta_2,U_2,\zeta_2} \SP \ZZ = \text{Aut}(\PP^1_\ZZ)
$$
is the group scheme $\mathbf{PGL}_{2}$.  Thus $U_2$ is isomorphic
to the quotient 
stack $[\SP \ZZ/\mathbf{PGL}_{2}]$.  

\mni
There is a similar atlas for $U_2$.  
Let $V = \ZZ\{\mb{e}_0,\mb{e}_1\}$ be a free module of
rank $2$.  Choose dual coordinates $y_0,y_1$ for $V^\vee$.  Let
$\PP^1_\ZZ = \PP(V)$ be the projective space with homogeneous
coordinates $y_0, y_1$.  Let $\AAA^1_\ZZ$ be the affine space with
coordinate $x$.  Denote by $Z\subset \AAA^1_\ZZ \times \PP^1_\ZZ$ the
closed subscheme $\mathbb{V}(x,y_1)$, i.e., the image of the section
$(0,[1,0])$.  Let $\nu:C\rightarrow \AAA^1_\ZZ \times \PP^1_\ZZ$ be
the blowing-up along $Z$.  Denote by $E\subset C$ the exceptional
divisor.
Define $\pi:C\rightarrow \AAA^1_\ZZ$ to be $\text{pr}_{\AA^1}\circ
\nu$.  This family defines a $1$-morphism
$\zeta_1:\AAA^1_\ZZ \rightarrow U_1$. 
This is a smooth surjective morphism.  The $2$-fibered product
$$
\AAA^1_\ZZ \times_{\zeta_1,U_1,\zeta_1} \AAA^1_\ZZ =
\text{Isom}_{\AAA^2_\ZZ}(\text{pr}_1^*C, \text{pr}_2^*C)
$$
is equivalent to a group scheme $G$ over $\AAA^1_\ZZ$ whose
restriction to $U =\AAA^1_\ZZ - \{0\}$ is $U \times \mathbf{G}_m \times
\mathbf{PGL}_2$ and whose restriction to $\{0\} \cong \SP \ZZ$ is the
wreath product $G_0: =(B\times B)\rtimes \mf{S}_2$. Here $B\subset
\mathbf{PGL}_2$ is a Borel subgroup, i.e., the stabilizer of a point
in $\PP^1$, and $\mf{S}_2$ acts by interchanging the two components of
$C_0$.  Altogether $U_1$ is isomorphic to the quotient stack
$[\AAA^1_\ZZ/G]$.  

\mni
Denote by $\pi:\mf{C}\rightarrow \mf{M}_{0,0}$ the
universal family.  This is a proper, flat 1-morphism of Artin stacks,
representable by algebraic spaces.  
Denote by $\text{Pic}_\pi$ the stack parametrizing proper, flat
families of connected, at-worst-nodal, arithmetic genus $0$ curves
together with a section of the relative Picard functor of the family.
This is also an Artin stack, and the natural 1-morphism
$\text{Pic}_\pi \rightarrow \mf{M}_{0,0}$ is representable by (highly
nonseparated) \'etale group schemes, cf. \cite[Prop. 9.3.1]{Raynaud}. 

\mni
In fact $\text{Pic}_\pi$ has a covering by open substacks, each of
which maps isomorphically to an open substack in $\mf{M}_{0,0}$.  The
combinatorics of how these open pieces are glued to form
$\text{Pic}_\pi$ is straightforward.  In particular, the inverse image
in $\text{Pic}_\pi$ of $U_2$ is canonically isomorphic
to $U_2\times \ZZ$.  

\mni
The intersection $U_1\cap \Delta$ parametrizes nodal, arithmetic genus $0$
curves with two components, $C=C'\cup C''$.  
For every pair of integers
$e',e''$ there is a unique invertible sheaf $L$ on $C$ whose
restriction to $C'$, resp. $C''$, has degree $e'$, resp. $e''$.
The rule
$[C]\mapsto [(C,L)]$ defines 
a 1-morphism $U_1\cap \Delta \rightarrow
\text{Pic}_\pi$ representable by open immersions.
Denote by $\Delta_{(e',e'')}$ the
closure in $\text{Pic}_\pi$ of the image of this 1-morphism.  
This is a Cartier divisor in
$\text{Pic}_\pi$.  The union over all $(e',e'')$ of the Cartier
divisors $\Delta_{(e',e'')}$ is
precisely the inverse image in $\text{Pic}_\pi$ of $\Delta$.  

\mni
More generally, for every integer $r\geq 0$, denote by
$\Pi^r(\text{Pic}_\pi)$ the $r$-fold $2$-fibered product of $\text{Pic}_\pi$
with itself over $\mf{M}_{0,0}$.  Equivalently, $\Pi^r(\text{Pic}_\pi)$ is
the stack of families of genus $0$ curves together with $r$ sections
of the relative Picard functor of the family.  Let
$(e'_1,e''_1,\dots,e'_r,e''_r)$ be an ordered $2r$-tuple of integers.
Define $\Delta_{(e'_1,e''_1,\dots,e'_r,e''_r)}$ to be the iterated $2$-fiber
product
$$
\Delta_{(e'_1,e''_1,\dots,e'_r,e''_r)} :=
\Delta_{(e'_1,e''_1)}\times_\Delta \dots \times_\Delta
\Delta_{(e'_r,e''_r)}.
$$
This is a Cartier divisor in $\Pi^r(\text{Pic}_\pi)$.  

% Let $r\geq 0$ be an integer, and let $e_1,\dots,e_r$ be integers.
% Denote by $\mf{M}_{\ZZ}(\tau_{0,0}(e_1,\dots,e_r))$ the 2-fibered
% product,
% $$
% \mf{M}_{\ZZ}(\tau_{0,0}(e_1)) \times_{\mf{M}_{0,0}}
% \mf{M}_{\ZZ}(\tau_{0,0}(e_2)) \times_{\mf{M}_{0,0}} \dots
% \times_{\mf{M}_{0,0}} \mf{M}_{\ZZ}(\tau_{0,0}(e_r)).
% $$
% For each $i=1,\dots,r$, let $(e'_i,e''_i)$ be a pair of integers such
% that $e'_i + e''_i = e_i$.  Let $\sigma$ be the modular graph with two
% vertices $v',v''$ with $g(v')=g(v'')=0$, one edge connecting $v',
% v''$, and no tails.  Let $\phi:\sigma \rightarrow \sigma_{0,0}$ be the
% canonical contraction.  Denote by
% $\zeta:\mf{M}^{\text{strict}}(\sigma) \rightarrow
% \mf{M}_{\ZZ}(\tau_{g,0}(e_1,\dots,e_r))$ the 1-morphism whose
% projection to the $i^{\text{th}}$ factor is determined via
% Lemma~\ref{lem-pts} by the lifting $\psi_i:(\sigma,\alpha_i)
% \rightarrow \tau_{g,0}(e_i)$ of $\phi$ such that $\alpha_i(v') = e'_i,
% \alpha_i(v'')=e_i''$.  Define $\Delta_{(e'_1,e''_1,\dots,e'_r,e''_r)}$
% to be the effective Cartier divisor on
% $\mf{M}_{\ZZ}(\tau_{0,0}(e_1,\dots,e_r))$ that is the closure of the
% image of $\zeta$.

\mni
Let $\pi:C\rightarrow M$ be a flat 1-morphism representable
by proper algebraic spaces whose geometric fibers are connected,
at-worst-nodal curves of arithmetic genus $0$.  This defines a
1-morphism $\xi_0:M \rightarrow \mf{M}_{0,0}$.  
Let $D_1,\dots,D_r$ be
Cartier divisor classes on $C$.  This defines a 1-morphism
$\xi:M\rightarrow \Pi^r(\text{Pic}_\pi)$ lifting $\xi_0$.  
Let $f(e'_1,e''_1,\dots,e'_r,e''_r)$ be a function on $\ZZ^{2r}$ with
values in $\ZZ$, resp. $\QQ$, etc.  

\begin{notat} \label{notat-boundary}
\marpar{notat-boundary}
Denote by
$$
\sum_{(\beta',\beta'')} f(\langle D_1,\beta' \rangle, \langle
D_1, \beta'' \rangle, \dots, \langle D_r, \beta' \rangle, \langle D_r,
\beta'' \rangle) \Delta_{\beta',\beta''}
$$
the Cartier divisor class, resp. $\QQ$-Cartier divisor class, etc., 
that is the pullback by $\xi$ of the Cartier divisor class, etc.,
$$
\sum_{(e'_1,e''_1,\dots,e'_r,e''_r)} f(e'_1,e''_1,\dots,e'_r,e''_r)
\Delta_{(e'_1,e''_1,\dots,e'_r,e''_r)},
$$
the summation over all sequences $(e'_1,e''_1,\dots,e'_r,e''_r)$.  
If $f(e'_1,e''_1,\dots,e'_r,e''_r) = f(e''_1,e'_1,\dots,e''_r,e'_r)$,
denote by,
$$
{\sum_{(\beta',\beta'')}}^\prime f(\langle D_1,\beta' \rangle, \langle
D_1, \beta'' \rangle, \dots, \langle D_r, \beta' \rangle, \langle D_r,
\beta'' \rangle) \Delta_{\beta',\beta''}
$$
the pullback by $\xi$ of,
$$
{\sum_{(e'_1,e''_1,\dots,e'_r,e''_r)}}^\prime
f(e'_1,e''_1,\dots,e'_r,e''_r)
\Delta_{(e'_1,e''_1,\dots,e'_r,e''_r)},
$$
where the summation is over equivalence classes of sequences
$(e'_1,e''_1,\dots,e'_r,e''_r)$ such that $e'_i+e''_i=e_i$ 
under the equivalence relation
$(e'_1,e''_1,\dots,e'_r,e''_r) \sim (e''_1,e'_1,\dots,e''_r,e'_r)$.  
\end{notat}

\mni
Of course these are all infinite sums, and so appear not to give
well-defined Cartier divisors.  
However, every
quasi-compact open subset of $\Pi^r(\text{Pic}_\pi)$ intersects only
finitely many of the divisors.  On this subset, this is a
finite sum and thus well-defined.
Since $\Pi^r(\text{Pic}_\pi)$ has a covering by
quasi-compact open subsets, the Cartier divisor is defined by gluing.  

\begin{ex} \label{ex-boundary}
\marpar{ex-boundary}
Let $n\geq 0$ be an integer and let $(A,B)$ be a partition of
$\{1,\dots,n\}$.  For the universal family over $\mf{M}_{0,n}$, denote
by $s_1,\dots,s_n$ the universal sections.  Then
$$
\sum_{\beta',\beta''} \prod_{i\in A} \langle \text{Image}(s_i),\beta' \rangle
\cdot \prod_{j\in B} \langle \text{Image}(s_j), \beta'' \rangle
\Delta_{\beta',\beta''} 
$$
is the Cartier divisor class of the boundary divisor $\Delta_{(A,B)}$.  
\end{ex}


\section{The functor $Q_\pi$} \label{sec-Qpi}
\marpar{sec-Qpi}

\noindent
Let $M$ be an Artin stack, and let $\pi:C \rightarrow M$ be a flat
1-morphism, relatively representable by proper algebraic spaces whose
geometric fibers are connected, at-worst-nodal curves of arithmetic
genus $0$.  There exists an invertible dualizing sheaf $\omega_\pi$,
and the relative trace map, $\text{Tr}_\pi: R\pi_* \omega_\pi[1]
\rightarrow \OO_M$, is a quasi-isomorphism.  In particular,
$\text{Ext}^1_{\OO_C}(\omega_\pi,\OO_C)$ is canonically isomorphic to
$H^0(M,\OO_M)$.  Therefore $1\in H^0(M,\OO_M)$ determines an extension
class, i.e., a short exact sequence,
$$
\begin{CD}
0 @>>> \omega_\pi @>>> E_\pi @>>> \OO_C @>>> 0.
\end{CD}
$$
The morphism $\pi$ is perfect, so for every complex $F^\bullet$ perfect
of bounded amplitude on $C$, $R\pi_* F^\bullet$ is a perfect complex of
bounded amplitude on $M$.  By ~\cite{detdiv}, the
determinant of a perfect complex of bounded amplitude is defined.

\begin{defn} \label{defn-E}
\marpar{defn-E}
For every complex $F^\bullet$ perfect of bounded amplitude on $C$, 
define $Q_\pi(F^\bullet) = \text{det}(R\pi_* E_\pi\otimes F^\bullet)$.
\end{defn}

\mni
There is another interpretation of $Q_\pi(F^\bullet)$.

\begin{lem} \label{lem-interp}
\marpar{lem-interp}
For every complex $F^\bullet$ perfect of bounded amplitude on $C$, 
$$
Q_\pi(F^\bullet) \cong
\text{det}(R\pi_*(F^\bullet)) \otimes 
\text{det}(R\pi_*((F^\bullet)^\vee))^\vee.
$$
\end{lem}

\begin{proof}
By the short exact sequence for $E_\pi$, $Q_\pi(F^\bullet) \cong
\text{det}(R\pi_*(F^\bullet)) \otimes \text{det}(R\pi_*(\omega_\pi
\otimes F^\bullet))$.  The lemma follows by duality.
\end{proof}

\mni
It is straightforward to compute $F^\bullet$ whenever there exist
cycle class groups for $C$ and $M$ such that Chern classes are defined
for all perfect complexes of bounded amplitude and such that
Grothendieck-Riemann-Roch holds for $\pi$.

\begin{lem} \label{lem-GRR}
\marpar{lem-GRR}
If there exist cycle class groups for $C$ and $M$ such that Chern
classes exist for all perfect complexes of bounded amplitude and such
that Grothendieck-Riemann-Roch holds for $\pi$, then modulo $2$-power
torsion, the first Chern class of $Q_\pi(F^\bullet)$ is
$\pi_*(C_1(F^\bullet)^2 - 2C_2(F^\bullet))$.
\end{lem}

\begin{proof}
Denote the Todd class of $\pi$ by $\tau = 1 + \tau_1 + \tau_2 +
\dots$.  Of course $\tau_1 = -C_1(\omega_\pi)$.  By GRR,
$\text{ch}(R\pi_* \OO_C) = \pi_*(\tau)$.  The canonical map $\OO_M
\rightarrow R\pi_*\OO_C$ is a quasi-isomorphism.  Therefore
$\pi_*(\tau_2)=0$, modulo $2$-power torsion.  By additivity of the
Chern character, $\text{ch}(E_\pi) = 2 + C_1(\omega_\pi) +
\frac{1}{2}C_1(\omega_\pi)^2 + \dots$.  Therefore,
$$
\text{ch}(E_\pi)\cdot \tau = 2 + 2\tau_2 + \dots
$$
So for any complex $F^\bullet$ perfect of bounded amplitude,
$$
\begin{array}{c}
\text{ch}(E_\pi\otimes F^\bullet)\cdot \tau = \text{ch}(F^\bullet) \cdot
\text{ch}(E_\pi) \cdot \tau = \\
(\text{rk}(F^\bullet) + C_1(F^\bullet) +
\frac{1}{2}(C_1(F^\bullet)^2 - 2C_2(F^\bullet)) +\dots)(2 + 2\tau_2+
\dots). 
\end{array}
$$
Applying $\pi_*$ gives,
$$
2\pi_*(C_1(F^\bullet)) + \pi_*(C_1(F^\bullet)^2 - 2C_2(F^\bullet)) + \dots
$$
Therefore the first Chern class of $\text{det}(R\pi_*(E_\pi\otimes
F^\bullet))$ is $\pi_*(C_1(F^\bullet)^2 - 2C_2(F^\bullet))$, modulo
$2$-power torsion.
\end{proof}

\begin{rmk} \label{rmk-Q}
The point is this.  In every reasonable case, $Q_\pi$ is just
$\pi_*(C_1^2-2C_2)$.  Moreover $Q_\pi$ is compatible with base-change
by arbitrary 1-morphisms.  This allows to reduce certain computations
to the Artin stack of all genus $0$ curves.  As far as we are aware,
no one has written a definition of cycle class groups for all locally
finitely presented Artin stacks that has Chern classes for all perfect
complexes of bounded amplitude, has pushforward maps and
Grothendieck-Riemann-Roch for perfect 1-morphisms representable by
proper algebraic spaces, and has pullback maps by arbitrary
1-morphisms for cycles coming from Chern classes.  Doubtless such a
theory exists; whatever it is, $Q_\pi = \pi_*(C_1^2-2C_2)$.
\end{rmk}

\mni
Let the following diagram be 2-Cartesian,
$$
\begin{CD}
C' @> \zeta_C >> C \\
@V \pi' VV @VV \pi V \\
M' @> \zeta_M >> M
\end{CD}
$$
together with a 2-equivalence $\theta:\pi\circ \zeta_C \Rightarrow
\zeta_M \circ \pi'$.  

\begin{lem} \label{lem-pullback}
\marpar{lem-pullback}
For every complex $F^\bullet$ perfect of bounded amplitude on $C$, 
$\zeta_M^* Q_\pi(F^\bullet)$ is isomorphic to $Q_{\pi'}(\zeta_C^*
F^\bullet)$.
\end{lem}

\begin{proof}
Of course $\zeta_C^* E_\pi = E_{\pi'}$.  And $\zeta_M^* R\pi_*$ is
canonically equivalent to $R(\pi')_* \zeta_C^*$ for perfect
complexes of bounded amplitude.  Therefore $\zeta_M^*
Q_\pi(F^\bullet)$ equals $\text{det}(\zeta_M^* R\pi_*(E_\pi\otimes
F^\bullet))$ equals $\text{det}(R(\pi')_* \zeta_C^*(E_\pi \otimes
F^\bullet))$ equals $\text{det}(R(\pi')_* E_{\pi'}\otimes \zeta_C^*
F^\bullet)$ equals $Q_{\pi'}(\zeta_C^* F^\bullet)$.
\end{proof}

\begin{lem} \label{lem-inv}
\marpar{lem-inv}
Let $L$ be an invertible sheaf on $C$ of relative degree $e$ over
$M$.  For every invertible sheaf $L'$ on $M$, $Q_\pi(L\otimes \pi^*L')
\cong Q_\pi(L)\otimes (L')^{2e}$.  In particular, if $e=0$,
$Q_\pi(L\otimes \pi^* L') \cong Q_\pi(L)$.
\end{lem}

\begin{proof}
To compute the rank of $R\pi_*(E_\pi\otimes F^\bullet)$ over any
connected component of $M$, it suffices to base-change to the spectrum
of a field mapping to that component.  Then, by
Grothendieck-Riemann-Roch, the rank is $2\text{deg}(C_1(F^\bullet))$.
In particular, $R\pi_*(E_\pi \otimes L)$ has rank $2e$.  

\mni
By the projection formula, $R\pi_*(E_\pi\otimes L\otimes \pi^* L')
\cong R\pi_*(E_\pi \otimes L)\otimes L'$.  Of course
$\text{det}(R\pi_*(E_\pi \otimes L)\otimes L') = Q_\pi(L)\otimes
(L')^\text{rank}$.  This follows from the uniqueness of $\text{det}$:
for any invertible sheaf $L'$ the association $F^\bullet \mapsto
\text{det}(F^\bullet \otimes L')\otimes
(L')^{-\text{rank}(F^\bullet)}$ also satisfies the axioms for a
determinant function and is hence canonically isomorphic to
$\text{det}(F^\bullet)$.  Therefore $Q_\pi(L\otimes \pi^*L') =
Q_\pi(L)\otimes (L')^{2e}$.
\end{proof}

\section{Local computations} \label{sec-local}
\marpar{sec-local}

\noindent
This section contains 2 computations: $Q_\pi(\omega_\pi)$ and
$Q_\pi(L)$ for every invertible sheaf on $C$ of relative degree $0$.
Because of Lemma~\ref{lem-pullback} the first computation reduces to
the universal case over $\mf{M}_{0,0}$.  Because of
Lemma~\ref{lem-pullback} and Lemma~\ref{lem-inv}, the second
compuation reduces to the universal case over $\text{Pic}^0_\pi$.  

\subsection{Computation of $Q_\pi(\omega_\pi)$} \label{subsec-Qomega}
\marpar{subsec-Qomega}

\mni
Associated to $\pi_C:C\rightarrow M$, there is a 1-morphism $\zeta_M:M
\rightarrow \mf{M}_{0,0}$, a 1-morphism $\zeta_C: C \rightarrow
\mc{C}$, and a 2-equivalence $\theta:\pi_{\mc{C}} \circ \zeta_C
\Rightarrow \zeta_M\circ \pi_C$ such that the following diagram is
2-Cartesian,
$$
\begin{CD}
C @> \zeta_C >> \mc{C} \\
@V \pi_C VV @VV \pi_{\mc{C}} V \\
M @> \zeta_M >> \mf{M}_{0,0}
\end{CD}
$$
Of course $\omega_{\pi_C}$ is isomorphic to $\zeta_C^*
\omega_{\pi_\mc{C}}$.  By Lemma~\ref{lem-pullback},
$Q_{\pi_C}(\omega_{\pi_C}) \cong \zeta_M^*
Q_{\pi_{\mc{C}}}(\omega_{\pi_{\mc{C}}})$.  So the computation of
$Q_{\pi_C}(\omega_{\pi_C})$ is reduced to the universal family
$\pi:\mf{C} \rightarrow \mf{M}_{0,0}$.

% \mni
% Let $U_1$ be $\mf{M}_{0,0}-\Delta'$ and let $U_2$ be
% $\mf{M}_{0,0}-\Delta$.  
% Let the open substack $U_1\subset \mf{M}_{0,0}$ be the complement of
% the union of the images of $\mf{M}(\phi):\mf{M}(\sigma) \rightarrow
% \mf{M}_{0,0}$ as $\phi:\sigma \rightarrow \sigma_{0,0}$ ranges over
% all contractions such that $\#\text{Vertex}(\sigma) \geq 3$.  Let $U_2
% \subset U_1$ be the open substack
% $\mf{M}^{\text{strict}}(\sigma_{0,0})$.

\begin{prop} \label{prop-Qomega}
\marpar{prop-Qomega}
\begin{enumerate}
\item[(i)]
Over the open substack $U_1$, $\omega_\pi^\vee$ is $\pi$-relatively
ample.  
\item[(ii)]
Over $U_1$,
$R^1\pi_*\omega_\pi^\vee|_{U_1} = (0)$ and $\pi_*
\omega_\pi^\vee|_{U_1}$ is locally free of rank 3.  
\item[(iii)]
Over $U_2$, there is
a canonical isomorphism $i:\text{det}(\pi_* \omega_\pi^\vee|_{U_2})
\rightarrow \OO_{U_2}$. 
\item[(iv)]
The image of $\text{det}(\pi_*\omega_\pi^\vee|_{U_1}) \rightarrow
\text{det}(\pi_*\omega_\pi^\vee|_{U_2}) \xrightarrow{i} \OO_{U_2}$ is
$\OO_{U_1}(-\Delta) \subset \OO_{U_2}$.
\item[(v)]
Over $U_1$, $Q_\pi(\omega_\pi)|_{U_1} \cong \OO_{U_1}(-\Delta)$.
Therefore on all of $\mf{M}_{0,0}$, $Q_\pi(\omega_\pi) \cong
\OO_{\mf{M}_{0,0}}(-\Delta)$.  
\end{enumerate}
\end{prop}

\begin{proof}
Recall the $1$-morphism $\zeta_1:\AAA^1_\ZZ
\rightarrow U_1$ from 
Section ~\ref{subsec-bound}.
Because $\zeta_1$ is smooth and surjective,  
(i) and (ii) can be
checked after base-change by $\zeta_1$.  Also (iv) will reduce to a
computation over $\AAA^1_\ZZ$ after base-change by $\zeta_1$.

\mni
\textbf{(i) and (ii):} Denote by $\PP^2_\ZZ$ the projective space with
coordinates $u_0,u_1,u_2$.  There is a rational transformation
$f:\AAA^1_\ZZ \times \PP^1_\ZZ \dashrightarrow \AAA^1_\ZZ \times
\PP^2_\ZZ$ by
$$
\begin{array}{ccc}
f^*x & = & x, \\
f^*u_0 & = & xy_0^2, \\
f^*u_1 & = & y_0y_1, \\
f^*u_2 & = & y_1^2
\end{array}
$$
By local computation, this extends to a morphism $f:C\rightarrow
\AAA^1_\ZZ \times \PP^2_\ZZ$ that is a closed immersion and whose
image is $\mathbb{V}(u_0u_2-xu_1^2)$.  By the adjunction formula,
$\omega_\pi$ is the pullback of $\OO_{\PP^2}(-1)$.  In particular,
$\omega_\pi^\vee$ is very ample. Moreover, because
$H^1(\PP^2_\ZZ,\OO_{\PP^2}(-1)) = (0)$, also $H^1(C,\omega_\pi^\vee) =
(0)$.  By cohomology and base-change results,
$R^1\pi_*(\omega_\pi^\vee) = (0)$ and $\pi_*(\omega_\pi^\vee)$ is
locally free of rank 3.  

\mni
\textbf{(iii):} The curve $\PP^1_\ZZ = \PP(V)$ determines a morphism
$\eta: \SP(\ZZ) \rightarrow U_2$.  This is smooth and surjective on
geometric points.  Moreover it gives a realization of $U_2$ as the
classifying stack of the group scheme $\text{Aut}(\PP(V)) =
\textbf{PGL}(V)$.  Taking the exterior power of the Euler exact
sequence, $\omega_{\PP(V)/\ZZ} = \bigwedge^2(V^\vee)\otimes
\OO_{\PP(V)}(-2)$.  Therefore $H^0(\PP(V),\omega_{\PP(V)/\ZZ}^\vee)$
equals $\bigwedge^2(V)\otimes \text{Sym}^2(V^\vee)$ as a
representation of $\textbf{GL}(V)$.  The determinant of this
representation is the trivial character of $\textbf{GL}(V)$.
Therefore it is the trivial character of $\textbf{PGL}(V)$.  This
gives an isomorphism of $\text{det}(\pi_*\omega_\pi|_{U_2})$ with
$\OO_{U_2}$.

\mni
\textbf{(iv):} This can be checked after pulling back by $\zeta_1$.  The
pullback of $U_2$ is $\mathbb{G}_{m,\ZZ} \subset \AAA^1_\ZZ$.  The
pullback of $i$ comes from the determinant of
$H^0(\mathbb{G}_{m,\ZZ}\times \PP^1_\ZZ,\omega_\pi^\vee) =
\bigwedge^2(V) \otimes \text{Sym}^2(V^\vee) \otimes
\OO_{\mathbb{G}_m}$.  By the adjunction formula, $\omega_{C/\AAA^1} =
\nu^*\omega_{\AAA^1\times \PP^1/\AAA^1}(E)$.  Hence
$\nu_*\omega_{C/\AAA^1}^\vee = I_Z \omega_{\AAA^1\times
\PP^1/\AAA^1}$.  Therefore the canonical map,
$$
H^0(C,\omega_{C/\AAA^1}^\vee) \rightarrow H^0(\AAA^1_\ZZ \times
\PP^1_\ZZ, \omega^\vee_{\AAA^1\times \PP^1/\AAA^1}),
$$ 
is given by,
$$
\begin{array}{l}
\OO_{\AAA^1}\{\mb{f}_0,\mb{f}_1,\mb{f}_2\} \rightarrow
\bigwedge^2(V)\otimes \text{Sym}^2(V^\vee) \otimes \OO_{\AAA^1}, \\
\mb{f}_0 \mapsto x\cdot (\mb{e}_0\wedge\mb{e}_1)\otimes y_0^2, \\
\mb{f}_1 \mapsto  \ \ \ \     (\mb{e}_0\wedge \mb{e}_1) \otimes y_0y_1, \\
\mb{f}_2 \mapsto  \ \ \ \      (\mb{e}_0\wedge \mb{e}_1) \otimes y_1^2
\end{array}
$$
It follows that $\text{det}(\pi_*\omega_\pi^\vee) \rightarrow
\OO_{\mathbb{G}_m}$ has image $\langle x \rangle \OO_{\AAA^1}$, i.e.,
$\zeta_1^* \OO_{U_1}(-\Delta)$.

\mni
\textbf{(v):} By the short exact sequence for $E_\pi$,
$Q_\pi(\omega_\pi) = \text{det}(R\pi_*\omega_\pi)\otimes
\text{deg}(R\pi_* \omega_\pi^2)$.  Because the trace map is a
quasi-isomorphism, $\text{det}(R\pi_* \omega_\pi) = \OO_{U_1}$.  By
(ii) and duality, $$ \text{det}(R\pi_* \omega_\pi^2) \cong
\text{det}(R^1\pi_*\omega_\pi^2)^vee \cong \text{det}(\pi_*
\omega_\pi^\vee).
$$  
By (iv), this is $\OO_{U_1}(-\Delta)$.  Therefore $Q_\pi(\omega_\pi)
\cong \OO_{U_1}(-\Delta)$ on $U_1$.  Because $\mf{M}_{0,0}$ is
regular, and because the complement of $U_1$ has codimension $2$, this
isomorphism of invertible sheaves extends to all of $\mf{M}_{0,0}$.
\end{proof}

\mni
The sheaf of relative differentials $\Omega_\pi$ is a pure coherent
sheaf on $\mc{C}$ of rank $1$, flat over $\mf{M}_{0,0}$ and is
quasi-isomorphic to a perfect complex of amplitude $[-1,0]$.  

\begin{lem} \label{lem-Omega}
\marpar{lem-Omega}
The perfect complex $R\pi_*\Omega_\pi$ has rank $-1$ and determinant
$\cong \OO_{\mf{M}_{0,0}}(-\Delta)$.  The perfect complex $R\pi_*
R\textit{Hom}_{\OO_{\mc{C}}}(\Omega_\pi,\OO_{\mc{C}})$ has rank $3$
and determinant $\cong \OO_{\mf{M}_{0,0}}(-2\Delta)$.
\end{lem}

\begin{proof}
There is a canonical injective sheaf homomorphism $\Omega_\pi
\rightarrow \omega_\pi$ and the support of the cokernel, $Z\subset
\mc{C}$, is a closed substack that is smooth and such that
$\pi:Z\rightarrow \mf{M}_{0,0}$ is unramified and is the normalization
of $\Delta$.  Over $U_1$, the lemma immediately follows from this and
the arguments in the proof of Proposition~\ref{prop-Qomega}.  As in
that case, it suffices to establish the lemma over $U_1$.
\end{proof}

\subsection{Computation of $Q_\pi(L)$ for invertible sheaves of degree
  $0$} 
\label{subsec-QL}
\marpar{subsec-QL}

\noindent
Let $M$ be an Artin stack, let $\pi:C\rightarrow M$ be a flat
1-morphism, relatively representable by proper algebraic spaces whose
geometric fibers are connected, at-worst-nodal curves of arithmetic
genus $0$.  Let $L$ be an invertible sheaf on $C$ of relative degree
$0$ over $M$.  This determines a $1$-morphism to the relative Picard of
the universal family over $\mf{M}_{0,0}$,
$$
\zeta_M:M \rightarrow \text{Pic}^0_\pi.
$$ 
The pullback of the universal family $\mf{C}$ is
equivalent to $C$ and the pullback of the universal bundle
$\OO_{\mf{C}}(\mc{D})$ differs from $L$ by $\pi^*L'$ for an invertible
sheaf $L'$ on $M$.  By Lemma~\ref{lem-pullback} and
Lemma~\ref{lem-inv}, $Q_\pi(L) \cong \zeta_M^*
Q_\pi(\OO_{\mf{C}}(\mc{D}))$.

\begin{prop} \label{prop-lcomp}
\marpar{prop-lcomp}
Over $\text{Pic}^0_\pi$, $\pi_*E_\pi(\mc{D}) = (0)$ and 
$R^1\pi_*
E_\pi(\mc{D})$ is a sheaf supported on the inverse image of
$\Delta$.  The stalk of
$R^1\pi_* E_\pi(\mc{D})$ at the generic point of $\Delta_{(a,-a)}$ is a
torsion sheaf of length $a^2$.  The filtration by order of vanishing
at the generic point has associated graded pieces of length $2a-1,
2a-3, \dots, 3,1$.  
\end{prop}

\begin{proof}
Over the open complement of $\Delta$, the divisor $\mc{D}$ is $0$.  So
the first part of the proposition reduces to the statement that
$R\pi_* E_\pi$ 
is quasi-isomorphic to $0$.  By definition of $E_\pi$, there is an
exact triangle,
$$
\begin{CD}
R\pi_* E_\pi @>>> R\pi_* \OO_{\mc{C}} @> \delta >> R\pi_* \omega_\pi
[1] @>>> 
R\pi_* E_\pi[1].
\end{CD}
$$
Of course the bundle $E_\pi$ and the canonical isomorphism $R\pi_*
\OO_{\mc{C}} \cong \OO_{\mf{M}}$
were defined so that the composition of $\delta$ with the trace
map, which is a quasi-isomorphism in this case, would be the
identity.  Therefore $\delta$ is a quasi-isomorphism, so $R\pi_*
E_\pi$ is quasi-isomorphic to $0$.  

\mni
The second part can be proved (and perhaps only makes sense)
after smooth base-change to a scheme.  Let $\PP^1_s$ be a copy of
$\PP^1$ with homogeneous coordinates $S_0,S_1$.  Let $\PP^1_x$ be a
copy of $\PP^1$ with homogeneous coordinates $X_0,X_1$.  Let $\PP^1_y$
be a copy of $\PP^1$ with homogeneous coordinates $Y_0,Y_1$.  Denote
by $C \subset \PP^1_s \times \PP^1_x \times \PP^1_y$ the divisor with
defining equation $F=S_0X_0Y_0 - S_1X_1Y_1$.  The projection
$\text{pr}_s:C \rightarrow \PP^1_s$ is a proper, flat morphism whose
geometric fibers are connected, at-worst-nodal curves of arithmetic
genus $0$.  Denote by $L$ the invertible sheaf on $C$ that is the
restriction of $\text{pr}_x^*\OO_{\PP^1_x}(a)\otimes
\text{pr}_y^*\OO_{\PP^1_y}(-a)$.  This is an invertible sheaf of
relative degree $0$.  Therefore there is an induced $1$-morphism
$\zeta:\PP^1_s \rightarrow \mf{M}_{\ZZ}(\tau_{0,0}(0))$.  

\mni
It is straightforward that $\zeta$ is smooth, and the image intersects
$\Delta_{b,-b}$ iff $b=a$.  Moreover, $\zeta^* \Delta_{a,-a}$ is the reduced
Cartier divisor $\mathbb{V}(S_0S_1) \subset \PP^1_s$.  There is an
obvious involution $i:\PP^1_s\rightarrow \PP^1_s$ by
$i(S_0,S_1)=(S_1,S_0)$, and $\zeta\circ i$ is $2$-equivalent to
$\zeta$.  Therefore the length of the $R^1\text{pr}_{s,*}
E_{\text{pr}_s}\otimes L$ is $2$ times the length of the stalk of
$R^1\pi_* E_{\pi}(\mc{D})$ at the generic point of $\Delta_a$; more
precisely, the length of the stalk at each of $(1,0),(0,1)\in \PP^1_s$
is the length of the stalk at $\Delta_a$.  Similarly for the lengths
of the associated graded pieces of the filtration.

\mni
Because $E_{\text{pr}_s}$ is the extension class of the trace mapping,
$R^1\text{pr}_{s,*}E_{\text{pr}_s} \otimes L$ is the cokernel of the
$\OO_{\PP^1_s}$-homomorphisms,
$$
\gamma:\text{pr}_{s,*}(L) \rightarrow
\text{Hom}_{\OO_{\PP^1_s}}(\text{pr}_{s,*}(L^\vee), \OO_{\PP^1_s}),
$$
induced via adjointness from the multiplication map,
$$
\text{pr}_{s,*}(L) \otimes \text{pr}_{s,*}(L^\vee) \rightarrow
\text{pr}_{s,*}(\OO_C) = \OO_{\PP^1_s}.
$$

\mni
On $\PP^1_s\times \PP^1_x \times \PP^1_y$ there is a locally free
resolution of the push-forward of $L$, resp. $L^\vee$,
$$
\begin{array}{c}
0 \rightarrow \OO_{\PP^1_s}(-1)\boxtimes \OO_{\PP^1_x}(a-1)\boxtimes
\OO_{\PP^1_y}(-a-1) \xrightarrow{F} \OO_{\PP^1_s}(0)\boxtimes
\OO_{\PP^1_x}(a) \boxtimes \OO_{\PP^1_y}(-a) \rightarrow L \rightarrow
0, \\
0 \rightarrow \OO_{\PP^1_s}(-1)\boxtimes \OO_{\PP^1_x}(-a-1)\boxtimes
\OO_{\PP^1_y}(a-1) \xrightarrow{F} \OO_{\PP^1_s}(0)\boxtimes
\OO_{\PP^1_x}(-a) \boxtimes \OO_{\PP^1_y}(a) \rightarrow L^\vee
\rightarrow 0
\end{array}
$$
Hence $R\text{pr}_{s,*}L$ is the complex,
$$
\OO_{\PP^1_s}(-1)\otimes_k H^0(\PP^1_x,\OO_{\PP^1_x}(a-1)) \otimes_k
H^1(\PP^1_y,\OO_{\PP^1_y}(-a-1)) \xrightarrow{F} \OO_{\PP^1_s}
\otimes_k H^0(\PP^1_x,\OO_{\PP^1_x}(a))\otimes_k
H^1(\PP^1_y,\OO_{\PP^1_y}(-a)).
$$
A similar result holds for $R\text{pr}_{s,*}L^\vee$.  
It is possible to write out
this map explicitly in terms of bases for $H^0$ and $H^1$, but for the
main statement just observe the complex has rank $1$ and degree $-a^2$.
A similar result holds for $R\text{pr}_{s,*}L^\vee$.  
Therefore $R^1\pi_* E_\pi(L)$
is a torsion sheaf of length $2a^2$.  Because it is equivariant for
$i$, the localization at each of $(0,1)$ and $(1,0)$ has length $a^2$.

\mni
The lengths of the associated graded pieces of the filtration by order
of vanishing at $\mathbb{V}(S_0S_1)$ can be computed from the
complexes for $R\text{pr}_{s,*}L$ and $R\text{pr}_{s,*}L^\vee$.  This
is left to the reader.
\end{proof}

\begin{cor} \label{cor-lcomp}
\marpar{cor-lcomp}
In the universal case, $Q_\pi(\mc{D}) = -\sum_{a\geq 0} a^2 \Delta_a$.
Therefore in the general case of $\pi:C\rightarrow M$ and an
invertible sheaf $L$ of relative degree $0$,
$$
Q_\pi(L) = {\sum_{\beta',\beta''}}^\prime \langle C_1(L),\beta'
\rangle \langle 
C_1(L), \beta'' \rangle \Delta_{\beta',\beta''}.
$$
\end{cor}

\section{Proof of Proposition ~\ref{prop-dcr}} \label{sec-div}
\marpar{sec-div}

\noindent
As usual, let $M$ be an Artin stack and let
$\pi:C\rightarrow M$ be a flat 1-morphism, relatively representable by
proper algebraic spaces whose geometric fibers are connected,
at-worst-nodal curves of genus $0$.

\begin{hyp} \label{hyp-GRR}
\marpar{hyp-GRR}
There are cycle class groups for $C$ and $M$ admitting Chern classes
for locally free sheaves, and such that Grothendieck-Riemann-Roch
holds for $\pi$.  In particular, this holds if $M$ is a
Deligne-Mumford stack whose coarse moduli space is quasi-projective.
\end{hyp}

% \begin{lem} \label{lem-rel1}
% For every Cartier divisor class $D$ on $C$ of relative degree $\langle
% D, \beta \rangle$ over $M$, modulo $2$-power torsion,
% $$
% \pi_*(D\cdot D) + \langle D,\beta \rangle \pi_*(D\cdot
% C_1(\omega_\pi)) = {\sum_{\beta',\beta''}}^\prime \langle D,\beta'
% \rangle \langle D,\beta'' \rangle \Delta_{\beta',\beta''}.
% $$
% \end{lem}

\begin{proof}[Proof of Proposition~\ref{prop-dcr}(i)]
Define $D' = 2D + \langle D,\beta \rangle C_1(\omega_\pi)$.  This is a
Cartier divisor class of relative degree $0$.  By
Corollary~\ref{cor-lcomp}, 
$$
Q_\pi(D') = {\sum_{\beta',\beta''}}^\prime (\langle 2D,\beta' \rangle
-\langle D,\beta \rangle)(\langle 2D,\beta'' \rangle - \langle D,\beta
\rangle) \Delta_{\beta',\beta''}.
$$
By Lemma~\ref{lem-GRR} this is,
$$
\begin{array}{c}
4\pi_*(D\cdot D) +4\langle D,\beta \rangle \pi_*(D\cdot
C_1(\omega_\pi) + (\langle D,\beta \rangle)^2 Q_\pi(C_1(\omega_\pi)) =
\\
{\sum_{\beta',\beta''}}^\prime (4\langle D,\beta' \rangle \langle
D,\beta'' \rangle - (\langle D,\beta \rangle)^2)
\Delta_{\beta',\beta''}.
\end{array}
$$
By Proposition~\ref{prop-Qomega}, $Q_\pi(\omega_\pi) =
-{\sum_{\beta',\beta''}}' \Delta_{\beta',\beta''}$.  Substituting this
into the equation, simplifying, and dividing by 4 gives the relation.
\end{proof}

\begin{lem} \label{lem-rel2}
\marpar{lem-rel2}
For every pair of Cartier divisor classes on $C$, $D_1,D_2$, of
relative degrees $\langle D_1,\beta \rangle$, resp. $\langle D_2,\beta
\rangle$, modulo $2$-power torsion,
$$
\begin{array}{c}
2\pi_*(D_1\cdot D_2) + \langle D_1,\beta \rangle \pi_*(D_2\cdot
C_1(\omega_\pi)) + \langle D_2,\beta \rangle \pi_*(D_1\cdot
C_1(\omega_\pi)) = 
\\
\\
{\sum_{\beta',\beta''}}^\prime (\langle D_1,\beta' \rangle \langle
D_2,\beta'' 
\rangle + \langle D_2,\beta' \rangle \langle D_1,\beta'' \rangle)
\Delta_{\beta',\beta''}.
\end{array}
$$
\end{lem}

\begin{proof}
This follows from Proposition~\ref{prop-dcr}(i) and the polarization identity
for quadratic forms.
\end{proof}

\begin{lem} \label{lem-rel3}
\marpar{lem-rel3}
For every section of $\pi$, $s:M\rightarrow C$, whose image is
contained in the smooth locus of $\pi$,
$$
s(M)\cdot s(M) + s(M)\cdot C_1(\omega_\pi).
$$
\end{lem}

\begin{proof}
This follows by adjunction since the relative dualizing sheaf of
$s(M)\rightarrow M$ is trivial.
\end{proof}

\begin{lem} \label{lem-rel4}
\marpar{lem-rel4}
For every section of $\pi$, $s:M\rightarrow C$, whose image is
contained in the smooth locus of $\pi$ and for every Cartier divisor
class $D$ on $C$ of relative degree $\langle D,\beta \rangle$ over
$M$, modulo $2$-power torsion,
$$
\begin{array}{c}
2\langle D,\beta \rangle s^*D  -\pi_*(D\cdot D) - \langle D,\beta
\rangle^2 \pi_*(s(M)\cdot s(M)) = \\
\\
{\sum_{\beta',\beta''}}^\prime
(\langle D,\beta' \rangle^2 \langle s(M),\beta'' \rangle + \langle
D,\beta'' \rangle^2 \langle s(M),\beta' \rangle )
\Delta_{\beta',\beta''}.
\end{array}
$$
\end{lem}

\begin{proof}
By Proposition~\ref{prop-dcr}(i),
$$
\begin{array}{c}
2s^* D + \pi_*(D\cdot C_1(\omega_\pi)) + \langle D,\beta \rangle
\pi_*(s(M)\cdot C_1(\omega_\pi)) = \\
\\
{\sum}^\prime (\langle D,\beta' \rangle \langle s(M),\beta'' \rangle +
\langle D,\beta'' \rangle \langle s(M),\beta' \rangle
)\Delta_{\beta',\beta''}.
\end{array}
$$
Multiplying both sides by $\langle D,\beta \rangle$,
$$
\begin{array}{c}
2\langle D,\beta \rangle s^* D + \langle D,\beta \rangle \pi_*(D \cdot
C_1(\omega_\pi)) + \langle D, \beta \rangle^2 \pi_*(s(M)\cdot
C_1(\omega_\pi)) = \\
\\
{\sum}^\prime(\langle D,\beta \rangle \langle D,\beta' \rangle \langle
s(M),\beta'' \rangle + \langle D,\beta \rangle \langle D,\beta''
\rangle \langle s(M),\beta' \rangle ) \Delta_{\beta',\beta''}.
\end{array}
$$
First of all, by Lemma~\ref{lem-rel4}, $\langle D,\beta \rangle^2
\pi_*(s(M) \cdot C_1(\omega_\pi)) = - \langle D,\beta \rangle^2
\pi_*(s(M)\cdot s(M))$.  Next, by Proposition~\ref{prop-dcr}(i),
$$
\langle D,\beta \rangle \pi_*(D\cdot C_1(\omega_\pi)) = -\pi_*(D\cdot
D) + {\sum}^\prime \langle D,\beta' \rangle \langle D,\beta''
\Delta_{\beta',\beta''}.
$$
Finally,
$$
\begin{array}{c}
\langle D,\beta \rangle \langle D,\beta' \rangle \langle s(M),\beta''
\rangle + \langle D,\beta \rangle \langle D,\beta'' \rangle \langle
s(M),\beta' \rangle = \\ \\
(\langle D,\beta' \rangle + \langle D,\beta'' \rangle) \langle
D,\beta' \rangle \langle s(M),\beta'' \rangle + (\langle D,\beta'
\rangle + \langle D,\beta'' \rangle) \langle D,\beta'' \rangle \langle
s(M), \beta' \rangle = \\ \\
\langle D,\beta' \rangle^2 \langle s(M),\beta'' \rangle + \langle
D,\beta'' \rangle^2 \langle s(M),\beta' \rangle + \langle D,\beta'
\rangle \langle D,\beta'' \rangle (\langle s(M),\beta' \rangle +
\langle s(M),\beta'' \rangle) = \\ \\
\langle D,\beta' \rangle^2 \langle s(M),\beta'' \rangle + \langle
D,\beta'' \rangle^2 \langle s(M),\beta' \rangle + \langle D,\beta'
\rangle \langle D,\beta'' \rangle.
\end{array}
$$
Plugging in these 3 identities and simplifying gives the relation.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop-dcr}(ii)]
Let $\pi:\mf{C}\rightarrow\mf{M}_{0,0}$ denote the universal family.  
Let
$\mf{C}_\text{smooth}$ denote the smooth locus of $\pi$.  The
2-fibered product $\text{pr}_1:\mf{C}_\text{smooth}
\times_{\mf{M}_{0,0}} \mf{C} 
\rightarrow \mf{C}_\text{smooth}$ together with the diagonal
$\Delta:\mf{C}_\text{smooth} \rightarrow \mf{C}_\text{smooth}
\times_{\mf{M}_{0,0}} \mf{C}$ determine a 1-morphism
$\mf{C}_\text{smooth} \rightarrow \mf{M}_{1,0}$.  This extends to a
1-morphism $\mf{C} \rightarrow \mf{M}_{1,0}$.  The pullback of the
universal curve is a 1-morphism $\pi':\mf{C}' \rightarrow \mf{C}$ that
factors through $\text{pr}_1:\mf{C}\times_{\mf{M}_{0,0}} \mf{C}
\rightarrow \mf{C}$.  Denote the pullback of the universal section by
$s:\mf{C} \rightarrow \mf{C}'$.  Now $\mf{C}$ is regular, and the
complement of $\mf{C}_\text{smooth}$ has codimension $2$. 
Therefore $s^*\OO_{\mf{C}'}(s(\mf{C}))$ can be computed on
$\mf{C}_\text{smooth}$.  But the restriction to $\mf{C}_\text{smooth}$
is clearly $\omega^\vee_\pi$.  Therefore $s^*\OO_{\mf{C}'}(s(\mf{C}))
\cong 
\omega_\pi^\vee$ on all of $\mf{C}$.

\mni
Pulling this back by $\zeta_C:C\rightarrow \mf{C}$ gives a 1-morphism
$\pi':C'\rightarrow C$ that factors through $\text{pr}_1:C\times_M
C\rightarrow C$.  Let $D$ be a Cartier divisor class on $C$ and
consider the pullback to $C'$ of $\text{pr}_2^* D$ on $C\times_M C$.
This is a Cartier divisor class $D'$ on $C'$.  Of course $s^* D' =
D$.  Moreover, by the projection formula the pushforward to $C\times_M
C$ of $D'\cdot D'$ is $\text{pr}_2^*(D\cdot D)$.  Therefore
$(\pi')_*(D'\cdot D')$ is $(\text{pr}_1)_*\text{pr}_2^*(D\cdot D)$,
i.e., $\pi^*\pi_*(D\cdot D)$.  Finally, denote by,
$$
\sum_{\beta',\beta''} \langle D,\beta'' \rangle^2
\widetilde{\Delta}_{\beta',\beta''},
$$
the divisor class on $C$,
$$
{\sum_{\beta',\beta''}}^\prime (\langle D,\beta'' \rangle^2 \langle
s,\beta' \rangle + \langle D,\beta' \rangle^2 \langle s,\beta''
\rangle)\Delta_{\beta',\beta''}.
$$
The point is this: if $\pi$ is smooth over every generic point of $M$,
then the divisor class $\widetilde{\Delta}_{\beta',\beta''}$ is the
irreducible component of $\pi^{-1}(\Delta_{\beta',\beta''})$
corresponding to the vertex $v'$, i.e., the irreducible component with
``curve class'' $\beta'$.
Therefore Proposition ~\ref{prop-dcr}(ii) follows from
Lemma~\ref{lem-rel4}. 
\end{proof}

% \begin{lem} \label{lem-rel5}
% \marpar{lem-rel5}
% For every Cartier divisor class $D$ on $C$ of relative degree $\langle
% D,\beta \rangle$ over $M$,
% $$
% \begin{array}{c}
% 2\langle D,\beta \rangle D - \pi^*\pi_*(D\cdot D) + \langle D,\beta
% \rangle^2 C_1(\omega_\pi) = \\ \\
% \sum_{\beta',\beta''} \langle D,\beta'' \rangle^2
% \widetilde{\Delta}_{\beta',\beta''}.
% \end{array}
% $$
% In particular, the relative Picard group of $\pi$ is generated by
% $C_1(\omega_\pi)$ and the boundary divisor classes
% $\widetilde{\Delta}_{\beta',\beta''}$.  
% \end{lem}

\begin{rmk} \label{rmk-rel5}
\marpar{rmk-rel5}
If $\langle D,\beta \rangle \neq 0$ then, at least up to torsion,
Proposition~\ref{prop-dcr}(i) follows from Proposition~\ref{prop-dcr}(ii) 
by intersecting
both sides of the relation by $D$ and then applying $\pi_*$.  This was
pointed out by Pandharipande, who also proved Lemma~\ref{lem-rel4} up
to numerical equivalence in ~\cite[Lem. 2.2.2]{QDiv} (by a very
different method).   
\end{rmk}

\begin{lem} \label{lem-rel6}
\marpar{lem-rel6}
Let $s,s':M\rightarrow C$ be sections with image in the smooth locus
of $\pi$ such that $s(M)$ and $s'(M)$ are disjoint.  Then,
$$
\pi_*(s(M)\cdot s(M)) + \pi_*(s'(M)\cdot s'(M)) = -
\sum_{\beta',\beta''} \langle s(M),\beta' \rangle \langle
s'(M),\beta'' \rangle \Delta_{\beta',\beta''}.
$$
\end{lem}

\begin{proof}
Apply Lemma~\ref{lem-rel2} and use $s(M)\cdot s'(M)=0$ and
Lemma~\ref{lem-rel3}.  
\end{proof}

\begin{lem} \label{lem-rel7}
Let $r\geq 2$ and $s_1,\dots,s_r:M\rightarrow C$ be sections with image in the
smooth locus of $\pi$ and which are pairwise disjoint.  Then,
$$
-\sum_{i=1}^r \pi_*(s_i(M)\cdot s_i(M)) = (r-2)\pi_*(s_1(M)\cdot
s_1(M)) + \sum_{\beta',\beta''} \langle s_1(M),\beta' \rangle \langle
s_2(M) + \dots + s_r(M),\beta'' \rangle \Delta_{\beta',\beta''}.
$$
\end{lem}

\begin{proof}
This follows from Lemma~\ref{lem-rel6} by induction.
\end{proof}

\begin{lem} \label{lem-rel8}
\marpar{lem-rel8}
Let $r\geq 2$ and let $s_1,\dots,s_r:M \rightarrow C$ be sections with
image in the smooth locus of $\pi$ and which are pairwise disjoint.
Then,
$$
-\sum_{i=1}^r \pi_*(s_i(M)\cdot s_i(M)) = r(r-2) \pi_*(s_1(M)\cdot
s_1(M)) + \sum_{\beta',\beta''} \langle s_1(M),\beta' \rangle \langle
s_2(M) + \dots + s_r(M),\beta'' \rangle^2 \Delta_{\beta',\beta''}.
$$
Combined with Lemma~\ref{lem-rel7} this gives,
$$
\begin{array}{c}
(r-1)(r-2)\pi_*(s_1(M)\cdot s_1(M)) = \\ \\
-\sum_{\beta',\beta''} \langle
s_1(M),\beta' \rangle \langle s_2(M) + \dots + s_r(M),\beta'' \rangle
(\langle s_2(M)+\dots +s_r(M), \beta'' \rangle - 1)
\Delta_{\beta',\beta''},
\end{array}
$$
which in turn gives,
$$
\begin{array}{c}
-(r-1)\sum_{i=1}^r\pi_*(s_i(M)\cdot s_i(M)) = \\ \\
\sum_{\beta',\beta''} \langle s_1(M),\beta' \rangle \langle s_2(M) +
\dots + s_r(M), \beta'' \rangle (r- \langle s_2(M) +\dots + s_r(M),
\beta'' \rangle) \Delta_{\beta',\beta''}.
\end{array}
$$
In the notation of Example~\ref{ex-boundary}, this is,
$$
-(r-1)(r-2)\pi_*(s_1(M)\cdot s_1(M)) = \sum_{(A,B), \ 1\in A}
\#B(\#B-1) \Delta_{(A,B)},
$$
and
$$
-(r-1)\sum_{i=1}^r \pi_*(s_i(M)\cdot s_i(M)) = \sum_{(A,B), \ 1\in A}
\#B(r-\#B) \Delta_{(A,B)}.
$$
\end{lem}

\begin{proof}
Denote $D=\sum_{i=2}^r s_i(M)$.  Apply Lemma~\ref{lem-rel4} to get,
$$
\begin{array}{c}
2(r-1)\cdot 0 - \sum_{i=2}^r \pi_*(s_i(M)\cdot s_i(M)) -
(r-1)^2\pi_*(s_1(M)\cdot s_1(M)) = \\ \\
\sum_{\beta',\beta''} \langle s_1(M),\beta' \rangle \langle
s_2(M)+\dots + s_r(M),\beta'' \rangle^2 \Delta_{\beta',\beta''}.
\end{array}
$$
Simplifying,
$$
-\sum_{j=1}^r \pi_*(s_i(M)\cdot s_i(M)) = r(r-2) \pi_*(s_1(M)\cdot
s_1(M)) + \sum \langle s_1(M),\beta' \rangle \langle s_2(M)+ \dots
+s_r(M),\beta'' \rangle^2 \Delta_{\beta',\beta''}.
$$
Subtracting from the relation in Lemma~\ref{lem-rel7} gives the
relation for $(r-1)(r-2)\pi_*(s_1(M)\cdot s_1(M))$.  
Multiplying the first relation by $(r-1)$, plugging in the second
relation and simplifying gives the third relation.
\end{proof}

\begin{lem} \label{lem-rel9}
\marpar{lem-rel9}
Let $r\geq 2$ and let $s_1,\dots,s_r:M\rightarrow C$ be everywhere
disjoint sections with image in the smooth locus.  For every $1\leq i
< j \leq r$, using the notation from Example~\ref{ex-boundary},
$$
\sum_{(A,B), \ i\in A} \#B(r-\#B) \Delta_{(A,B)} = \sum_{(A',B'), j \in
  A} \#B'(r-\#B') \Delta_{(A',B')}.
$$
\end{lem}

\begin{proof}
This follows from Lemma~\ref{lem-rel8} by permuting the roles of $1$
with $i$ and $j$.
\end{proof}

\begin{lem} \label{lem-rel10}
\marpar{lem-rel10}
Let $r \geq 2$ and let $s_1,\dots,s_r:M \rightarrow C$ be everywhere
disjoint sections with image in the smooth locus of $\pi$.  For every
Cartier 
divisor class $D$ on $C$ of relative degree $\langle D,\beta \rangle$,
$$
2(r-1)(r-2)\langle D,\beta \rangle s_1^*D = (r-1)(r-2)\pi_*(D\cdot D)
+ \sum_{\beta',\beta''} \langle s_1(M),\beta' \rangle a(D,\beta'')
\Delta_{\beta',\beta''},
$$
where, 
$$
a(D,\beta'') = (r-1)(r-2)\langle D,\beta'' \rangle^2 -
\langle D,\beta \rangle^2 \langle s_2(M)+\dots +s_r(M), \beta'' \rangle (
\langle s_2(M)+\dots + s_r(M),\beta'' \rangle -1).
$$
In particular, if $r\geq 3$, then modulo torsion $s_i^*D$ is in the
span of $\pi_*(D\cdot D)$ and boundary divisors for every
$i=1,\dots,r$.
\end{lem}

\begin{proof}
This follows from Lemma~\ref{lem-rel4} and Lemma~\ref{lem-rel8}.
\end{proof}

\begin{lem} \label{lem-rel11}
\marpar{lem-rel11}
Let $r\geq 2$ and let $s_1,\dots,s_r:M\rightarrow C$ be everywhere
disjoint sections with image in the smooth locus of $\pi$.  Consider
the sheaf $\mc{E} = \Omega_\pi(s_1(M)+\dots+s_r(M))$.  The perfect
complex $R\pi_*R\textit{Hom}_{\OO_C}(\mc{E},\OO_C)$ has rank $3-r$ and
the first Chern class of the determinant is $-2\Delta
-\sum_{i=1}^r(s_i(M)\cdot s_i(M))$.  In particular, if $r\geq 2$, up
to torsion,
$$
\begin{array}{c}
C_1(\text{det}R\pi_*
R\textit{Hom}_{\OO_C}(\Omega_\pi(s_1(M)+\dots+s_r(M)),\OO_C)) = \\ \\
-2\Delta + \frac{1}{r-1}\sum_{(A,B),\ 1\in A} \#B(r-\#B)
\Delta_{(A,B)}.
\end{array}
$$
\end{lem}

\begin{proof}
There is a short exact sequence,
$$
\begin{CD}
0 @>>> \Omega_\pi @>>> \Omega_\pi(s_1(M)+\dots+s_r(M))
@>>> \oplus_{i=1}^r (s_i)_*\OO_M @>>> 0.
\end{CD}
$$
Combining this with Lemma~\ref{lem-Omega}, Lemma~\ref{lem-rel8}, and
chasing through exact sequences gives the lemma.
\end{proof}

\section{Proof of Theorem~\ref{thm-main}} \label{sec-vircan}
\marpar{sec-vircan}

\noindent
Let $k$ be a field, let $X$ be a connected, smooth algebraic space
over $k$ of dimension $n$, let
$M$ be an Artin stack over $k$, let $\pi:C\rightarrow M$ be a flat
1-morphism, 
representable by proper algebraic spaces whose geometric
fibers are connected, at-worst-nodal curves of arithmetic genus $0$,
let $s_1,\dots,s_r:M\rightarrow C$ be pairwise disjoint sections with
image contained in the smooth locus of $\pi$ (possibly $r=0$, i.e.,
there are no sections), and let $f:C\rightarrow
X$ be a 1-morphism of $k$-stacks.  In this setting, Behrend and
Fantechi introduced a perfect complex $E^\bullet$ on $M$ of amplitude
$[-1,1]$ and a morphism to the cotangent complex,
$\phi:E^\bullet \rightarrow L_M^\bullet$, ~\cite{BM}.  
If $\text{char}(k)=0$ and $M$ is the Deligne-Mumford stack of stable
maps to $X$, Behrend and Fantechi prove $E^\bullet$ has amplitude
$[-1,0]$,
$h^0(\phi)$ is an isomorphism
and $h^{-1}(\phi)$ is surjective.  In
many interesting cases, $\phi$ is a quasi-isomorphism.  Then
$\text{det}(E^\bullet)$ is an invertible 
dualizing sheaf for $M$. 
Because of this, 
$\text{det}(E^\bullet)$ is called the \emph{virtual canonical
  bundle}.  In this section the relations from Section~\ref{sec-div}
are used to give a formula for the divisor class of the virtual
canonical bundle.  Assume that Hypothesis ~\ref{hyp-GRR} holds for $\pi$.

\mni
Denote by $L_{(\pi,f)}$ the cotangent complex of the morphism
$(\pi,f):C\rightarrow M\times X$.  This is a perfect complex of
amplitude $[-1,0]$.  There is a distinguished triangle,
$$
\begin{CD}
L_\pi @>>> L_{(\pi,f)} @>>> f^* \Omega_X[1] @>>> L_\pi[1].
\end{CD}
$$
There is a slight variation $L_{(\pi,f,s)}$ taking into account the
sections which fits into a distinguished triangle,
$$
\begin{CD}
L_\pi(s_1(M)+\dots+s_r(M)) @>>> L_{(\pi,f,s)} @>>> f^*\Omega_X[1] @>>>
L_\pi(s_1(M)+\dots+s_r(M))[1]. 
\end{CD}
$$
The complex $E^\bullet$ is defined
to be $(R\pi_*(L_{(\pi,f,s)}^\vee)[1])^\vee$, where $(F^\bullet)^\vee$
is $R\textit{Hom}(F^\bullet,\OO)$.  In particular,
$\text{det}(E^\bullet)$ is the determinant of
$R\pi_*(L_{(\pi,f,s)}^\vee)$.  From the distinguished triangle,
$\text{det}(E^\bullet)$ is
$$
\text{det}(R\pi_*
R\textit{Hom}_{\OO_C}(\Omega_\pi(s_1(M)+\dots+s_r(M)),\OO_C)) 
\otimes \text{det}(R\pi_* f^*T_X)^\vee.
$$
By Lemma~\ref{lem-rel11}, the first term is known.  
The second term
follows easily from Grothendieck-Riemann-Roch.  

\begin{lem} \label{lem-TX}
\marpar{lem-TX}
Assume that the relative degree of $f^*C_1(\Omega_X)$ is nonzero.
Then $R\pi_*f^*T_X[-1]$ has rank $\langle -f^*C_1(\Omega_X),\beta
\rangle + n$, and up to torsion the first Chern class of the
determinant is,
$$
\begin{array}{c}
\frac{1}{2\langle -f^*C_1(\Omega_X),\beta \rangle} \lt[ 2\langle
-f^*C_1(\Omega_X), \beta \rangle \pi_* f^* C_2(\Omega_X) \rt. \\
\\
 - (\langle
-f^* C_1(\Omega_X),\beta \rangle + 1) \pi_* f^* C_1(\Omega_X)^2 +
\\ \\
\lt.
{\sum}' \langle -f^*C_1(\Omega_X),\beta' \rangle \langle
-f^*C_1(\Omega_X),\beta'' \rangle \Delta_{\beta',\beta''}\rt].
\end{array}
$$
\end{lem}

\begin{proof}
The Todd class $\tau_\pi$ of $\pi$ is $1 - \frac{1}{2}C_1(\omega_\pi)
+ \tau_2 + \dots$, 
where $\pi_*\tau_2 = 0$.  The Chern character of $f^*T_X$ is,
$$
n - f^*C_1(\Omega_X) + \frac{1}{2}(f^*C_1(\Omega_X)^2 -
2f^*C_2(\Omega_X)) + \dots
$$
Therefore $\text{ch}(f^*T_X)\cdot \tau_\pi$ equals,
$$
n -
\lt[f^*C_1(\Omega_X)+\frac{n}{2}C_1(\Omega_\pi) \rt] + \frac{1}{2}\lt[
f^*C_1(\Omega_X)^2 - 2f^*C_2(\Omega_X) + f^*C_1(\Omega_X)\cdot
C_1(\omega_\pi) \rt] + n\tau_2 + \dots
$$
Applying $\pi_*$ and using that $\pi_*\tau_2 = 0$, the rank is
$n+\langle -f^*C_1(\Omega_X),\beta \rangle$, and the determinant has
first Chern class,
$$
\frac{1}{2}\pi_*\lt[ f^*C_1(\Omega_X)^2 - 2f^*C_2(\Omega_X) \rt] +
\frac{1}{2} \pi_*(f^*C_1(\Omega_X)\cdot C_1(\omega_\pi)).
$$
Applying Proposition~\ref{prop-dcr} and simplifying gives the relation.
\end{proof}  

\mni
Putting the two terms together gives the formulas in Theorem
~\ref{thm-main}.  

% \begin{prop} \label{prop-vc}
% \marpar{prop-vc}
% The rank of $E^\bullet$ is $\langle -f^*C_1(\Omega_X),\beta \rangle +
% n + r - 3.$  The following divisor class reations hold modulo torsion.
% If $\langle -f^*C_1(\Omega_X),\beta \rangle\neq 0$ and $r=0$,
% the first Chern class of the virtual canonical bundle
% is,
% \begin{equation}
% \begin{array}{c}
% \frac{1}{2\langle -f^*C_1(\Omega_X),\beta \rangle} \lt[ 2\langle
% -f^*C_1(\Omega_X), \beta \rangle \pi_* f^* C_2(\Omega_X) \rt. \\
% \\
% - (\langle
% -f^* C_1(\Omega_X),\beta \rangle + 1) \pi_* f^* C_1(\Omega_X)^2 +
% \\ \\
% \lt.
% {\sum}' (\langle -f^*C_1(\Omega_X),\beta' \rangle \langle
% -f^*C_1(\Omega_X),\beta'' \rangle -4\langle -f^*C_1(\Omega_X),\beta
% \rangle) \Delta_{\beta',\beta''}\rt].
% \end{array}
% \end{equation}
% If $\langle -f^*C_1(\Omega_X),\beta \rangle \neq 0$ and $r=1$, 
% the first Chern class of the virtual canonical bundle is,
% \begin{equation}
% \begin{array}{c}
% \frac{1}{2\langle -f^*C_1(\Omega_X),\beta \rangle} \lt[ 2\langle
% -f^*C_1(\Omega_X), \beta \rangle \pi_* f^* C_2(\Omega_X) \rt. \\
% \\
% - (\langle
% -f^* C_1(\Omega_X),\beta \rangle + 1) \pi_* f^* C_1(\Omega_X)^2 +
% \\ \\
% \lt.
% {\sum}' (\langle -f^*C_1(\Omega_X),\beta' \rangle \langle
% -f^*C_1(\Omega_X),\beta'' \rangle -4\langle -f^*C_1(\Omega_X),\beta
% \rangle) \Delta_{\beta',\beta''}\rt] \\
% \\
% -\pi_*(s_1(M)\cdot s_1(M)).
% \end{array}
% \end{equation}
% If $\langle -f^*C_1(\Omega_X),\beta \rangle \neq 0$ and $r\geq 2$, 
% the first Chern class of the virtual canonical bundle is,
% \begin{equation}
% \begin{array}{c}
% \frac{1}{2\langle -f^*C_1(\Omega_X),\beta \rangle} \lt[ 2\langle
% -f^*C_1(\Omega_X), \beta \rangle \pi_* f^* C_2(\Omega_X) \rt. \\
% \\
% - (\langle
% -f^* C_1(\Omega_X),\beta \rangle + 1) \pi_* f^* C_1(\Omega_X)^2 +
% \\ \\
% \lt.
% {\sum}' (\langle -f^*C_1(\Omega_X),\beta' \rangle \langle
% -f^*C_1(\Omega_X),\beta'' \rangle -4\langle -f^*C_1(\Omega_X),\beta
% \rangle) \Delta_{\beta',\beta''}\rt] \\
% \\
% + \frac{1}{r-1} \sum_{(A,B), 1\in
%   A} \#B(r-\#B)\Delta_{(A,B)}.
% \end{array}
% \end{equation}
% \end{prop}

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