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\begin{document}

\title[Global sections]{Global sections of some vector bundles on
  Kontsevich moduli spaces} 

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


\begin{abstract}
On the Kontsevich moduli space of unpointed stable maps to $\PP^1$ of
genus $0$ and degree $e$, there is a tautological vector bundle of
rank $e-1$.  Global sections of tensor powers of this vector bundle
arise when considering holomorphic contravariant tensors on Kontsevich
spaces of stable maps to more general projective varieties.  The
computation of the global sections is reduced to an explicit
combinatorial problem. 
\end{abstract}

\maketitle


\section{Introduction}~\label{sec-intro}
\marginpar{sec-intro}

\noindent
This note concerns the computation of the global sections of tensor
powers of a tautological
vector bundle on the Kontsevich moduli space of unpointed stable maps
to $\PP^1$ of genus $0$ and degree $e$.  This computation is a small
part of a larger project whose goal is to prove the conjecture
(apparently due to Fano) that there exist smooth Fano hypersurfaces 
that are not unirational.  

\medskip\noindent
The strategy of the proof is as follows (cf. ~\cite{S1}).
For a Fano hypersurface $X$ and every integer $e>0$, the idea is to prove 
that on $\Kbm{0,0}{X,e}$ there are many \emph{holomorphic
  contravariant tensors}, i.e. sections of
tensor powers of the cotangent bundle.  Because the restriction of
a holomorphic contravariant tensor to a rational curve is $0$, these
tensors ``bound'' the rational curves on $\Kbm{0,0}{X,e}$, which in turn
``bound'' the rational surfaces on $X$.  The goal is to produce enough
tensors to prove that through a very general point of $X$ there is no
rational surface, thus proving $X$ is not unirational.  In her thesis,
Beheshti has done this for $e=1$ and $X$ a smooth hypersurface of
degree $n$ or $n-1$
in $\PP^n$: she proves that for a point of $X$ in the complement of
countably many codimension $2$ subvarieties, there is no rational
surface ruled by lines contained in $X$ containing the point.
Beheshti's result appears to extend to the case $e \leq n$.  

\medskip\noindent
Two techniques are known
for producing tensors on $\Kbm{0,0}{X,e}$.
The first uses the correspondence coming from the universal
stable map over $\Kbm{0,0}{X,e}$ to obtain holomorphic $(p,0)$-forms
on $\Kbm{0,0}{X,e}$ from holomorphic $(p+1,1)$-forms on $X$,
cf. ~\cite{dJS2}.  The second produces pluricanonical forms on
$\Kbm{0,0}{X,e}$ by computing the canonical divisor class and proving
it is \emph{big}, cf. ~\cite{S1}.  Neither of these techniques produce
enough tensors when $e$ is large.

\medskip\noindent
A third technique, to which this note is directly relevant, is to
produce tensors \emph{from the inside out}.  There is a closed
substack $Y$ of $\Kbm{0,0}{X,e}$ parametrizing degree $e$ covers of
lines on $X$.  
The technique is to produce tensors on the formal neighborhood of $Y$
in $\Kbm{0,0}{X,e}$. The idea is to prove these tensors
algebraize, or to at least understand the formal tensors well enough to
guess which come from algebraic tensors (hopefully suggesting some
fourth technique for proving these tensors algebraize).  

\medskip\noindent
Behind 
this is the fact that $Y$ is \emph{much simpler} than 
$\Kbm{0,0}{X,e}$:
The stack $Y$ is the total space of a fibration whose
base is the Fano variety of lines on $X$, and whose fiber is
isomorphic to $\Kbm{0,0}{\PP^1,e}$.  Everything is known
about the Fano variety of lines on $X$.  And quite a bit is known
about $\Kbm{0,0}{\PP^1,e}$.  The restriction of the cotangent bundle
of $\Kbm{0,0}{X,e}$ to $Y$ has a filtration where the subquotients are
the pullback of the cotangent bundle on the Fano variety of lines, the
relative cotangent bundle of the fibration, and the conormal bundle of
$Y$ in $\Kbm{0,0}{X,e}$. 
The conormal bundle is essentially the tensor
product of the pullback from the Fano variety of lines of a
well-understood vector bundle with the vector bundle $\mc{R}_Y^\vee$
considered in this note.  

\medskip\noindent
The vector bundle $\mc{R}_Y^\vee$ is a relative version of a
tautological vector bundle $\mc{R}^\vee$ on $\Kbm{0,0}{\PP^1,e}$.  
The vector bundle $\mc{R}^\vee$ is very
natural, but it has not been studied before.  The question here is the
most basic one: What are the global sections of $\mc{R}^\vee$? More
generally,
what
are the global sections of a tensor power of
$\mc{R}^\vee$, and what are the global sections of the tensor product of
a power of $\mc{R}^\vee$ with a power of the cotangent bundle?
Forming the direct sum of global sections of every power of
$\mc{R}^\vee$ gives a (non-commutative) ring, and forming the direct
sum of global sections of the tensor product of every power of
$\mc{R}^\vee$ with a fixed power of the cotangent bundle gives a
left-module over this ring.  The question is to describe this ring and
to describe the module over this ring: in what degrees is the
ring generated, in what degrees are the relations, what are the
dimensions of the graded pieces of this ring, in what degrees is the
module generated, etc?

% \subsection{The vector bundle $\mc{R}^\vee$}\label{subsec-R}
% \marginpar{subsec-R}
% \noindent
% Let $B$ be an affine scheme such that $H^0(B,\OO_B)$ contains $\QQ$.
% Let $E$ be
% a locally free $\OO_B$-module of rank $2$.  Denote by $\Kbm{0,0}{\PP
% E/B,e} \rightarrow B$ the $1$-morphism of Deligne-Mumford stacks
% parametrizing Kontsevich 
% stable maps of genus $0$ and degree $e$ into fibers of
% $\PP E \rightarrow B$.  Denote by $\pi_{\PP E}: C_{\PP E} \rightarrow
% \Kbm{0,0}{\PP E/B,e}$ the universal curve.  Denote by $g_{\PP
%   E}:C_{\PP E} \rightarrow \PP E$ the universal map.  Denote by
% $\mc{P}$ the coherent sheaf $(\pi_{\PP E})_* (g_{\PP E})^* \OO_{\PP
%   E}(1)$.  This is a locally free sheaf.  Moreover, the natural map of
% coherent sheaves,
% $$
% E^\vee \otimes_{\OO_B} \OO \rightarrow \mc{P},
% $$
% is injective and the cokernel is locally free of rank $e-1$.  Denote
% the cokernel by $\mc{R} = \mc{R}_{\PP E,e-1}$.  Roughly speaking, the
% dual $\mc{R}^\vee$ is close to being ample.  The goal of this note is
% to describe for every pair of positive integers $e,a$, 
% the $H^0(B,\OO_B)$-module of global
% sections of $(\mc{R}^\vee)^{\otimes a}$ on
% $\Kbm{0,0}{\PP E/B,e}$.  

\section{The general construction}\label{sec-gc}
\marginpar{sec-gc}

\subsection{Unpointed curves}\label{subsec-unptd}
\marginpar{subsec-unptd}
\noindent
Let $M$ be a Deligne-Mumford stack over $B$.  Let $\pi:C\rightarrow B$
be a proper, flat, representable $1$-morphism of relative dimension
$1$.  Let $g:C \rightarrow \PP E$ be a $1$-morphism.  There is a
natural map of coherent $\OO_M$-modules,
$$
\phi^\dagger_{\pi,g}:E^\vee \otimes_{\OO_B} \OO_M \rightarrow \pi_*
g^* \OO_{\PP E}(1),
$$
constructed as follows.
%Denote by $\mc{R}_{\pi,g}$ the cokernel of $\phi^\dagger_{\pi,g}$.
There is a natural map of coherent sheaves on $M\times_B \PP E$,
$$
(\pi,g)^\#:  \OO_{M\times_B \PP E} \rightarrow (\pi,g)_* \OO_C.
$$ 
This induces a map of coherent sheaves,
$$
(\pi,g)^\#_1: \text{pr}_{\PP E}^* \OO_{\PP E}(1) \rightarrow (\pi,g)_*
g^* \OO_{\PP E}(1).
$$
The map $\phi^\dagger_{\pi,g}$ is the map obtained by applying
$(\text{pr}_M)_*$ to $(\pi,g)_1^\#$.
The subject of this note is the coherent sheaf,
$$
\mc{R}_{\pi,g} = \text{Coker}(\phi^\dagger_{\pi,g}).
$$
Assume that $(\pi,g): C \rightarrow M \times_B \PP E$ is surjective
and generically finite of degree $e$.  


\begin{prop} \label{prop-1}
\marginpar{prop-1}
Suppose the total derived pushforward $R(\pi,g)_*\OO_C$ is a perfect,
bounded complex.  Suppose that for every geometric point $p$ of $M$,
$h^0(C_p,\OO_{C_p}) =1$ and $h^1(C_p,\OO_{C_p})=0$.
Then 
\begin{enumerate}
\item[(i)]
$(\pi,g)^\#$ is injective and the cokernel is a locally free sheaf of
rank $e-1$,
\item[(ii)]  
$\phi^\dagger_{\pi,g}$ is injective and $\mc{R}_{\pi,g}$ is locally
free of rank $e-1$, and
\item[(iii)]
the cokernel of $(\pi,g)_1^\#$ is isomorphic to 
$\text{pr}_M^* \mc{R}_{\pi,g}$.
\end{enumerate}
\end{prop}

\noindent
\begin{proof}
\textbf{(i)}
The claim is that $R(\pi,g)_*\OO_C$ is quasi-isomorphic to a locally
free sheaf concentrated in degree $0$; in particular $(\pi,g)_*\OO_C$
is a locally free sheaf.  A perfect, bounded complex
concentrated in nonnegative degrees is quasi-isomorphic to a locally
free sheaf concentrated in degree $0$ iff all the higher cohomology
sheaves are zero.  By a standard downward induction argument, it
suffices to prove that for
every geometric point $p\in M$, all the higher cohomology sheaves of
$R(\pi,g)_* \OO_C \otimes^{\mathbf{L}}_{\OO_M} \kappa(p)$ are
zero.  

\medskip\noindent
Denote by $g_p:C_p \rightarrow \PP E_p$ the fiber of $(\pi,g)$
over $p$.  The higher cohomology sheaves are the same as the higher
direct image sheaves $R^i (g_p)_* \OO_{C_p}$.  
Every fiber of $g_p$ has dimension $\leq 1$.  Thus
$R^i (g_p)_* \OO_{C_p} = (0)$ for $i>1$.  It remains to prove that
$R^1(g_p)_* \OO_{C_p}$ is zero.  Since $g_p$ is finite over a dense
open subset of $\PP E_p$,
$R^1 (g_p)_* \OO_{C_p}$ is a torsion sheaf.  By the Leray spectral
sequence, $H^0(\PP E_p, R^1(g_p)_* \OO_{C_p})$ is a subspace of
$H^1(C_p,\OO_{C_p})$, which is 
zero.  A torsion sheaf on $\PP E_p$ with no global sections is the
zero sheaf.  So $R^1 (g_p)_* \OO_{C_p} = (0)$, proving the claim.    

\medskip\noindent
Item (i) claims that $(\pi,g)^\#$ is injective and the cokernel is
locally free. 
The domain and target of $(\pi,g)^\#$ are
locally free sheaves.  
So, by the local flatness criterion, 
it suffices to prove that for every geometric point $p$ 
of $M \times_B \PP E$,
the map $(\pi,g)^\# \otimes \kappa(p)$ is injective.  This map is the
inclusion of scalars $\kappa(p)$ in the ring of global sections of the
fiber $(\pi,g)^{-1}(p)$.  Because $(\pi,g)$ is surjective, the fiber
is nonempty.  Thus inclusion of scalars is injective.

\medskip\noindent
\textbf{(ii) and (iii)}
Because
$h^0(\OO)=1$ for every geometric fiber of $\pi$, by the same type of
argument as above, $(\text{pr}_M)_*(\pi,g)^\#$ is an isomorphism of
$\OO_M$-modules.  In particular, the cokernel of $(\pi,g)^\#$ is a
locally free sheaf of rank $e-1$ such that both $(\text{pr}_M)_*$ and
$R^(\text{pr}_M)*$ are zero.  Combined with Grothendieck's Lemma, this
implies that the cokernel is of the form $\text{pr}_M^*\mc{R}'\otimes
\text{pr}_{\PP E}^*\OO_{\PP E}(-1)$ for
some locally free $\OO_M$-module of rank $e-1$, $\mc{R}'$.  Therefore
there is a short exact sequence,
$$
0 \rightarrow \text{pr}_{\PP E}^* \OO_{\PP E}(1) \rightarrow (\pi,g)_*
g^*\OO_{\PP E}(1) \rightarrow \text{pr}_M^*\mc{R}' \rightarrow 0.
$$
Item (ii) follows by applying $(\text{pr}_M)_*$.  In particular,
$\mc{R}' = \mc{R}_{\pi,g}$, which is (iii). 
\end{proof}

\begin{rmk} \label{rmk-1}
\marginpar{rmk-1}
If $\pi$ is a local complete intersection morphism, then also
$(\pi,g)$ is a local complete intersection morphism so that
$R(\pi,g)_*\OO_C$ is a perfect, bounded complex.  In particular, if
$\pi$ is a proper, flat family of connected, at-worst-nodal curves of
arithmetic genus $0$, the hypotheses of Proposition~\ref{prop-1} hold.
\end{rmk}

\medskip\noindent
Let $\pi':C' \rightarrow M$ and $g':C'\rightarrow \PP E$ be a second
pair satisfying the hypotheses of Proposition~\ref{prop-1}.  Let
$h:C\rightarrow C'$ be a $1$-morphism of $M$-stacks
such that $g$ is equivalent to $g'\circ h$.  Then
there is an induced morphism of $\OO_M$-modules,
$h^*:\mc{R}_{\pi',g'} \rightarrow \mc{R}_{\pi,g}$.  

\medskip\noindent
Let
$u:N\rightarrow M$ be a $1$-morphism of Deligne-Mumford stacks.
Denote by $C_N$ the $2$-fiber product, $N \times_M C$.  Denote by
$\pi_N:C_N \rightarrow N$ and $g_N:C_N \rightarrow \PP E$ the induced
$1$-morphisms.  There is a canonical isomorphism of $\OO_N$-modules,
$u^* \mc{R}_{\pi,g} \rightarrow \mc{R}_{\pi_N,g_N}$.  

\medskip\noindent
\begin{notat} \label{notat-1}
\marginpar{notat-1}
For $M = \Kbm{0,0}{\PP
  E/B,e}$ and for $\pi$ and $g$ as in Section~\ref{sec-intro}
denote $\mc{R}_{\pi,g}$ by $\mc{R}_{\PP E,e}$, or simply
$\mc{R}$ when there is no risk of confusion.
\end{notat}

\noindent
\subsection{Pointed curves}\label{subsec-ptd}
\marginpar{subsec-ptd}
Let $M$, $C$, $\pi$ and $g$ be as in Subsection~\ref{subsec-unptd}.
Let $\sigma:M\rightarrow C$ be a section of $\pi$.  Denote by $\tau:M
\rightarrow M\times_B \PP E$ the composition $(\pi,g)\circ \sigma$.
Denote by $\mc{I}$ the ideal sheaf of $\sigma(M)$ in $C$.
Denote by $\OO_{M\times_B \PP E}(-\tau)$ the invertible ideal sheaf of
$\tau(M)$ in $M\times_B \PP E$.

\medskip\noindent
There is a natural map of coherent $\OO_M$-modules,
$$
\phi^\dagger_{\pi,g,\sigma}:(\text{pr}_M)_*(\text{pr}_{\PP E}^*\OO_{\PP
  E}(1)\otimes(-\tau)) \rightarrow \pi_*( g^*\OO_{\PP E}(1)\otimes \mc{I}),
$$
constructed as follows.
There is a natural map of coherent sheaves on $M\times_B \PP E$,
$$
(\pi,g)_\sigma^\#: \OO_{M\times_B \PP E}(-\tau) \rightarrow (\pi,g)_*
\mc{I}. 
$$
This induces a map of coherent sheaves,
$$
(\pi,g)_{\sigma,1}^\#: \text{pr}_{\PP E}^* \OO_{\PP E}(1)(-\tau)
\rightarrow (\pi,g)_*( g^*\OO_{\PP E}(1)\otimes \mc{I}).
$$
The map $\phi^\dagger_{\pi,g,\sigma}$ is the map obtained by applying
$(\text{pr}_M)_*$ to $(\pi,g)_{\sigma,1}^\#$.
As above, assume that $(\pi,g)$ is surjective and generically finite
of degree $e$.

\begin{prop} \label{prop-2}
\marginpar{prop-2}
Suppose the total derived pushforward $R(\pi,g)_*\OO_C$ is a perfect
bounded complex.  Suppose that for every geometric point $p$ of $M$,
$h^0(C_p,\OO_{C_p}) = 1$ and $h^1(C_p,\mc{I}\otimes \kappa(p))=0$.
Then
\begin{enumerate}
\item[(i)]
$(\pi,g)_\sigma^\#$ is injective and the cokernel is canonically 
isomorphic to the
cokernel of $(\pi,g)^\#$, and
\item[(ii)]
$\phi^\dagger_{\pi,g,\sigma}$ is injective and the cokernel is
canonically isomorphic to $\mc{R}_{\pi,g}$.
\end{enumerate}
\end{prop}

\noindent
\begin{proof}
Because $\OO_C$ and $\sigma_*\OO_M$ are flat over $M$,
also $\mc{I}$ is flat over $M$.  Therefore $\mc{I}\otimes\kappa(p)$ is
the ideal sheaf of $\sigma(p)$ in $C_p$.  Because $h^1$ of this sheaf
is zero, also $h^1(C_p,\OO_{C_p})=0$.  So the hypotheses of
Proposition~\ref{prop-1} hold.  Therefore $(\pi,g)_*\OO_C$ is locally
free.

\medskip\noindent
Associated to the short exact sequence,
$$
0 \rightarrow \mc{I} \rightarrow \OO_C \rightarrow \sigma_*\OO_M
\rightarrow 0,
$$
there is a long exact sequence of higher direct images,
$$
0 \rightarrow (\pi,g)_* \mc{I} \rightarrow (\pi,g)_* \OO_C \rightarrow
\tau_* \OO_M \rightarrow R^1(\pi,g)_* \mc{I} \rightarrow 0.
$$
To prove that $R^1(\pi,g)_* \mc{I}$ is zero, it suffices to prove that
for every geometric point $p$ of $M$, the fiber
$R^1(\pi,g)_*\mc{I}\otimes \kappa(p)$ is zero.  The fiber is
$R^1(g_p)_*(\mc{I}\otimes \kappa(p))$.  Because $g_p$ is generically
finite, this is a torsion sheaf.  By the Leray spectral sequence, the
space of global sections of this sheaf is a subspace of
$H^1(C_p,\mc{I}\otimes \kappa(p))$, which is zero by hypothesis.  A
torsion sheaf on $\PP E_p$ with no global sections is the zero sheaf.
Therefore $R^1(\pi,g)_* \mc{I} = (0)$.  

\medskip\noindent
There is a commutative diagram with exact rows,
$$
\begin{CD}
0 @>>> \OO_{M \times_B \PP E}(-\tau) @>>> \OO_{M \times_B \PP E} @>>>
\tau_* \OO_M @>>> 0 \\
& & @V (\pi,g)_\sigma^\# VV @VV (\pi,g)^\# V @VV = V \\
0 @>>> (\pi,g)_* \mc{I} @>>> (\pi,g)_* \OO_C @>>> \tau_* \OO_M @>>> 0
\end{CD}
$$
By Proposition~\ref{prop-1}, $(\pi,g)^\#$ is injective and the
cokernel is canonically isomorphic to $\pi_M^* \mc{R}\otimes \pi_{\PP
  E}^* \OO_{\PP E}(-1)$.  Therefore, by the Snake Lemma,
$(\pi,g)_{\sigma}^\#$ is injective and the cokernel is canonically
isomorphic to $\pi_M^* \mc{R} \otimes \pi_{\PP E}^* \OO_{\PP E}(-1)$.
This proves (i), and (ii) follows by applying $(\text{pr}_M)_*$.
\end{proof}

\section{The case $e=2$} \label{sec-e2}
\marginpar{sec-e2}

\noindent
In the special case that $e=2$, the Deligne-Mumford stack
$\Kbm{0,0}{\PP E/B,e}$ has a particularly simple description.  This
leads to a complete description of the global sections of each
$(\mc{R}^\vee)^{\otimes a}$.  The symmetric square, $\text{Sym}^2(\PP
E/M)$ is canonically isomorphic to the Hilbert scheme of degree $2$
divisors on fibers of $\PP E$, i.e. $\PP\text{Sym}^2(E^\vee)$.
There is a \emph{branch morphism}, i.e. a $1$-morphism
$\text{br}: \Kbm{0,0}{\PP
  E/B,2} \rightarrow \PP\text{Sym}^2(E^\vee)$, cf. ~\cite{FaP}.  

\medskip\noindent
The
morphism $(\pi,g)$ is finite and flat of degree $2$.  The trace
morphism gives a splitting of the short exact sequence,
$$
0 \rightarrow \OO_{M \times_B \PP E} \rightarrow (\pi,g)_* \OO_C
\rightarrow \text{pr}_M^* \mc{R} \otimes \text{pr}_{\PP E}^*\OO_{\PP E}(-1)
\rightarrow 0.
$$
Then the multiplication map on $(\pi,g)_* \OO_C$ induces a morphism,
$$
(\text{pr}_M^* \mc{R} \otimes \text{pr}_{\PP E}^* \OO_{\PP
  E}(-1))^{\otimes 2} \rightarrow \OO_{M \times_B \PP E}.
$$
Computing \'etale locally, or even formally locally, this morphism is
an isomorphism onto the ideal sheaf of the branch divisor.  By
definition of the branch morphism, this ideal sheaf is,
$$
\text{pr}_M^* \text{br}^* \OO_{\PP \text{Sym}^2(E^\vee)}(-1) \otimes
\text{pr}_{\PP E}^* \OO_{\PP E}(-2).
$$
The upshot is that there is a canonical isomorphism of coherent
sheaves,
$$
\text{pr}_M^* (\mc{R})^{\otimes 2} \rightarrow \text{pr}_M^*
\text{br}^* \OO_{\PP \text{Sym}^2(E^\vee)}(-1).
$$
Applying $(\text{pr}_M)_*$ gives an isomorphism of invertible sheaves
on $\Kbm{0,0}{\PP E/B,2}$,
$$
\mc{R}^{\otimes 2} \rightarrow \text{br}^* \OO_{\PP
  \text{Sym}^2(E^\vee)}(-1).
$$
Therefore there is a $1$-morphism from $\Kbm{0,0}{\PP E/B,2}$ to the
Deligne-Mumford stack $Y$ over $\PP \text{Sym}^2(E^\vee)$ parametrizing
square roots of $\OO(-1)$.

\medskip\noindent
Conversely, on $Y \times_B \PP E$, one can reverse the logic above to
construct an algebra whose relative Spec is a finite flat scheme over
$Y \times_B \PP E$, say $C_Y \rightarrow Y \times_B \PP E$.  Computing
locally, this is a family of stable maps to $\PP E$.  So there is an
induced $1$-morphism $Y \rightarrow \Kbm{0,0}{\PP E/B,2}$.  It is
straightforward to compute that these $1$-morphisms define an
equivalence of stacks.  In particular, the space of global sections of
$(\mc{R}^\vee)^{\otimes a}$ is canonically isomorphic to the space of
global sections of the same power of the square root of $\OO(1)$.
Therefore,
$$
H^0(\Kbm{0,0}{\PP E/B,2}, (\mc{R}^\vee)^{\otimes a}) \cong \lt\{
  \begin{array}{ll}
\text{Sym}^{\frac{a}{2}}(\text{Sym}^2 E), & a \ \text{is even} \\
(0), & a \ \text{is odd}
\end{array} \rt.
$$

\section{The global quotient description of an open
  substack}\label{subsec-gq}
\marginpar{subsec-gq}

\noindent
Unfortunately, for $e\geq 3$, there is no such simple description of
$\Kbm{0,0}{\PP E/B,e}$.  However, there is a dense open substack that
is a global quotient by the action of $\textbf{PGL}_2$ of a dense open
subscheme of a projective space.  The idea is to first compute the
spaces of sections over this open subset by invariant theory, and then
to determine which sections extend to regular sections on all of
$\Kbm{0,0}{\PP E/B,e}$.  

\medskip\noindent
Let $e$ be an integer $\geq 2$.
For $i=1,\dots,\lfloor \frac{e}{2} \rfloor$, denote by $\Delta_i
\subset \Kbm{0,0}{\PP E/B,e}$ the irreducible Cartier divisor whose
general point parametrizes a stable map whose domain has two
irreducible components, one of degree $i$ over $\PP E$ and one of
degree $e-i$ over $\PP E$.  Define $U \subset \Kbm{0,0}{\PP E/B,e}$ to
be the complement of $\Delta_2 \cup \dots \cup \Delta_{\lfloor
  \frac{e}{2} \rfloor}$.  In case $e=2$ or $3$, $U$ is all of
$\Kbm{0,0}{\PP E/B,e}$.  In every case, $U$ intersects $\Delta_1$.  

\medskip\noindent
Let $F$ be a locally free sheaf of rank $2$ on $B$.  This sheaf
is auxiliary; the simplest choice is $F = \OO_B \oplus \OO_B$.
Denote by $P = P_{e_1}$ the Hilbert scheme over $B$ of flat families
of closed subschemes of $\PP E \times \PP F$ whose fibers are in the
complete linear system of $\text{pr}_{\PP E}^* \OO_{\PP E}(1) \otimes
\text{pr}_{\PP F}^* \OO_{\PP F}(e)$, i.e.
$$
P = \PP H^0(\PP E\times_B \PP F, \text{pr}_{\PP E}^* \OO_{\PP E}(1) \otimes
\text{pr}_{\PP F}^* \OO_{\PP F}(e) ) = \PP( E^\vee \otimes
\text{Sym}^e F^\vee).
$$
Denote by $\mc{D} = \mc{D}_e$ the Cartier divisor in
$P \times \PP E \times \PP F$ that is the universal closed subscheme.
There is a resolution of the structure sheaf,
$$
0 \rightarrow \text{pr}_P^* \OO_P(-1) \otimes \text{pr}_{\PP E}^*
\OO_{\PP E}(-1) \otimes \text{pr}_{\PP F}^* \OO_{\PP F}(-e)
\rightarrow \OO \rightarrow \OO_{\mc{D}} \rightarrow 0.
$$

\medskip\noindent
The morphism $(\text{pr}_P,\text{pr}_{\PP E}):\mc{D} \rightarrow P
\times \PP E$ is finite and flat of degree $e$.  There is a maximal
open subscheme of $P$ over which $\mc{D}$ is a family of stable maps
to $\PP E$.  Denote this open subscheme by $V$.  The complement of $V$
in $P$ is a closed set that has codimension $\geq 3$.  There is an
induced $1$-morphism $f:V \rightarrow \Kbm{0,0}{\PP E/B,e}$.  By
consideration of divisors on $\PP E \times \PP F$, the image of $f$ is
in $U$.  Thus consider $f$ as a $1$-morphism, $f:V \rightarrow U$.  

\medskip\noindent
There is an induction argument that proves that $f:V
\rightarrow U$ is surjective on geometric points (the induction is on
the number of irreducible components of the domain curve).  
There is an action $m$
of $\text{Aut}(\PP F) = \textbf{PGL}(F)$ on $P$.  The open subscheme
$V$ is $\textbf{PGL}(F)$-invariant.  The $1$-morphism $f$ is
$\textbf{PGL}(F)$-invariant, in the sense that there is a
$2$-isomorphism between the following $1$-morphisms,
\begin{eqnarray*}
\textbf{PGL}(F)\times_B V \xrightarrow{\text{pr}_V} V
\xrightarrow{f} U, \\
\textbf{PGL}(F)\times_B V \xrightarrow{m} V \xrightarrow{f} U.
\end{eqnarray*}
There is an induced $1$-morphism $f':\text{PGL}(F) \times_B V \rightarrow
V \times_U V$ such that $\text{pr}_1 \circ f' = m$ and $\text{pr}_2
\circ f' = \text{pr}_V$.  By a straightforward (but tedious) argument,
$f'$ is an isomorphism of schemes.  
So there is an induced $1$-morphism, 
$$
u:[\textbf{PGL}(F) \backslash V]
\rightarrow U.
$$

\medskip\noindent
The claim is that $u$ is an equivalence of stacks.  This can be
checked after base-change by the surjective, 
smooth morphism $f:V\rightarrow U$.
But the base change is the identity morphism $\text{Id}_V$ by
construction.  Because $u$ is an equivalence of stacks,
there is a canonical isomorphism between the space of
sections of $(\mc{R}^\vee)^{\otimes a}$ over $U$ and the space of
$\text{PGL}(F)$-invariant
sections of $f^*(\mc{R}^\vee)^{\otimes a}$ over $V$.  This can be
computed by invariant theory.

\section{Invariant theory description} \label{sec-invt}
\marginpar{sec-invt}

\noindent
The morphisms $\text{pr}_P: \mc{D} \rightarrow P$ and 
$\text{pr}_{\PP E}: \mc{D} \rightarrow \PP E$ satisfy the hypotheses of
Proposition~\ref{prop-1}.  Denote by $\mc{R}_P$ the associated locally
free sheaf.
Twisting the short exact sequence in the last section, there is a short
exact sequence,
$$
0 \rightarrow \text{pr}_P^* \OO_P(-1) \otimes \text{pr}_{\PP
  F}^*\OO_{\PP F}(-e) \rightarrow \text{pr}_{\PP E}^* \OO_{\PP E}(1)
\rightarrow \text{pr}_{\PP E}^* \OO_{\PP E}(1) \otimes \OO_{\mc{D}}
\rightarrow 0.
$$
Applying $(\text{pr}_P)_*$, gives an isomorphism of $\mc{R}_P$ with
$\OO_P(-1) \otimes_{\OO_B} H^1(\PP F,\OO_{\PP F}(-e)).$  Therefore the
dual, $\mc{R}_P^\vee$ is isomorphic to,
$$
\mc{R}_P^\vee \cong \OO_P(1) \otimes_{\OO_B} H^0(\PP F,\OO_{\PP F}(e)
\otimes \omega_{\PP F/B}) \cong \OO_P(1) \otimes_{\OO_B}\lt[
\text{Sym}^{e-2}(F^\vee) \otimes \wedge^2(F^\vee) \rt].
$$
This uses the canonical isomorphism, $\omega_{\PP F/B} \cong \OO_{\PP
  F}(-2) \otimes_{\OO_B} \wedge^2(F^\vee)$.

\noindent
By definition of $f$, there is an isomorphism of $V \times_P \mc{D}$
with the pullback by $f$ of the universal stable map.  Therefore there
is a canonical isomorphism of $f^* \mc{R}^\vee$ with the restriction
of $\mc{R}_P$ to $V$.  So there is a canonical isomorphism between the
space of sections of $(\mc{R}^\vee)^{\otimes a}$ on $U$ with the space
of $\textbf{PGL}(F)$-invariant sections of $(\mc{R}_P^\vee)^{\otimes a}$
on $V$.  Because the complement of $V$ has codimension $\geq 3$, every
section on $V$ extends to a global section.  So there is a canonical
isomorphism,
$$
\begin{array}{l}
H^0(U,(\mc{R}^\vee)^{\otimes a}) \cong H^0(P,(\mc{R}_P^\vee)^{\otimes
  a})^{\textbf{PGL}(F)} =\\
\lt(\text{Sym}^a(E\otimes \text{Sym}^e F)\otimes
\lt[ \text{Sym}^{e-2}(F^\vee) \otimes \wedge^2(F^\vee) \rt]^{\otimes
  a}\rt)^{\textbf{PGL}(F)}.
\end{array}
$$

\medskip\noindent
A few words about this. First,
there is an obvious action of $\textbf{GL}(F)$ on
the space of global sections.  By inspection the center of
$\textbf{GL}(F)$ acts trivially on this space, so there is an induced
action of $\textbf{PGL}(F)$.  But it is more convenient to consider
this as a
$\textbf{GL}(F)$-representation and compute the
$\textbf{GL}(F)$-invariants. 

\medskip\noindent
Second, there is an obvious action of the symmetric group
$\mathfrak{S}_a$.  A full description of $H^0(U,(\mc{R}^\vee)^{\otimes
  a})$ should include a description as a
$\mathfrak{S}_a$-representation.  Likewise, there is an action of
$\textbf{GL}(E)$, and a full description should describe this action.
Finally, there is an obvious product map,
$$
H^0(U,(\mc{R}^\vee)^{\otimes a}) \otimes H^0(U,
(\mc{R}^\vee)^{\otimes b}) \rightarrow H^0(U, (\mc{R}^\vee)^{\otimes
  (a+b)}).
$$
A full description should describe this product map.

\medskip\noindent
In some sense the formula,
$$
H^0(U,(\mc{R}^\vee)^{\otimes a}) \cong 
\lt(\text{Sym}^a(E\otimes \text{Sym}^e F)\otimes
\lt[ \text{Sym}^{e-2}(F^\vee) \otimes \wedge^2(F^\vee) \rt]^{\otimes
  a}\rt)^{\textbf{GL}(F)},
$$
satisfies all of these conditions.  In another sense, it satisfies
none of them.  By inspection, the center of $\textbf{GL}(E)$ acts by
the character $t\mapsto t^a$.  Thus the irreducible decomposition of
this representation is of the form,
$$
\bigoplus_{u=0}^{\lfloor \frac{a}{2} \rfloor} \text{Sym}^{a-2u}(E)
\otimes (\wedge^2 E)^{\otimes u} \otimes_\QQ W_{a,u},
$$
where $W_{a,u}$ is a finite dimensional
$\mathfrak{S}_a$-representation over $\QQ$.  The irreducible
$\mathfrak{S}_a$-representations are in $1$-to-$1$ correspondence with
partitions $\lambda = (\lambda_1,\dots,\lambda_k)$ of $a$; denote by
$V_\lambda$ the corresponding representation.  There is an
irreducible decomposition,
$$
W_{a,u} = \bigoplus_\lambda W_{a,u,\lambda}\otimes V_\lambda,
$$
where $W_{a,u,\lambda}$ is a trivial $\mathfrak{S}_a$-representation
over $\QQ$.  

\medskip\noindent
The first goal is to compute, for each $(a,u,\lambda)$, the dimension of the
$\QQ$-vector space $W_{a,u,\lambda}$.  The second goal is to compute
the multiplication map on the direct summands.
For each pair of
integers
$a, b\geq
0$, for each triple of integers
$0 \leq u \leq \lfloor \frac{a}{2} \rfloor$, $0
\leq v \leq \lfloor \frac{b}{2} \rfloor$, $0 \leq w \leq \lfloor
\frac{a+b}{2} \rfloor$, and for each triple of a partition $\lambda$
of $a$, $\mu$ of $b$ and $\nu$ of $a+b$, there is a map
$T_{(a,b),(u,v,w),(\lambda, \mu,\nu)}$,
$$
\begin{array}{c}
\text{Sym}^{a-2u}(E) \otimes
\text{Sym}^{b-2v}(E) 
\otimes (\wedge^2 E)^{\otimes (u+v)} \otimes V_\lambda \otimes V_\mu
\otimes W_{a,u,\lambda} \otimes W_{b,v,\mu} \\
\rightarrow \text{Sym}^{a+b-2w}(E) \otimes (\wedge^2 E)^{\otimes w}
\otimes V_\nu \otimes W_{a+b,w,\nu}.
\end{array}
$$
The multiplication map is the sum of all these maps.

\medskip\noindent
This map is equivariant for all the obvious groups, so the map can be
decomposed by plethysm into a sum of smaller maps.
For the triple of partitions $\lambda,\mu,\nu$,
denote by $N_{\lambda,\mu,\nu}$ the corresponding
Littlewood-Richardson coefficient, so that,
$$
V_\lambda \otimes V_\mu \cong \bigoplus_\nu (V_\nu)^{\oplus
  N_{\lambda,\mu,\nu}}.
$$
Without loss of generality, suppose $a-2u \geq b-2v$.
There is an irreducible
decomposition,
$$
\text{Sym}^{a-2u}(E) \otimes \text{Sym}^{b-2v}(E) =
\bigoplus_{k=0}^{b-2v} \text{Sym}^{a+b-2u-2v-2k}(E) \otimes (\wedge^2
E)^{\otimes k}.
$$
It is clear that $T_{(a,b),(u,v,w),(\lambda,\mu,\nu)}$ is zero unless
$u+v \leq w \leq u+b-v$ and $N_{\lambda,\mu,\nu}\neq 0$.  And in this
case, the map is equivalent to a map of $\QQ$-vector spaces,
$$
T'_{(a,b),(u,v,w),(\lambda,\mu,\nu)}: (W_{a,u,\lambda}\otimes
M_{b,v,\mu})^{\oplus W_{\lambda,\mu,\nu}} \rightarrow W_{a+b,w,\nu}.
$$
The second goal is to compute each of these maps.

\medskip\noindent
The final goal is to compute $H^0(\Kbm{0,0}{\PP E/B,e},
(\mc{R}^\vee)^{\otimes a})$, not the space of sections
 $H^0(U,(\mc{R}^\vee)^{\otimes a})$.  This requires understanding 
for each $i=2,\dots,\lfloor \frac{e}{2} \rfloor$ the map sending
a section over $U$ to its \emph{principal part} along $\Delta_i$.
Because of this, the description of the space of sections should be
as explicit as possible.

\section{Computations} \label{sec-comps}
\marginpar{sec-comps}

\noindent
To compute the $\QQ$-vector spaces $W_{a,w,\lambda}$, it
suffices to consider the case $B=\SP \QQ$.  From now on $B=\SP \QQ$.  
Let $F$ be $\OO_B\oplus \OO_B$.  Let $\alpha, \beta$ be a basis for
$F$.  To simplify notation, the invertible sheaves spanned by
$\alpha$ and $\beta$ will also be written as $\alpha$ and $\beta$.  This
basis determines a maximal torus $\mathbb{G}_m(\alpha) \times
\mathbb{G}_m(\beta) \subset \textbf{GL}(F)$.  Every finite dimensional
representation $M$ of $\textbf{GL}(F)$ has an eigenspace
decomposition, or \emph{weight decomposition},
with respect to this torus,
$$
M = \bigoplus_{l,m \in \ZZ} M_{l,m}
$$
where the torus acts on $M_{l,m}$ by the character $(s,t) \mapsto s^l
t^m$.  Of course $M^{\textbf{GL}(F)} \subset M_{0,0}$.  Moreover, the
subspace $\oplus_{l+m=0} M_{l,m}$ is invariant under the action of
$\textbf{SL}(F)$.  The intersection of $\mathbb{G}_m(\alpha)\times
\mathbb{G}_m(\beta)$ with $\textbf{SL}(F)$ is a maximal torus.  Choose a
positive root
for this maximal torus.  The action of the positive root
determines an injective map,
$$
\chi: M_{-1,1} \rightarrow M_{0,0},
$$
and the following composition is an isomorphism,
$$
M^{\textbf{GL}(F)} \hookrightarrow M_{0,0} \rightarrow
\text{Coker}(\chi). 
$$

\medskip\noindent
In the case of $M = H^0(P,(\mc{R}_P^\vee)^{\otimes a})$, there is an action
of $\textbf{GL}(E)\times \mathfrak{S}_a$ on each of the weight spaces
$M_{l,k}$.  Because the category of representations of
$\textbf{GL}(E)\times \mathfrak{S}_a$ is semisimple, to determine the
representation $M^{\textbf{GL}(F)}$, it suffices to determine the
representations $M_{0,0}$ and $M_{-1,1}$.  The goal of this section is
to give a simple combinatorial description of the representations
$M_{0,0}$ and $M_{-1,1}$.

\medskip\noindent
As a representation of $\mathbb{G}_m(\alpha)\times
\mathbb{G}_m(\beta)$, there is a weight decomposition,
$$
\text{Sym}^e(F) \cong \alpha^e \oplus \dots \oplus
\alpha^{e-i}\beta^{i} \oplus \dots \oplus \beta^e.
$$
Therefore there is a decomposition,
$$
\text{Sym}^a(E\otimes \text{Sym}^e(F)) \cong \bigoplus_{\underline{i} \in
  I_{e,a}} \text{Sym}^{\underline{i}}(E) \otimes
\alpha^{l(\underline{i})} \beta^{m(\underline{i})} \otimes
\mathbf{b}^{\underline{i}},
$$
where $\underline{i}$ is a sequence of nonnegative integers
$(i_0,\dots,i_e)$, where $I_{e,a}$ is the set of sequences such that
$\sum_{k=0}^e i_k = a$, where $l(\underline{i})$ is defined to be
$\sum_{k}(e-k)i_k$, where $m(\underline{i})$ is defined to be
$\sum_{k} k i_k$, where $\mathbf{b}^{\underline{i}}$ is just a
placeholder $\mathbf{b}_0^{i_0} \cdot \dots \cdot \mathbf{b}_e^{i_e}$,
and where
$$
\text{Sym}^{\underline{i}}(E) = \bigotimes_{k=0}^e \text{Sym}^{i_k}(E).
$$
Of course $I_{e,a}$ is in bijection with the set of Young tableaux with at
most $a$ rows and at most $e$ columns, and $m(\underline{i})$ is the
number of squares in the corresponding tableaux.  

\medskip\noindent
Similarly, there is a weight decomposition,
$$
\text{Sym}^{e-2}(F^\vee) \otimes \wedge^2(F^\vee) \cong A^{e-1}B
\oplus \dots \oplus A^{e-j}B^j \oplus \dots \oplus A B^{e-1},
$$
where $(A,B)$ is the dual ordered basis to $(\alpha,\beta)$.  Therefore
there is a decomposition,
$$
\lt[ \text{Sym}^{e-2}(F^\vee) \otimes \wedge^2(F^\vee) \rt]^{\otimes
  a} \cong 
\bigoplus_{\underline{j} \in J_{e,a}} A^{ea-|\underline{j}|}
B^{|\underline{j}|} \otimes \mathbf{c}_{\underline{j}},
$$
where $\underline{j}$ is a sequence of integers $(j_1,\dots,j_a)$ such
that each $1\leq j_\kappa \leq e-1$, where $|\underline{j}| = j_1 +
\dots + j_a$, and 
where
$\mathbf{c}_{\underline{j}}$ is a placeholder $\mathbf{c}_{j_1}\otimes
\dots \otimes \mathbf{c}_{j_a}$.  The action of $\mathfrak{S}_a$ is
the obvious one: for $\sigma \in \mathfrak{S}_a$,
$$
\sigma \cdot \mathbf{c}_{j_1}\otimes \dots \otimes
\mathbf{c}_{j_\kappa} \otimes \dots
\otimes\mathbf{c}_{j_a} =
\mathbf{c}_{j_{\sigma(1)}} \otimes \dots \otimes
\mathbf{c}_{j_{\sigma(\kappa)}} \otimes \dots \otimes
\mathbf{c}_{j_{\sigma(a)}}.
$$

\medskip\noindent
Putting this together, the weight-$(0,0)$ subspace of
$M=H^0(P,(\mc{R}_P^\vee)^{\otimes a})$ is,
$$
M_{0,0} = \bigoplus_{(\underline{i},\underline{j})\in K_{e,a}}
\text{Sym}^{\underline{i}}(E) \otimes \alpha^0\beta^0 \otimes
\mathbf{b}^{\underline{i}} \otimes \mathbf{c}_{\underline{j}},
$$
where, as before, $\mathbf{b}^{\underline{i}}$ and
$\mathbf{c}_{\underline{j}}$ are placeholders, and where $K_{e,a}$ is,
$$
K_{e,a} = \lt\{ (\underline{i},\underline{j})\in I_{e,a} \times
J_{e,a} | m(\underline{i}) = |\underline{j}| \rt\}.
$$
And the weight-$(-1,1)$ subspace is,
$$
M_{-1,1} = \bigoplus_{(\underline{i},\underline{j}) \in K'_{e,a}} 
\text{Sym}^{\underline{i}}(E) \otimes \alpha^{-1}\beta^1 \otimes
\mathbf{b}^{\underline{i}} \otimes \mathbf{c}_{\underline{j}},
$$
where,
$$
K'_{e,a} = \lt\{ (\underline{i},\underline{j}) \in I_{e,a} \times
J_{e,a} | m(\underline{i}) = 1 + |\underline{j}| \rt\}.
$$

\medskip\noindent
To relate this to the spaces $W_{a,u}$, choose an ordered basis
$(\epsilon,\eta)$ of $E$.  This determines a maximal torus
$\mathbb{G}_m(\epsilon) \times \mathbb{G}_m(\eta)$ in $\text{GL}(E)$.
The weights of $M_{0,0}$ and $M_{-1,1}$ with respect to this torus are
$(a-w,w)$, $w=0,\dots,a$.  The weight decompositions are,
\begin{eqnarray*}
M_{0,0} = \bigoplus_{u=0}^a
\bigoplus_{(\underline{h},\underline{i},\underline{j}) \in K_{e,a,u}}
\epsilon^{a-u} \eta^u \otimes \alpha^0 \beta^0 \otimes
\mathbf{a}^{\underline{h}} \otimes \mathbf{b}^{\underline{i}} \otimes
\mathbf{c}_{\underline{j}} \\
M_{-1,1} = \bigoplus_{u=0}^a
\bigoplus_{(\underline{h},\underline{i},\underline{j}) \in K'_{e,a,u}}
\epsilon^{a-u} \eta^u \otimes \alpha^{-1} \beta^1 \otimes
\mathbf{a}^{\underline{h}} \otimes \mathbf{b}^{\underline{i}} \otimes
\mathbf{c}_{\underline{j}},
\end{eqnarray*}
where $\underline{h}$ is a sequence of nonnegative integers
$(h_0,\dots,h_e)$, where $\mathbf{a}^{\underline{h}}$ is a placeholder
$\mathbf{a}_0^{h_0} \cdot \dots \cdot \mathbf{a}_e^{h_e}$, and where,
\begin{eqnarray*}
K_{e,a,u} = \{ (\underline{h},\underline{i},\underline{j}) |
(\underline{i},\underline{j}) \in K_{e,a},  \text{ each } 0\leq h_k \leq
i_k, \text{ and } \sum_k h_k = u \}, \\
K'_{e,a,u} = \{ (\underline{h},\underline{i},\underline{j}) |
(\underline{i},\underline{j}) \in K'_{e,a}, \text{ each } 0\leq h_k \leq
i_k, \text{ and } \sum_k h_k = u \}.
\end{eqnarray*}

\medskip\noindent
For each $e,a,u$, there is a $\mathfrak{S}_a$-representation
$W'_{e,a,u}$, unique up to isomorphism, such that,
$$
\bigoplus_{(\underline{h},\underline{i},\underline{j})\in K_{e,a,u}}
\mathbf{a}^{\underline{h}} \otimes \mathbf{b}^{\underline{i}} \otimes
\mathbf{c}_{\underline{j}} 
\cong W'_{e,a,u} \oplus 
\bigoplus_{(\underline{h},\underline{i},\underline{j})\in K'_{e,a,u}}
\mathbf{a}^{\underline{h}} \otimes \mathbf{b}^{\underline{i}} \otimes
\mathbf{c}_{\underline{j}}.
$$ 
Finally, the relation to $W_{a,u}$ is,
$$
W'_{e,a,u} \cong \bigoplus_{0\leq u' \leq u} W_{a,u}.
$$
Because the category of $\mathfrak{S}_a$-representations is
semisimple, determining all the representations $W'_{e,a,u}$
determines $W_{a,u}$.  Therefore, determining all the
$\mathfrak{S}_a$-sets $K_{e,a,u}$ and $K'_{e,a,u}$ determines all the
representations $W_{a,u}$.
In particular,
$$
\text{dim}_\QQ W_{a,u} = (\#K_{e,a,u} - \#K'_{e,a,u}) -
(\#K_{e,a,u-1} - \#K'_{e,a,u-1}),
$$
where $K_{e,a,-1}$ and $K'_{e,a,-1}$ are defined to be the empty set.

\section{The case $e=2$ and the case $a=1$}\label{sec-ce2}
\marginpar{sec-ce2}

\subsection{The case $e=2$}\label{subsec-ce2}
\marginpar{subsec-ce2}
In the case $e=2$, the sets $K_{2,a,u}$ and $K'_{2,a,u}$ can be
computed explicitly.  For each $a$, $J_{2,a} = \{ (1,\dots,1) \}$.  In
particular, the action of $\mathfrak{S}_a$ is trivial.  Also,
\begin{eqnarray*}
K_{2,a} = \lt\{ (i_0,a-2i_0,i_0) | 0 \leq i_o \leq \lfloor \frac{a}{2}
\rfloor \rt\}, \\
K'_{2,a} = \lt\{ (i_0,a-1-2i_0,i_0+1) | 0 \leq i_o \leq \lfloor \frac{a-1}{2}
\rfloor \rt\}.
\end{eqnarray*}
For each $\underline{i} = (i_0,i_1,i_2)$, the number of
$\underline{h}$'s is $(i_0+1)(i_1+1)(i_2+1)$.  Therefore,
\begin{eqnarray*}
\text{dim}_\QQ M_{0,0} = \sum_{\i=0}^{\lfloor \frac{a}{2} \rfloor}
(i_0+1)^2(a+1 - 2i_0), \\
\text{dim}_\QQ M_{-1,1} = \sum_{\i=0}^{\lfloor \frac{a-1}{2} \rfloor}
(i_0+1)(i_0+2)(a - 2i_0).
\end{eqnarray*}
If $a$ is odd, say $a=2b+1$, then $\lfloor \frac{a}{2} \rfloor =
\lfloor \frac{a-1}{2} \rfloor = b$, and the difference of dimensions
is,
$$
\text{dim}_\QQ M_{0,0} - \text{dim}_\QQ M_{-1,1} = \sum_{i=0}^b
(i_0+1)(3i_0-2b).
$$
After some tedious calculation, this comes out as $0$ (since it is a
cubic polynomial in $b$, it suffices to consider $b=0,1,2$).  
If $a$ is even, say $a=2b$, then $\lfloor \frac{a}{2} \rfloor = b$ and
$\lfloor \frac{a-1}{2} \rfloor = b-1$.  So the difference of
dimensions is,
$$
\text{dim}_\QQ M_{0,0} - \text{dim}_\QQ M_{-1,1} = (b+1)^2 +
\sum_{i_0=0}^{b-1} (3i_0^2 - (2b-4)i_0 - (2b-1)).
$$
After more tedious calculation, this comes out as,
$$
\text{dim}_\QQ M_{0,0} - \text{dim}_\QQ M_{-1,1} =
\frac{1}{2}(b+2)(b+1),
$$
(since it is a cubic polynomial in $b$, again it suffices to consider
$b=0,1,2$).  
This is precisely the dimension of $\text{Sym}^b(\text{Sym}^2 E)$, so
the computation agrees with the result of Section~\ref{sec-e2}.  

\subsection{The case $a=1$} \label{subsec-a1}
The case $a=1$ is also simple.  The elements of $K_{e,1}$ are
$s^j = (\underline{i}^j,(j))$, $1\leq
j \leq e-1$, where 
$$
i^j_k = \lt\{ \begin{array}{ll}
1, & k = j \\
0, & \text{otherwise}
\end{array} \rt.
$$
The elements of $K'_{e,1}$ are $t^j = (\underline{i}^{j}, (j-1))$,
$2\leq j \leq e$.  In particular $K_{e,1}$ and $K'_{e,1}$ have the
same number of elements.  For each element, there are $2$ choices for
$\underline{h}$.  Thus $K_{e,1,u}$ and $K'_{e,1,u}$ have the same
number of elements for $u=0,1$.  Therefore,
$$
H^0(\Kbm{0,0}{\PP E/B,e}, \mc{R}^\vee) = (0).
$$

\marginpar{subsec-a1}




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