text:
\section{Introduction and description of results}

In heterotic $E_8\times E_8$ string theory
models of $N=1$ supersymmetry in 4D arise by compactification
on a Calabi-Yau threefold $X$ with vector bundle $V$.
Originally the case of $V$ the tangent bundle was considered
which led to an unbroken gauge group $E_6$ (times a hidden $E_8$).
The generalisation to an $SU(n)$ bundle $V$ gives unbroken GUT groups like
$SO(10)$ and $SU(5)$ (we will in the following focus on the visible sector
and may assume an $E_8$ bundle $V_2$ embedded in the second $E_8$).
To be able to handle the $SU(n)$ bundle $V$ most explicitly we assume that $V$
arises by the spectral cover construction
for bundles on an $X$ which has an elliptic fibration $\pi:X\rightarrow B$.
This description uses a surface $C\subset X$ given by
an $n$-fold (ramified) cover of the base $B$ and a line bundle $L$ on $C$.
Assuming $C$ ample the continuous moduli of $V$
come just from the deformations of $C$ in $X$; these
are given by the polynomial
coefficients entering the defining equation of $C$.

For a holomorphic curve $b$, arising as support of a
world-sheet instanton, to contribute to the superpotential
one assumes that $b$ is isolated and rational.
In the following we want to bring to bear the explicit information about the
bundle $V$ provided by the spectral cover construction; we will
therefore restrict us to the case of horizontal curves, i.e.~curves $b$
lying in $B$.
If the world-sheet instanton
contribution $W_b$ supported on $b$ is generically
nonzero one wants to have the finer information how this contribution
depends on the bundle (and possibly complex structure)
moduli. This is described by the Pfaffian prefactor
$Pfaff$ of the classical instanton contribution $e^{i\int_b J}$.
$Pfaff$ is given by a generally complicated determinantal
expression in the bundle moduli.

Because of holomorphic dependence crucial is here
the sublocus in the moduli space where $Pfaff$ vanishes.
This leads to an identification of $Pfaff$ with a geometrical
determinantal expression whose vanishing controls the vanishing of $Pfaff$.
In some cases $Pfaff$ vanishes
identically in the moduli for well-understood reasons.
More interesting is the case where $Pfaff$ is generically nonvanishing
in the moduli. Especially interesting, and the raison d'\^{e}tre of the
present paper, is the case where the Pfaffian shows some structure,
i.e.~is not as complicated as generically under the given circumstances.
This means more precisely that one has a nontrivial factorisation
like $Pfaff=f^k$ with $k>1$, or $Pfaff=fg$ or even $Pfaff=f^kg$.
This can simplify the search for zeroes of the superpotential,
and in a case with a multiplicity $k>1$ also of its derivative.

A small set of three examples of such behavior was found [\ref{BDO}]
by using computer calculation of some large determinants.
This had the character (like a fourth example of vanishing Pfaffian)
of a surprising simplification arising by an intransparent
(purely numerical) brute-force computation.
Our goal here is to get a conceptual understanding
(i.e.~beyond doing algebra for concrete matrix expressions)
of the way such a simplifying structure arises.
In the present paper we explain the case of the vanishing Pfaffian
and the occurrence of the factor $f$; here explaining means we give
a conceptual, non-computational reason
for the examples and generalise them to further cases;
the question of multiplicity $k$ of $f$ will be dealt with in a separate paper
as will be the treatment of the somewhat differently behaved
second factor $g$ (the Giant $Q_{11}$ in Example 1, c.f.~below).

In the rest of this {\em Introduction} we will first recall more precisely
the numerical results of [\ref{BDO}]; then we will describe what
is derived conceptually in the present paper and state the generalisations.
In {\em sect.~\ref{spectralcovermethod}} we recall some facts about
the spectral cover construction of bundles.
In {\em sect.~\ref{restriction to E}} we apply this to our case by restricting
the construction to the elliptic surface ${\cal E}$
lying above the instanton curve $b\subset B$ in the elliptic fibration,
providing a spectral curve $c$ and a line bundle $l$ on it.
We investigate some special loci in moduli space
for one of our main example classes, the $SU(3)$ bundles.
In {\em sect.~\ref{WS instantons}}
we recall from [\ref{Wittenhet}] and [\ref{BDO}]
some general conditions for world-sheet instantons to contribute to the
superpotential and make these conditions explicit in the parameters of the
construction.

In {\em sect.~\ref{section reduction}}
we explain the main idea of the paper: how a factor in the Pfaffian
can be explained by reduction to a different line bundle $\bar{l}$
which is 'simpler' than $l$.
In {\em sect.~\ref{Equality condition}} we describe how this
idea can be practically implemented if one can imposes
a certain ('equality') condition such that the resulting question for
$\bar{l}$ is again controlled by a determinantal expression.
Here we give an (in a certain sense) exhaustive list of cases
where the reasoning described applies.
In {\em sect.~\ref{The vanishing condition}} we describe how the case of
a generically vanishing Pfaffian fits into the set-up outlined so far.
In {\em sect.~\ref{difference subsection}}, which has the character of an
insert, we show how the argument described so far
has to be supplemented if some assumptions are weakened;
this will be relevant for Example 1.
In {\em sect.~\ref{Examples}} we discuss in detail the Examples of [\ref{BDO}]
and corresponding generalisations to which they give rise.

The {\em Appendices} collect auxiliary investigations.
App.~\ref{Polynomial factors} gives the polynomial factors of $Pfaff$
in the Examples.
App.~\ref{rational curves} collects facts about
(horizontal) rational curves in the Calabi-Yau threefold $X$.
App.~\ref{Lemmata} gives Lemmata
conc.~the cohomological contribution criteria for instantons.
App.~\ref{augmentation} points to a different cohomological method,
alternative to the procedure in sect.~\ref{section reduction}.
App.~\ref{negative lambda} shows how the case of a negative bundle parameter
$\lambda$ is related to the usual procedure for $-\lambda$
(cf.~Example 1). App.~\ref{Reduction cases}
shows how one obtains the exhaustive lists
for cases of reduction or vanishing Pfaffian.
App.~\ref{concrete matrix representations} gives
details connecting the structural investigations and concrete
matrix representations.
App.~\ref{chi = 0 app} illustrates the special case $\chi=0$ and shows a
case of $Pfaff\equiv 0$ algebraically.
App.~\ref{sym} establishes needed facts and notation for symmetrized
tensor products and resultants.

\subsection{\label{Some experimental results}Some experimental results}
Some Examples were found experimentally via
computer evaluations of the occurring large determinants [\ref{BDO}].
The cases concerned spectral $SU(3)$ bundles
of bundle parameters $\lambda\in \frac{1}{2}{\bf Z}$ and
$r=\eta b\in {\bf Z}$ (they are defined in
sect.~\ref{spectralcovermethod} and \ref{coordinates for curve case}
and are needed here only to display the results in an overview)
with $b$ the base of the ${\bf P^1}$-fibration on $B={\bf F_k}$
\begin{center}
\begin{tabular}{|c||c|c|c||c|}
\hline
Example & $\lambda$ & $r$ & ${\bf F_k}$ & $Pfaff$\\
\hline\hline
$1$ & $-5/2$ & $4$ & ${\bf F_1}$ & $f^{11}g$\\
\hline
$2$ & $3/2$  & $5$ & ${\bf F_1}$ & $f^4$\\
\hline
$3$ & $3/2$  & $1$ & ${\bf F_2}$ & $0$\\
\hline
$4$ & $3/2$  & $2$ & ${\bf F_2}$ & $f^4$\\
\hline
\end{tabular}
\end{center}

The detailed polynomial expressions of the factors of $Pfaff$ are
given in app.~\ref{Polynomial factors}.

\subsection{Conceptual explanations and generalisations}

When we make
in sect.~\ref{restriction to E} the transition from
a description of $V$ over $X$ via the spectral surface $C$ and the line
bundle $L$ on it to the corresponding notions on ${\cal E}$ we will,
besides the spectral curve $c:=C|_{{\cal E}}$, also introduce the corresponding
line bundle $L|_c$. And as we are able, under our assumptions, to describe
the line bundle $L$ on $C$ as a restriction $L=\underline{L}|_C$
from a line bundle $\underline{L}$ on $X$, we will similarly be able to describe
the line bundle $L|_c$ on $c$ as a restriction $L|_c=l|_c$
from a line bundle $l$ on ${\cal E}$; actually, of course, $l=\underline{L}|_{{\cal E}}$.
For all these relations cf.~sect.~\ref{twist class}.

We note that when we wish to emphasize the dependence on a modulus $t$
we write $V_t$, $C_t$ and so on for the corresponding objects
(cf.~sect.~\ref{Overview and summary}). The
abstract modulus $t$ will actually turn out to be a modulus of the
surface $C$, respectively the curve $c$, and will be given concretely
by polynomial coefficients of its defining equation
(this will be studied very explicitly for the case of $SU(3)$ bundles,
cf.~(\ref{equ spec curve}) and (\ref{coefficient expressions}), for example).

We usually take $\chi=c_1(B)\cdot b$ to be $0$ or $1$,
with $1$ the relevant case, cf.~sect.~\ref{chi subsection}; the class
in ${\cal E}=\pi^{-1}(b)$
of the spectral cover curve $c$ over $b$ is $ns+rF$ with $s=\sigma|_{{\cal E}}$
the section of the elliptic surface ${\cal E}$ and $F$ the class
of the elliptic fibre.

We will explain and generalise the experimental results
in sect.~\ref{Some experimental results}.
For the degenerate case of Example 3 a conceptual interpretation
from the existence of nontrivial sections of $l(-F)|_c$,
cf.~sect.~\ref{The vanishing condition} and below, will be given
(besides an algebraic explanation, cf.~sect.~\ref{chi = 0 app}).
Example 1 has a somewhat exceptional status, cf.~sect.~\ref{Exceptional} and
\ref{Ex 1 sect}.
Examples 2 and 4 are covered below
(full details for Examples 2,3,4 are given in the sections indicated
below).

\newpage

{\em \underline{Vanishing Pfaffian:} $Pfaff\equiv 0$}

Let us begin with the case where the Pfaffian vanishes identically.
This happens in the following cases (in sect.~\ref{red cases for bar al > n},
\ref{The vanishing condition} and app.~\ref{cases al > n} it is described
under which assumptions these cases give an exhaustive description; a
corresponding remark applies below)
\begin{itemize}
\item $SU(n)$ bundles with $\chi=0, r=1, \lambda \in \frac{1}{2}+{\bf Z}^{\geq 1}$
\item $SU(3)$ bundles with $r\not\equiv \chi (2)$ and $\lambda =3/2$
\end{itemize}

This constitutes a vast generalisation of Example 3.

{\em \underline{Factorising Pfaffian:} $Pfaff=fg$}
\hspace{2cm}(including the case $g=f^m$)

In the main case the Pfaffian factorises such that $f|Pfaff$.
More precisely we will identify a concrete factor
$f=\det \bar{\iota}_1$ in $Pfaff=\det \iota_1$
where\footnote{with $\beta = \frac{r+\chi}{2}-\lambda(r-n\chi)-1$ and
$\bar{\beta}=(r+\chi)-\frac{1}{2}\frac{n+1}{n-1}r-\frac{1}{2}(r-n\chi)-1$}
(cf.~sect.~\ref{red cases for bar al = n} and app.~\ref{list subsection})
\begin{eqnarray}
\iota_1: H^1\Big({\cal E}, l(-F-c)\Big)\longrightarrow H^1\Big({\cal E}, l(-F)\Big) ,&& \;\;
l(-F)={\cal O}_{{\cal E}}\Big(n(\lambda + \frac{1}{2})s+\beta F\Big)\;\;\\
\bar{\iota}_1: H^1\Big({\cal E}, \bar{l}(-F-c)\Big)\longrightarrow H^1\Big({\cal E}, \bar{l}(-F)\Big),
&&\;\;
\bar{l}(-F)={\cal O}_{{\cal E}}\Big(ns+\bar{\beta}F\Big)\;\;
\end{eqnarray}
\begin{itemize}
\item $SU(3)$ bundles
\begin{center}
\begin{tabular}{|c||c|c|c||c|c||c|c|}
\hline
$\sharp$ &$\chi$ & $r$ & $\lambda $ & $l(-F)={\cal O}_{{\cal E}}(\cdot)$
& $\bar{l}(-F)={\cal O}_{{\cal E}}(\cdot)$ & $\deg Pfaff$ & $\deg f$ \\
\hline
\hline
1 & $0$ & $2$& $ \frac{3}{2}, \frac{5}{2}, ...$
& $3(\lambda + \frac{1}{2})s-2\lambda F$ & $3s-2F$ & $6(\lambda^2-\frac{1}{4})$ & $3$ \\
\hline
2 & $\geq 1$ & $5\chi$ & $\frac{3}{2}$
& $6s-F$ & $3s-F$ & $20\chi$ & $5\chi$ \\
\hline
3 & $\chi$ & $\geq 5\chi, \equiv \chi(2)$ & $\frac{3}{2}$
&$6s-(r-5\chi+1)F$ &$3s-(\frac{r-5\chi}{2}+1)F$ & $6r-10\chi$ &
$\frac{3r-5\chi}{2}$\\
\hline
4 & $0$ & $4$ & $\frac{5}{2}$ & $9s-9F$ & $3s-3F$& $72$ & $6$\\
\hline
5 & $1$ & $5$ & $\frac{5}{2}$ & $9s-3F$ & $3s-F$ & $62$ & $5$\\
\hline
\end{tabular}
\end{center}
Case 3 constitutes a vast generalisation of Example 2 and 4
(why case 2 is listed separately besides case 3 will only become clear
later and is of no concern here).
\item $SU(4)$ bundles
\begin{center}
\begin{tabular}{|c||c|c|c||c|c||c|c|}
\hline
$\sharp$ & $\chi$ & $r$ & $\lambda $ & $l(-F)={\cal O}_{{\cal E}}(\cdot)$
& $\bar{l}(-F)={\cal O}_{{\cal E}}(\cdot)$ & $\deg Pfaff$ & $\deg f$ \\
\hline
\hline
1 & $\geq 1$  & $9\chi$ & $1$ & $6s-F$ & $4s-F$ & $20\chi$ & $9\chi$ \\
\hline
2 & $0$ & $3$ & $\frac{3}{2}$ & $8s-4F$ & $4s-2F$ & $24$ & $4$\\
\hline
\end{tabular}
\end{center}
\end{itemize}
So case 5 of $SU(3)$ and case 1 of $SU(4)$ have a second factor
(there certainly $Pfaff\neq f^k$).

\subsection{\label{Overview and summary}Overview and summary}

As the issue in question - the vanishing behaviour of the Pfaffian prefactor
(of a world-sheet instanton superpotential) in dependence
on the vector bundle moduli -
necessarily uses a heavy amount of algebraic-geometric notions it may be
useful to provide here also a nontechnical overview of the more detailled
investigations which follow in the later sections
(herein we allow ourselves to give an only approximate description of various
issues whose more detailed aspects are dealt with in the main text).

According to the main contribution criterion which will be recalled in
sect.~\ref{contribution criteria} one has as precise vanishing criterion
for the Pfaffian
\begin{eqnarray}
Pfaff(t)=0& \Longleftrightarrow & \Gamma\Big(b, V_t|_b\otimes {\cal O}_b(-1)\Big)
\; \cong \; \Gamma\Big(c_t, l(-F)|_{c_t}\Big)\; \neq \; 0
\end{eqnarray}

Clearly, if the line bundle\footnote{the notations for this
and the following line bundles are used -
with their here indicated meaning - only in this subsection}
$L={\cal O}_c(D)$ over $c$
whose possible sections are concerned
here is a tensor product $L_1\otimes L_2={\cal O}_c(D_1)\otimes {\cal O}_c(D_2)$
of line bundles which themselves {\em have}
a nontrivial section,
then also their product bundle, in question here, would have such a section.
For example, for one - auxiliary - factor $L_2$ the existence of such a section
might be assured by a general argument, like that the line bundle is
associated to an effective divisor; thereby the problem of existence of a
nontrivial section would have been reduced from the original problem
for $L$ to the other factor $L_1$. Because $D-D_1$ is, as assumed, effective
one might say that the problem is reduced to a smaller line bundle
\begin{eqnarray}
D-D_1 \;\; \mbox{effective}\;\;\;\;\;\; & \longrightarrow &
\;\;\;\;\;\; \mbox{reduction from $L$ to $L_1$}
\end{eqnarray}
Actually
there will be an even more important sense of such a 'reduction in size'
when going from $L$ to $L_1$ described later (the relevant vanishing condition
will be expressed by a determinant of a matrix of smaller size; we start
to develop this technically in sect.~\ref{section reduction}).

Now, one of the easiest possibilities for an accessible criterion for the
existence of a section would arise if for $L_1$ a similar determinantal
expression could be found (whose vanishing controls the existence of a
nontrivial section) as for $L$.
For this one considers for $L_1$ similar cohomological sequences as for $L$
and has to see whether again, under certain conditions, one can relate
the space of sections in question to a map between $H^1$-cohomologies
of line bundles over ${\cal E}=\pi^{-1}(b)$ where the latter spaces have
{\em equal dimension}; if that {\em equality condition} (which we study
in sect.~\ref{Equality condition})
is satisfied one gets indeed again a determinantal
expression which controls the issue in question here, the existence
of a nontrivial section of $L_1$
\begin{eqnarray}
\mbox{equality condition}\;\;\;&\longrightarrow &
\;\;\; \mbox{determinantal expression controlling $h^0(c, L_1)\neq 0$}
\;\;\;\;\;\;
\end{eqnarray}

Occasionally, if another special condition is fulfilled, it may also
happen that the line bundle $L_1$ does always\footnote{independently of any
special choice of moduli (such as making a determinantal expression vanish)}
have a section; then, of course, the same holds for $L$ and the Pfaffian
will vanish identically on the moduli space (the corresponding {\em vanishing
condition} is studied in sect.~\ref{The vanishing condition})
\begin{eqnarray}
\mbox{vanishing condition}\;\;\;\;\;\;&\longrightarrow &
\;\;\;\;\;\; Pfaff \equiv 0
\end{eqnarray}

In carrying out this program there occurs a difficulty which deserves to be
mentioned. In the reduction step one needed to show that a certain line bundle
(the bundle $L_2$) above is related to an {\em effective} divisor.
Now in many cases when line bundles on spectral surfaces $C$ in $X$
or on spectral curves $c$ in ${\cal E}$ are considered it is enough
to restrict attention to objects induced by restriction
from the ambient space, be it $X$ or ${\cal E}$. For example, in our case
where line bundles over the curve $c\subset {\cal E}$ are relevant,
by restricting generality in the manner indicated and looking only
to objects like ${\cal O}_{{\cal E}}(\alpha s + \beta F)$, where $b$ and $F$ are
the rational base and elliptic fibre of the elliptic surface ${\cal E}$,
the question of effectiveness reduces simply to the question whether
integral coefficients are non-negative. This way of doing things suffices
for some major examples, like the Examples 2 and 4 which are repeatedly
studied in this paper like in [\ref{BDO}]. However there are other examples
which show quite interesting behaviour where employing this restriction
is not sufficient, like in the Example 1. More precisely
what happens is this: although $L_2$ arises as a restriction from ${\cal E}$ to $c$
of a line bundle $L_2'={\cal O}_{{\cal E}}(D_2')$ on ${\cal E}$
it is not necessary that $D_2'$ is effective as a divisor on ${\cal E}$; this is
only sufficient
\begin{eqnarray}
D_2'\;\;\; \mbox{effective} \;\;\;
& \stackrel{\mbox{\normalsize{$\not\Longleftarrow$}}}{\Longrightarrow} &
\;\;\; D_2=D_2'|_c \;\;\; \mbox{effective}
\end{eqnarray}
(the difference between the easily applicable, but too strong
condition of effectiveness of $D_2'$ and the precise condition of
effectiveness of $D_2$ is studied in sect.~\ref{difference subsection}).
What one needs to study actually is whether $D_2$ is effective
as a divisor on $c$ which clearly is more difficult. Nevertheless
cases like the example 1 mentioned need for their 'factorisation reduction'
the employment of such more subtle line bundles.

After having gone in the required technical detail
through the steps described above
in sections \ref{section reduction} to \ref{difference subsection}
we will apply these methods to our main Examples 2 or 4 and 1
in sect.~\ref{n=3, la =3/2} and \ref{Ex 1 sect}, respectively.
All other cases are listed in sect.~\ref{red cases for bar al = n}
and app.~\ref{list subsection}.

\newpage

\section{\label{spectralcovermethod}Spectral $SU(n)$ bundles over the elliptic
Calabi-Yau threefold $X$}

\resetcounter

In case $X$ admits an elliptic fibration $\pi: X\rightarrow B$ with
a section $\sigma$ one can describe the bundle $V$
explicitly via the spectral cover $C$ of $B$:
the data of an $SU(n)$ bundle are encoded by
an $n$-fold ramified cover surface $C$
of the base $B$, which datum comes down to a class $\eta$ in $H^{1,1}(B)$,
and a line bundle $L$ over $C$; the latter,
under the standing assumption $h^{1,0}(C)=0$, reduces to the datum given
by a class $\gamma$ in $H^{1,1}(C)$ (whose non-triviality is crucial to get
chiral matter [\ref{C}]).

\subsection{\label{Elliptic CY}
The elliptic Calabi-Yau space $X$ over the surface $B$}

Before describing the bundle in greater detail let us first elucidate
the structure of the space $X$. Thre threefold is actually given as a
hypersurface in an ambient four-fold ${\cal W}_B$
which itself is defined as a ${\bf P^2}$-bundle over the base $B$
\begin{eqnarray}
{\cal W}_B&=& {\bf P}({\cal L}^2\oplus {\cal L}^3\oplus {\cal O})
\end{eqnarray}
where ${\cal L}=K_B^{-1}$.
The homogeneous coordinates in the fibre ${\bf P^2}$ are denoted by $x,y,z$
and $X$ is the divisor given by vanishing of the defining equation
$zy^2=4x^3-g_2xz^2-g_3z^3$ where $g_2$ and $g_3$ are sections of
${\cal L}^4$ and ${\cal L}^6$.

The base $B$ is actually either a del Pezzo surface
(including ${\bf P^2}$ and some blow-ups), a Hirzebruch surface
(plus some blow-ups) or an Enriques surface.
As we will have to consider rational curves, as support of the
world-sheet instanton, in $B$ we recall in app.~\ref{rational curves}
the relevant facts in this regard and list the possible cases.

\subsection{The spectral cover surface $C$ over $B$}

The idea of the spectral cover description of an $SU(n)$ bundle $V$ is to
consider first the bundle over an elliptic fibre $F$,
and then to paste together these descriptions allowing global twisting data.
$V$ (assumed to be fibrewise semistable)
decomposes fibrewise as a direct sum of line bundles of degree zero,
encoded by a set of $n$ points summing up to
zero in the group law:
this is the point $p_0=(0,1,0)$, the point
at infinity $x=y=\infty$ in affine coordinates with $z=1$;
it is globalized by the section $\sigma$.
Letting
this vary over the base $B$ one gets a hypersurface $C \subset X$,
a ramified $n$-fold cover of $B$. Denoting the cohomology
class of $\sigma(B)\subset X$ by $\sigma$ one finds, allowing the twist by a
line bundle ${\cal M}={\cal O}_B(M)$ over $B$ with $c_1({\cal M})=[M]=\eta \in H^{1,1}(B)$
\begin{eqnarray}
C=n\sigma + \eta
\end{eqnarray}
For the $n$-tuple of points $\{ p_i \; | \; i=1, \dots , n\}$ (on a fibre)
there exists a unique (up to a factor in ${\bf C^*}$) meromorphic function
$w$ of divisor $(w)=\sum_{i=1}^n (p_i - p_0)$: this means
in the standard convention for divisors
of meromorphic functions that $w$ has zeroes at the $p_i$
(i.e.~$f$ would be holomorphic for $(f)$ effective). This $w$ is,
given in inhomogeneous form, a polynomial in $x$ and $y$
(which have a double and triple pole at $p_0$, respectively)
\begin{eqnarray}
\label{locus}
w & = & a_0+a_2x+a_3y+\dots \, a_nx^{(n-3)/2}y\; =\; 0
\end{eqnarray}
(this is for $n$ odd\footnote{for $n=3$ in sect.~\ref{Ri section}
we call $a_0=C, a_2=B, a_3=A$ and write
$D_m=\sum_{i=0}^m d_i u^i v^{m-i}$ for $D=C,B,A$ when restricting the
consideration to the projective line $b\subset B$
with its homogeneous coordinates $(u,v)$; the subscript $m$ denotes
the degree (identified as $r, r-2\chi, r-3\chi$
in sect.~\ref{coordinates for curve case}) of the homogeneous polynomial $D$
(the reader will not confuse the
$a_i$ in (\ref{locus}) with the coefficients of $A$)};
for $n$ even the last term reads $a_nx^{\frac{n}{2}}$).
Globally $C$ is given as the locus (\ref{locus})
with $w$ a section of ${\cal O}(\sigma)^n \otimes {\cal M}$
(here ${\cal M}$ being understood as pulled back to $X$)
and $a_i\in H^0(B, {\cal M} \otimes {\cal L}^{-i})$.

$\eta$ has to fulfill some conditions. As the spectral cover
is an actual surface one needs
\begin{eqnarray}
C \;\;\;\; \mbox{effective} \;\; \;\; \;\; \;\; \;\; \;\;
(\; \Longleftrightarrow \;\;\; \eta \geq 0)
\end{eqnarray}
(i.e. $\eta$ effective).
Second, to guarantee [\ref{FMWIII}]
that $V$ is a {\em stable} vector bundle, one needs [\ref{irred}]
\begin{eqnarray}
\label{irreducibility}
C \;\; \mbox{irreducible} & \Longleftrightarrow &\;\;\;\;\;\;\;
\{a_0=0\} \; \mbox{irreducible} \;\;\;\;\;\;\; \;\;\;
\;\;\; (\Longleftrightarrow \;  \eta \cdot b \geq 0)\\
\label{irreducibility effectivity}
&&\mbox{and} \,\; \{a_n=0\}=C\cdot \sigma \;\;
\mbox{effective}\;\;\; (\Longleftrightarrow \; \eta - n c_1 \geq 0)
\end{eqnarray}

\subsubsection{The moduli of $V$}

The isomorphism class of $V$ in this set-up will be determined by
$C$ and a certain line bundle $L$ over it, specified in the next subsection;
its cohomology class will have to take a specific form (to get $c_1(V)=0$).
Therefore an important subclass of cases
(and the one to which we restrict ourselves throughout)
is given by spectral cover surfaces with
the divisor $C$ not just effective but even ample (positive)
\begin{eqnarray}
C \;\;\;\; \mbox{ample} \;\; \;\; \;\; \;\; \;\; \;\;
(\; \Longrightarrow \;\;\; \eta -n c_1 >0)
\end{eqnarray}
$C$ will then have the property $h^{1,0}(C)=0$ (inherited from $X$)
and so line bundles on $C$ are characterised by their Chern classes.
That is, the (continuous) bundle moduli of $(X,V)$
are then given just by ${\bf P}H^0(X, {\cal O}_X(C))$,
i.e.~the different choices of $C$ which in turn are parametrised
by the polynomial coefficients of its
defining equation (in addition there is a discrete parameter $\lambda$
described in the next subsection).

\subsection{The line bundle $L=\underline{L}|_C$ over the spectral cover surface $C$}

One describes the $SU(n)$ bundle $V$ over $X$ by a line bundle
$L$ over $C$
\begin{eqnarray}
V=p_*(p_C^*L\otimes {\cal P})
\end{eqnarray}
with $p:X\times_B C\rightarrow X$ and $p_C: X\times_B C\rightarrow C$ the projections and
${\cal P}$ the global variant of (a symmetrized version of) the Poincare line
bundle over $F_1\times F_2$, i.e. the universal bundle which realizes $F_2$ as
moduli space of degree zero line bundles over $F_1$.
$L$ is specified by a half-integral number $\lambda$. This occurs as
$c_1(V)=\pi_*\big(c_1(L)+\frac{c_1(C)-c_1}{2}\big)=0$ implies
\begin{eqnarray}
\label{c_1(L)}
c_1(L) \; = \; -\frac{1}{2}\Big(c_1(C)-\pi_{C*}c_1\Big)+\gamma \;
= \; \frac{n\sigma + \eta + c_1}{2}|_C+\gamma
\end{eqnarray}
(we will omit usually the obvious pullbacks).
Here $\gamma$ denotes the only generally given class in the kernel of
$\pi_{C*}:H^{1,1}(C)\rightarrow H^{1,1}(B)$, i.e.
($\underline{\gamma} \in H^{1,1}(X)$)
\begin{eqnarray}
\label{gamma definition}
\gamma \; =  \; \underline{\gamma}|_C \;\;\;\; \mbox{with} \;\;\;\;
\underline{\gamma}  \; =  \; \lambda \, \Big(n\sigma-(\eta-nc_1)\Big)
\end{eqnarray}
This gives precise integrality conditions for $\lambda$:
if $n$ is odd, then one needs actually $\lambda\in \frac{1}{2}+{\bf Z}$;
if $n$ is even, then $\lambda \in \frac{1}{2}+{\bf Z}$ needs
$c_1 \equiv 0 \; \mbox{mod}\, 2$
and $\lambda \in {\bf Z}$ needs $\eta \equiv c_1 \; \mbox{mod}\, 2$.

Assuming $h^{1,0}(C)=0$ line bundles on $C$ are characterised by their
Chern classes. Therefore one can define line bundles $\underline{{\cal G}}$ and ${\cal G}$
on  $X$ and $C$, respectively, by
\begin{eqnarray}
c_1(\underline{{\cal G}}) \; = \; \underline{\gamma} \;\; ,
\;\; c_1({\cal G}) \; = \; \gamma
\end{eqnarray}
(when we want to make the $\lambda$-dependence explicit we denote these by
$\underline{{\cal G}}_{\lambda}$ and ${\cal G}_{\lambda}$). They are related to
corresponding divisor classes (modulo linear equivalence)
$\underline{G}$ and $G$ with
\begin{eqnarray}
\underline{{\cal G}} \; = \; {\cal O}_X\big( \underline{G}\big)\;\; , \;\;
{\cal G} \; = \; {\cal O}_C\big( G \big)
\end{eqnarray}
i.e. one has  $\underline{G}=\lambda (n\sigma - \pi^*(M + n K_B))$
and, for example, $G|_c=\lambda(ns-(r-n\chi)F)|_c$.

Note that all these considerations of $\underline{{\cal G}}$ and ${\cal G}$ apply strictly
only formally as the corresponding Chern classes will, taken alone for
themselves, be only half-integral in general; only the full combination in
(\ref{c_1(L)}) will be integral and define a proper line bundle.
Similar remarks apply to the formal decompositions written below
($K_B$ denotes here a line bundle and also the corresponding divisor class).

Explicitly one finds for the various incarnations of the spectral line bundle
\begin{eqnarray}
\label{cal L def}
\underline{L} & = & \Big( {\cal O}_X(\sigma)^n \otimes \pi^* {\cal M} \otimes
\pi^* K_B^{-1}\Big)^{1/2}\otimes
\underline{{\cal G}}\; = \;
{\cal O}_X\Big( \frac{C+\pi^*\, K_B^{-1}}{2}+\underline{G}\Big)\\
L & = & K_C^{1/2} \otimes\pi_C^*K_B^{-1/2}\otimes {\cal G}
\end{eqnarray}
Explicitly one has for the Chern class
\begin{eqnarray}
c_1(\underline{L}) & = &
n\Big(\lambda + \frac{1}{2}\Big)\sigma + \Big(\frac{1}{2}-\lambda\Big)\eta
+ \Big(\frac{1}{2}+n\lambda\Big)c_1
\end{eqnarray}

\section{\label{restriction to E}
Spectral $SU(n)$ bundles over the elliptic surface ${\cal E}$}

\subsection{\label{chi subsection}
The elliptic surface ${\cal E}$ over the instanton curve $b$}

\resetcounter

A horizontal curve lies in two surfaces in $X$: in the base $B$
and in ${\cal E}= \pi^{-1}b$, the elliptic surface over $b$.
Let us define the following expressions related to the restriction to ${\cal E}$
\begin{eqnarray}
s\; :=\; \sigma|_{{\cal E}}\;\;\;\;\; , \;\;\;\;\; \chi\; := \; c_1 \cdot b \in {\bf Z}
\end{eqnarray}
(with $c_1:=c_1(B)$).
With the adjunction relation
$\sigma^2=-c_1\sigma$ and (\ref{irreducibility})
one finds
\begin{eqnarray}
\label{chi k relation}
s^2\; =\; -\chi \leq 0
\end{eqnarray}
with $\chi \geq 0$ from our assumptions (\ref{resBdef}), (\ref{defE}).
The tangent bundle decomposes over $b$
\begin{eqnarray}
TX|_b & = & {\cal O}_b(2)\; \oplus {\cal O}_b(a_h)\; \oplus {\cal O}_b(a_v)
\end{eqnarray}
with the latter two terms comprising the normal bundle where $a_h+a_v=-2$.
If $b$ is {\em isolated} then $a_h=a_v=-1$.
Clearly for our horizontal curve $b$ one has
\begin{eqnarray}
a_h \; = \; (b|_B)^2\, , \;\;\;\;\;\; a_v \; = \; (b|_{{\cal E}})^2:= s^2=-\chi
\end{eqnarray}

The canonical class of the elliptic surface ${\cal E}$ is
\begin{eqnarray}
\label{K relation}
K_{{\cal E}}=\pi_{{\cal E}}^* \Big(K_b+{\cal O}_b(\chi)\Big)={\cal O}_{{\cal E}}\big( (\chi-2)F\Big)
\;
\Longrightarrow \; c_1({\cal E})=(2-\chi)F
\end{eqnarray}
Equivalently the relative dualizing sheaf is
\begin{eqnarray}
\omega_{{\cal E}/b}\; & = & \; K_{{\cal E}}\otimes \pi_{{\cal E}}^*\, K_b^{-1}\;
= \; \pi_{{\cal E}}^* \, {\cal O}_b(\chi)
\end{eqnarray}
Here
(cf. for example [\ref{AChor}]; note that the discriminant of $X$ over $B$ has
class $\Delta = 12 c_1$)
\begin{eqnarray}
\chi=\chi({\cal E}, {\cal O}_{{\cal E}})=\frac{1}{12}e({\cal E})=c_1\cdot b
\end{eqnarray}

So, all one needs to know about the position of $b$ in the base $B$
is encoded by the number $\chi$.
Beyond that we will need the rank $n$ of the vector bundle $V$
and its spectral data $\eta, \lambda$, or rather $r, \lambda$ after the
restriction to ${\cal E}$, cf. (\ref{def r}). One upshot of the discussion above
is that $\chi =1$ is the relevant case to consider; for contrast we
also consider $\chi=0$ (where the whole interpretation changes)
and so keep the parameter $\chi$ manifest throughout.

For $b$ the base in the ${\bf P^1}$-fibered Hirzebruch surface ${\bf F_k}=B$
one has the following cases (where $\chi=2-k\geq 0$ and $c_1({\cal E})=kF$)
\begin{center}
\begin{tabular}{|c||c|c|c|}
\hline
$B$ & ${\bf F_0}$ & ${\bf F_1}$ & ${\bf F_2}$\\
\hline
${\cal E}$ & $K3$  & $dP_9$ & $b\times F$ \\
\hline
\end{tabular}
\end{center}
The case $B={\bf P^2}$
of $c_1=3l$ and $c_1\cdot l=3$
or $ c_1\cdot 2l=6$ gives an ${\cal E}$ of Euler number $36$ or $72$, respectively.

\subsection{\label{coordinates for curve case}
The spectral cover curve $c$ over $b$}

The spectral surface $C$ of our bundle $V$ being an $n$-fold
cover of the base one has
\begin{eqnarray}
\label{big proj}
C \;\;\;\;\;\;\;\; & \hookrightarrow & \;\;\; X\nonumber \\
\;\;\; \pi_C\,\downarrow \; n:1& &\pi \,\downarrow \, F \;\;\; \;\;\; \\
B \;\;\;\;\;\;\;\; & = & \;\;\; B \nonumber
\end{eqnarray}
The support of the world-sheet instanton we
consider is a rational curve $b$ inside the base $B$
(in turn embedded in $X$ by the zero section $\sigma$); specifically
one may think of $b$ as given by the base ${\bf P^1_b}$
inside the ${\bf P^1_f}$-fibered surface $B={\bf F_k}$.
Let ${\cal E}:= \pi^{-1}b$ be the elliptic surface over $b$
and $c:=C|_{{\cal E}}$ the corresponding spectral curve of $V|_{{\cal E}}$
\begin{eqnarray}
\label{small proj}
c \;\;\;\;\;\;\;\;\; & \hookrightarrow & \;\;\;\;\; {\cal E}\nonumber \\
\pi_c \,\downarrow \; n:1& & \pi_{{\cal E}} \,\downarrow \, F  \;\;\;\; \\
b \;\;\;\;\;\;\;\;\; & = &\;\;\; \;\; b\nonumber
\end{eqnarray}
(which we assume irreducible as we did for $C$).
If the whole description is restricted from the
elliptic threefold $X\subset {\cal W}_B$ over $B$
(where the fourfold ${\cal W}_B$ is the ${\bf P^2}$-bundle
of Weierstrass coordinates over $B$)
to the elliptic surface ${\cal E}$ over $b$ one gets again the
equation (\ref{locus}) for $c\subset {\cal E} \subset {\cal W}_{b}$
(in the threefold given by the ${\bf P^2}$ bundle
of Weierstrass coordinates over $b\cong {\bf P^1}$);
what was ${\cal L}=K_B^{-1}$ for the situation over $X$,
becomes here ${\cal L}|_b={\cal O}_b(\chi)$ such that now
$a_i|_b\in H^0(b, {\cal O}_b(r-i\chi))$
(where $r:=\eta\cdot b$ such that ${\cal M}|_b={\cal O}_b(r)$, cf.~below).
The section $s$, i.e.~concretely the (group-)zero point
$p_0=(x_0, y_0, z_0)=(0,1,0)=(z)\cap F_t$
in each fibre over a point $t=u/v\in {\bf P^1}=b$,
consists, when restricted to $c$,
out of $s\cdot c=r-n\chi$ points (here $(z)$ is the locus where $z=0$).

Let us define the following expression related to the restriction to ${\cal E}$
\begin{eqnarray}
\label{def r}
r & := & \eta \cdot b \in {\bf Z}
\end{eqnarray}
So $C=n\sigma + \eta$ gives $c=ns+rF$. With $\eta = xb+yf$ on ${\bf F_k}$
one finds $r\geq 0$
with
$r=0 \Leftrightarrow \eta = x \, b_{\infty}$ where
$b_{\infty}=b+kf$. This case can occur only over ${\bf F_2}$ as
by (\ref{irreducibility effectivity})
\begin{eqnarray}
\label{r chi relation}
r\geq n\, \chi
\end{eqnarray}
For $C$ ample (as we will assume) one gets even (such that
in particular always $r>0$)
\begin{eqnarray}
\label{r chi relation sharp}
r> n\, \chi
\end{eqnarray}

The integer $r$ has an interpretation as an
instanton number\footnote{The class $\gamma$ would not occur
in the specification
of a spectral bundle $V_{{\cal E}}$ over the one-dimensional
base $b$ where $\pi_{c\, *}:H^{1,1}(c)\rightarrow H^{1,1}(b)$
is injective, cf.~[\ref{FMW}],
in accord with the fact that by (\ref{r instnr})
$c_2(V|_{{\cal E}})$ sees only the $\gamma$-free part of
$c_2(V)$ and not
$\omega =  -\frac{n^3-n}{24}c_1^2-\frac{n}{8}\eta(\eta-nc_1)
-\frac{1}{2}\pi_*(\gamma^2)\; = \;
-\frac{n^3-n}{24}c_1^2+(\lambda^2-\frac{1}{4})\frac{n}{2}\eta(\eta-nc_1)$.}
\begin{eqnarray}
\label{r instnr}
c_2(V)=\eta \sigma + \omega & \Longrightarrow & c_2(V|_{{\cal E}})=r
\end{eqnarray}

The canonical bundle of $c$ is given by (cf.~(\ref{K relation}))
\begin{eqnarray}
\label{can bund of c}
K_c&=&\Big( {\cal O}_{{\cal E}}(c)\otimes K_{{\cal E}}\Big)|_c\; =
\; {\cal O}_{{\cal E}}\Big( ns+(r+\chi-2)F\Big)|_c
\end{eqnarray}
For later use, cf.~the comments after (\ref{specialisation}) below,
we remark that (which is $\geq 0$ under our assumptions)
\begin{eqnarray}
\label{canonical degree}
\deg K_c^{1/2} =g_c-1& = & n (r - \frac{n-1}{2}\chi -1)
\end{eqnarray}

The cohomologically nontrivial line bundles on ${\cal E}$
which come from $X$, i.e.~are of the form ${\cal O}_{{\cal E}}(xs+yF)$, and
which become flat on $c$ are powers of\footnote{for $gcd(n, r-n\chi)=1$,
which is fulfilled automatically in the cases of application
(where $\chi=1, n\leq 5$)}
\begin{eqnarray}
\label{Lambda}
\Lambda&:=& {\cal O}_{{\cal E}}\Big( ns - (r-n\chi)F\Big)
\end{eqnarray}

\subsubsection{\label{specialisation subsection}Moduli of the spectral curve}

The coefficients of the homogeneous polynomials $a_i$
are (after one overall scaling)
moduli $m\in {\cal M}_{{\cal E}}(c)={\bf P}H^0({\cal E}, {\cal O}_{{\cal E}}(c))$
of external motions of $c$ in ${\cal E}$, that is of those part of the moduli
in ${\cal M}_{bun}(X,V)$ which is relevant in our consideration over ${\cal E}$.

{\em Behaviour over the special sublocus
$\Sigma_{\Lambda}=\{f_{\Lambda}=0\}$ of the moduli space ${\cal M}_{{\cal E}}(c)$}

Consider in the moduli space ${\cal M}_{{\cal E}}(c) (\ni m)$ the
specialisation locus $\Sigma_{\Lambda}$ where
\begin{eqnarray}
\label{Lambda triviality on c}
\Lambda|_c &\cong & {\cal O}_c
\end{eqnarray}
and let us assume that this locus is characterised by
a (set of) condition(s) $f_{\Lambda}(m)=0$,
say\footnote{\label{vector-valued fct}This locus will
in general (cf.~later Example $1$) have codimension higher
than one, so $f$ should be interpreted as a vector-valued 'function';
i.e.~two or more conditions have to be posed at the same time.}.
Now, one of the principal objects of our study (cf.~(\ref{prec criterion})),
the line bundle $l(-F)|_c={\cal O}_{{\cal E}}\Big(n(\lambda + \frac{1}{2})s+\beta F\Big)|_c\;
\cong  K_c^{1/2}\otimes \Lambda|_c^{\lambda}$,
cf.~(\ref{degree equalities}) and (\ref{l(-F)}),
becomes along $\Sigma_{\Lambda}$
\begin{eqnarray}
\label{specialisation}
l(-F)|_c & \buildrel \mbox{\scriptsize on} \;\Sigma_{\Lambda} \over \longrightarrow &
{\cal O}_{{\cal E}}\Big([(\lambda + \frac{1}{2})(r-n\chi)+\beta]F\Big)|_c=
{\cal O}_{{\cal E}}\Big((r-\frac{n-1}{2}\chi-1)F\Big)|_c\,
\buildrel \Sigma_{\Lambda} \over
\cong \, K_c^{1/2}
\;\;\;\;\;\;\;\;\;\;\;
\end{eqnarray}
Note that (\ref{specialisation}) is a bundle of integral degree on $c$.
We did exclude here the case $n$ even, $\chi$ odd, $\lambda\in {\bf Z}$
where the alternative evaluation ${\cal O}_{{\cal E}}(\frac{n}{2}s+\frac{r-1}{2}F)|_c$
applies.
So (with the mentioned alternative evaluation understood) we see
{\em that $l(-F)|_c$ becomes along $\Sigma_{\Lambda}$ effective}:
$l(-F) \cong  {\cal O}_{{\cal E}}(mF)|_c$ where $m=r-\frac{n-1}{2}\chi-1$
(and similarly for $n$ even).

\subsubsection{\label{Ri section}The case of $SU(3)$ bundles}

Here ($C=a_0, B=a_2, A=a_3$; $m\in  H^0\Big( {\cal E}, {\cal O}_{{\cal E}}(3s+[m]\chi F)\Big)$
for $m=z, x, y; [m]= 0, 2, 3$)
\begin{eqnarray}
\label{equ spec curve}
w\; = \; C_r\, z + B_{r-2\chi} \, x + A_{r-3\chi}\, y\;
& \in & H^0\Big( {\cal E}, {\cal O}_{{\cal E}}(3s+rF)\Big) \\
& \cong &
zH^0\Big(b, {\cal O}(r)\Big)\oplus xH^0\Big(b, {\cal O}(r-2\chi)\Big)
\oplus yH^0\Big(b, {\cal O}(r-3\chi)\Big) \nonumber
\end{eqnarray}

The defining equation for $c$ is
$w=C_r(t)z+B_{r-2\chi}(t)x+A_{r-3\chi}(t)y=0$
where $t=(u,v)\in b$ with homogeneous coordinates $u,v$ on $b$,
i.e.~(suitable pull-backs understood)
$u,v\in H^0\Big( {\cal E}, {\cal O}_{{\cal E}}(0s+1F)\Big)$. So
for the special case $n=3$ the number $r-n\chi$ of points of $s\cdot c$
coincides with the degree of $A_{r-3\chi}$,
and the points are just the $r-3\chi$ points $\{p_0, t_i\}\in {\cal E}$
in the fibres  $c_{t_i}:=F_{t_i}|_c$
over the zeroes $t_i$ of $A_{r-n\chi}=\prod_{i=1}^{r-n\chi}A_1^{(i)}$
\begin{eqnarray}
\label{s|_c generically}
s|_c&=&\sum_{A_{r-3\chi}(t_i)=0}\{p_0, t_i\}
\end{eqnarray}
Generically in $c$-moduli (for $B(t_i)\not = 0$)
these are {\em simple}\footnote{though a three-fold touching point of the line
$(z)_t$ and the elliptic curve $F_t$ in the
${\bf P}^{\bf 2}_t$ over $t\in b$}
points in the fibres $F_{t_i}$ w.r.t.~the intersection of $s=p_0$
with the defining line $(w)_{t_i}=\{C(t_i)z+B(t_i)x=0\}$ of $c$
(unequal to the line $(z)_{t_i}=\{z=0\}\subset {\bf P}^{\bf 2}_{t_i}$):
here $C(t_i)z+B(t_i)x=0$ determines $x$ from $z$, and $y$ from
the Weierstrass equation, giving the other two points $p_i^{\pm}$ of
$c_{t_i}=\{p_0+p_i^++p_i^-, t_i\}$.

\noindent
{\em The codimension one sublocus $Res(A,B)=0$ in ${\cal M}_c$ }

The generic result (\ref{s|_c generically})
changes at the specialisation locus $Res(A,B)=0$
(where $B_3(t_1)=0$, cf.~app.~\ref{resultant}):
the $(w)$ line (where $w=0$ in the ${\bf P}^{\bf 2}_{t_1}$-fibre in ${\cal W}_b$)
becomes  just the $(z)$ line and $p_0$ becomes a three-fold point of $c_{t_1}$
(where $\sim$ means linear equivalence)
\begin{eqnarray}
(z)|_c=3s|_c=\sum_i \{3p_0, t_i\} = F_{t_1}|_c + \sum_{i\geq 2}\{3p_0, t_i\}
\neq  F_{t_1}|_c +\sum_{i\geq 2}F_{t_i}|_c \sim (r-n\chi)F|_c
\end{eqnarray}
{\em The codimension $r-n\chi$ sublocus $A|B$ or $R_i=0, i=1, ..., r-n\chi$
in ${\cal M}_c$ }

Here we demand that {\em not one but all} $r-n\chi$ roots
of $A$ are in common with $B$, i.e.~$A|B$.
Now $f_{\Lambda}=0$ is (implied by) a set of $r-n\chi$ conditions
$R_i=0$ which are just the $r-n\chi$ conditions $B(t_i)=0$,
i.e.~$R_i=Res(B, A_1^{(i)})$: demanding that all $R_i=0$
the previous argument gives now in each $c_{t_i}$ a threefold $\{p_0, t_i\}$,
so ${\cal O}_{{\cal E}}(3s)|_c  \cong  {\cal O}_{{\cal E}}((r-3\chi)F)|_c$ or
$\Lambda|_c \cong  {\cal O}_c$ (cf.~sect.~\ref{specialisation subsection}),
so (here no converse, cf.~Example 4 in sect.~\ref{Ex 4 case})
\begin{eqnarray}
\label{Ri = 0 implication}
A|B & \Longleftrightarrow & R_i=0, \forall i \;\;\; \Longrightarrow \Lambda|_c \cong {\cal O}_c
\end{eqnarray}
So one has $(R_i)_i=0 \Longrightarrow f_{\Lambda}=0$.
Note further that the codim $r-n\chi$ locus
$\{R_i(m)=0, \forall i\} (\subset \Sigma_{\Lambda})$
is a subset
of the codim 1 locus $\{Res(B, A)(m)=0\}$,
i.e.~$(R_i)_i=0\Longrightarrow Res(B, A)=0$, so
(a power of) $Res$ is a
combination\footnote{\label{multiple resultant zero}From
$R_i=0, \forall i$ one expects
an $(r-3\chi)$-fold zero of the resultant; for a representation
$Res=P_{r-n\chi}(R_1, ..., R_{r-n\chi})$, with $P_{r-n\chi}$ a homogeneous
polynomial of degree $r-n\chi$ in the $R_i$,
cf.~(\ref{f in the R_i}),(\ref{Ex 4 resultant}).}
(in the ideal sense) of the $R_i$.

\subsection{\label{twist class}The line bundle $l|_c$
over the spectral cover curve $c$}

As a further datum describing $V$ beyond the
surface $C$, which encodes $V$ just fiberwise,
one has a line bundle $L$ over $C$ with  $V=p_*(p_C^*L\otimes {\cal P})$.
$L$ arises in the simplest case as a restriction $L=\underline{L}|_C$
to $C$ of a line bundle $\underline{L}$ on $X$.
We define also a corresponding restriction
$l:=\underline{L}|_{{\cal E}}$ to ${\cal E}$ (with $l|_c=L|_c$).
So one has the inclusions of line bundles
\begin{eqnarray}
L \;\;\;\;\;\; \hookrightarrow \;\;\; \underline{L} \;\;\;\;\;
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\;\;\;\;\;
\; l|_c \;\;\;\;\;\;  \hookrightarrow  \;\;\;\; l
\nonumber \\
\downarrow \;\;\;\;\; \;\;\;\;\;\;\;\;  \;\;\; \downarrow \;\;\;\;\;
& \;\; \mbox{and} \;\;& \;\;\;\;\;
\; \downarrow \;\;\;\;\;\;\;\;  \;\;\; \;\;\; \;\;\; \downarrow
\\
C \;\;\;\;\;\; \hookrightarrow  \;\;\; X\;\;\;\;\;
& \;\;\;\;\; &\;\;\;\;\;
\; c \;\;\;\;\;\;\;  \hookrightarrow  \;\;\;\; {\cal E}
\nonumber
\end{eqnarray}

The crucial fact is that one has for spectral bundles
\begin{eqnarray}
\label{bundle projection}
V|_{B}=\pi_{C*} L  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\mbox{such that}\;\;\;\;\;\;\;\; V|_b=\pi_{c*}\; l|_c
\end{eqnarray}
Similarly to (\ref{cal L def})
one has with $\underline{\gamma}|_{{\cal E}}=\lambda\big(ns-(r-n\chi)F\big)$
and ${\cal L}|_b=K_B^{-1}|_b={\cal O}_b(\chi)$
that\footnote{\label{K_c footnote}The equality to the second line
in (\ref{degree equalities}) follows from
$l^2|_c \otimes {\cal F}^{-2}\cong
{\cal O}_{{\cal E}}\Big( (ns+rF) + \chi F\Big)|_c={\cal O}_{{\cal E}}\Big( (ns+rF-kF)+2F\Big)|_c
\cong K_c \otimes \pi_c^*K_b^{-1}$
using (\ref{can bund of c}).}
\begin{eqnarray}
\label{l def}
l&=&\underline{L}|_{{\cal E}}\; \cong \;
{\cal O}_{{\cal E}}\Bigg( \frac{c+\pi_{{\cal E}}^{*}{\cal O}_b(\chi)}{2}+\underline{G}|_{{\cal E}}\Bigg)
\; = \;
{\cal O}_{{\cal E}}\Bigg( \frac{ns+(r+\chi)F}{2}\Bigg)\otimes \underline{{\cal G}}|_{{\cal E}}
\;\;\;\;\;\;\;\;\; \nonumber\\
&& \Longrightarrow \;\;\;\;\;\; l(-F) \; \cong \; \Big( K_{{\cal E}}\otimes {\cal O}_{{\cal E}}(c)\Big)^{1/2}
\otimes \Lambda^{\lambda}\\
\label{degree equalities}
l|_c&=&{\cal O}_c\Bigg( \frac{ns + (r+\chi)F}{2}|_c \Bigg)\otimes {\cal F} = L|_c \cong
\Big(K_C^{1/2} \otimes\pi_C^*K_B^{-1/2}\Big)|_c \otimes {\cal F}
\nonumber\\
&\cong& K_c^{1/2}\otimes\pi_c^*K_b^{-1/2}\otimes {\cal F}
\;\;\; \Longrightarrow \;\; l(-F)|_c\; \cong  K_c^{1/2}\otimes \Lambda^{\lambda}|_c
=\; K_c^{1/2}\otimes {\cal F}
\end{eqnarray}
Here we have introduced the flat (cf.~below) line bundle on $c$
given by the restriction
\begin{eqnarray}
{\cal F} & = & {\cal G}|_c\; = \; {\cal O}_c\big( G|_c\big)
\end{eqnarray}
Explicitly one has for the Chern class of the line bundle $l$ on ${\cal E}$
\begin{eqnarray}
c_1(l) & = & n\Big(\lambda+\frac{1}{2}\Big)s
+\Big( (\frac{1}{2}-\lambda)r+(\frac{1}{2}+n\lambda)\chi\Big) F
\end{eqnarray}
One notes
that the line bundle ${\cal F}$ over $c$ is flat as
\begin{eqnarray}
\label{ineffectivity}
\Big(ns-(r-n\chi)F\Big)(ns+rF)=0& \Longrightarrow & \deg \, G|_c=0
\end{eqnarray}
Note that the flat bundle ${\cal F}$ has continuous moduli corresponding
to $Jac(c)$ as $h^{1,0}(c)\neq 0$ whereas we did assume $h^{1,0}(C)= 0$
so that we had only the discrete twist datum $\gamma$
(we will not have to treat the ambiguities of $K_c^{1/2}$
in the present paper).

\section{\label{WS instantons}World-sheet instantons and
superpotential contribution}

\resetcounter

\subsection{General case: $SU(n)$ bundles over an instanton curve $b$}

To set the stage we recall first some versions of the criterion
for a world-sheet instanton to contribute to the superpotential $W$.
Let us assume that $V$ is an $SU(n)$ bundle, embedded in the first $E_8$ group.
A world-sheet instanton, supported on an isolated rational curve $b$,
contributes according to the following criterion [\ref{Wittenhet}]
(with $V|_b(-1)$ denoting $V|_b \otimes {\cal O}_b(-1)$)
\begin{eqnarray}
\label{main criterion}
W_b\neq 0 & \Longleftrightarrow & h^0\Big(b, V|_b(-1)\Big)=0
\end{eqnarray}
Considered with respect to the structure group $SO(2n) \supset SU(n)$
(with $n\leq 8$) one has
\begin{eqnarray}
V|_b&=&\bigoplus_{i=1}^n\, {\cal O}_b(\kappa_i)\oplus {\cal O}_b(-\kappa_i)
\;\;\;\;\;\;\;\;\;\;\;\;\;\; (\kappa_i\geq 0)
\end{eqnarray}
such that
$V|_b(-1)=\bigoplus_{i=1}^n\, {\cal O}_b(\kappa_i -1)\oplus {\cal O}_b(-\kappa_i -1)$
gives
\begin{eqnarray}
h^0(b, V|_b(-1))=\sum_{\kappa_i -1 \geq 0} \kappa_i=\sum \kappa_i
\end{eqnarray}
Therefore $b$ contributes precisely if $V|_b$ is trivial, i.e.
\begin{eqnarray}
\label{triviality criterion}
W_b\neq 0 & \Longleftrightarrow & V|_b = \bigoplus_{i=1}^n\, {\cal O}_b
\end{eqnarray}
A corresponding framing would give $n$ linearly independent
global sections such that
\begin{eqnarray}
\label{n global sections}
W_b\neq 0 & \Longrightarrow & h^0(b, V|_b)=n
\end{eqnarray}
(This is, of course, also directly a consequence of (\ref{main criterion}),
cf. app.~\ref{Lemmata}, Lemma 1). A counterexample to the converse
of (\ref{n global sections})
is given by $V|_b = {\cal O}_b\oplus {\cal O}_b(1)\oplus {\cal O}_b(-1)$.

Considered in $SU(n)$ (as we will henceforth do) one has
\begin{eqnarray}
\label{decomp over b}
V|_b&=&\bigoplus_{i=1}^n\, {\cal O}_b(k_i)\;\;\;\;\;\;\;\;\;\;\;\;
\mbox{with} \;\;\;\;
\sum k_i=0 \;\;\;\;\;\; (k_i\in {\bf Z})
\end{eqnarray}
where now
\begin{eqnarray}
h^0(b, V|_b(-1))=\sum_{k_i -1 \geq 0} k_i=\sum_{k_i\geq 0}k_i
\end{eqnarray}
such that one gets (noting $\sum k_i=0$) again (\ref{triviality criterion})
\begin{eqnarray}
h^0(b, V|_b(-1))=0&\Longleftrightarrow & k_i=0\;\;\;\;
\mbox{for all $i$ with $k_i\geq 0$}\nonumber\\
&\Longleftrightarrow & k_i=0 \;\;\;\;
\mbox{for all $i$}
\end{eqnarray}

\subsection{\label{contribution criteria}Elliptic case with
spectral bundles and a base curve $b$}

\noindent
{\bf \underline{Precise criterion for $W_b\neq 0$}}
\begin{eqnarray}
\label{prec criterion}
W_b\neq 0 \Longleftrightarrow
0=h^0\Big(b, V|_b(-1)\Big)=
h^0\Big(b, \pi_{c*}l|_c\otimes {\cal O}_b(-1)\Big)=h^0\Big(c, l(-F)|_c\Big)
\end{eqnarray}

Note that $l(-F)|_c$ occurs in the short exact sequence of sheaves on ${\cal E}$
\begin{eqnarray}
0 \longrightarrow l(-F-c) \buildrel \iota \over \longrightarrow l(-F) \longrightarrow l(-F)|_c \longrightarrow 0
\end{eqnarray}
(cf.~for the following also [\ref{BDO}])
which gives a long exact sequence of cohomology groups
\begin{eqnarray}
\label{long exact sequence}
0 \longrightarrow H^0\Big({\cal E}, \, l(-F-c)\Big) \buildrel \iota_0 \over
\longrightarrow H^0\Big({\cal E}, \, l(-F)\Big) \longrightarrow H^0\Big(c, \, l(-F)|_c\Big) \longrightarrow ...
\end{eqnarray}

\noindent
\underline{{\bf First, necessary criterion for $W_b\neq 0$}}
\begin{eqnarray}
\label{criterion}
W_b\neq 0 \;\;\;\; \Longrightarrow \;\;\;\;
dim \; \iota_0 \; H^0\Big({\cal E}, \, l(-F-c)\Big) \;
= \; dim \; H^0\Big({\cal E}, \, l(-F)\Big)
\end{eqnarray}

As $\iota_0$ is an embedding this amounts just to
the condition $h^0 \Big({\cal E}, \, l(-F-c)\Big)
= h^0\Big({\cal E}, \, l(-F)\Big)$.
From (\ref{criterion}) one gets as {\em sufficient criterion
for non-contribution of $b$}
\begin{eqnarray}
\label{further criterion}
h^0\Big({\cal E}, \, l(-F-c)\Big)= 0  \;\;\; , \;\;\;
h^0\Big({\cal E}, \, l(-F)\Big) >  0
\;\;\;\; \Longrightarrow  \;\;\;\; W_b=0
\end{eqnarray}

In the following we make a technical assumption
on the bundle parameter $\lambda$ (cf. below)
\begin{eqnarray}
\label{lambda assumption}
\lambda > 1/2
\end{eqnarray}
Note that then, as recalled after (\ref{technical assumption}),
the second cohomology groups
in the long exact sequence (\ref{long exact sequence}) vanish,
leaving only the three $H^0$- and the three $H^1$-terms
(cf.~Ex. after (\ref{H^1 reduction conditions})).
By $c_1(V|_b)=0$ and (\ref{index}) the two terms over
$c$ have equal dimension
such that
\begin{eqnarray}
\label{dimensions}
h^0\Big( {\cal E}, l(-F)\Big) - h^0\Big( {\cal E}, l(-F-c)\Big)=
h^1\Big( {\cal E}, l(-F)\Big) - h^1\Big( {\cal E}, l(-F-c)\Big)
\end{eqnarray}
If criterion (\ref{criterion}) is
fulfilled there is a further, more precise assertion.

\noindent
\underline{{\bf Second, (conditional,) precise criterion for $W_b\neq 0$}}\\
Let the necessary condition for contribution (\ref{criterion}) be
fulfilled and $\lambda > 1/2$. Then
\begin{eqnarray}
\label{criterion 2}
W_b\neq 0 \;\;\;\; \Longleftrightarrow \;\;\;\;
dim \; \iota_1 \; H^1\Big({\cal E}, l(-F-c)\Big) \;
= \; dim \; H^1\Big({\cal E}, l(-F)\Big)
\end{eqnarray}
This follows by noting that in the long exact sequence
(which can be written here as)
\begin{eqnarray}
0 \longrightarrow H^0\Big(c, l(-F)|_c\Big)\longrightarrow H^1\Big({\cal E}, l(-F-c)\Big)
\buildrel \iota_1 \over \longrightarrow
H^1\Big({\cal E}, l(-F)\Big) \longrightarrow H^1\Big(c, l(-F)|_c\Big)\longrightarrow 0 \nonumber
\end{eqnarray}
the outer (by $c_1(V|_b)=0$, cf.~(\ref{index})),
and so the inner, two spaces have equal dimension.
So (\ref{criterion 2}) amounts to $\iota_1$ being an isomorphism
(generically in the moduli).

Concretely the map $\iota_1$ is induced from multiplication
with a moduli-dependent element
\begin{eqnarray}
\label{moduli-dependent 1}
\tilde{\iota} & \in & H^0\Big({\cal E}, {\cal O}_{{\cal E}}(c)\Big)
\end{eqnarray}

\subsection{\label{Criteria Evaluation}Evaluation
of the extrinsic contribution criteria}

To evaluate concretely the contribution criteria note first
(by $c=ns+rF$ and (\ref{l def}))
\begin{eqnarray}
\label{l(-F)}
l(-F)&=&{\cal O}_{{\cal E}}\Big( \alpha s + \beta F\Big)\\
\label{l(-F-c)}
l(-F-c)&=&{\cal O}_{{\cal E}}\Big( (\alpha-n) s + (\beta - r) F\Big)
\end{eqnarray}
We use the numerical parameters $\alpha, \beta$
given by\footnote{The numerical specification of $\beta$ by (\ref{beta eval})
(which in the end goes back just to (\ref{c_1(L)}),
restricted to ${\cal E}\subset X$) expresses just that
we tuned parameters to get $c_1(V)=0$ (resp.~the
corresponding version restricted to ${\cal E}$). This was crucial to have
$h^1({\cal E}, l(-F-c))=h^1({\cal E}, l(-F))$, cf.~the proof after (\ref{criterion 2})
(working within the assumption $\alpha>n$,
so that the $H^2$-terms vanished;
for a converse  cf.~sect.~\ref{equ cond al > n}).}
(where $n\chi-r\leq 0$ by (\ref{r chi relation}))
\begin{eqnarray}
\label{al eval}
\alpha & = & n(\lambda+\frac{1}{2})\\
\label{beta eval}
\beta &=& \beta^{(\chi)}_{n,r}(\lambda) :=\frac{r+\chi}{2}-\lambda(r-n\chi)-1
\; = \; \Big(\frac{1}{2}-\lambda\Big)\, r + \Big(\frac{1}{2}+n\lambda\Big)\, \chi -1
\end{eqnarray}
It will be useful to keep on record the following rewriting
relating $\alpha$ and $\beta$
\begin{eqnarray}
\label{beta rewriting}
-(\beta+1)& = & \frac{\alpha -n}{n} \, (r-n\chi) - \frac{n+1}{2}\, \chi
\end{eqnarray}

Now $h^0\Big({\cal E}, {\cal O}_{{\cal E}}(p s + q F)\Big)$
vanishes, if $p > 0$, just for negative $q$
(cf. app.~\ref{Lemmata}, Lemma 2)
\begin{eqnarray}
\label{vanishing crit}
\underline{p >0:}\;\;\;\;\;\;\;\;
h^0\Big({\cal E}, {\cal O}_{{\cal E}}(p s + q F)\Big)=
h^0\Big(b, \pi_{{\cal E} *}{\cal O}_{{\cal E}}(p s + q F)\Big)
=0 \;\; \Longleftrightarrow \;\; q < 0 \;\;\;\;\;
\end{eqnarray}

In view of (\ref{l(-F-c)}) and (\ref{vanishing crit})
we will in the following assume
\begin{eqnarray}
\label{technical assumption}
\alpha - n > 0 \;\;\;\;\;\;\;\; \;\;\;\;\;
\mbox{, i.e.} \;\;\;\;\;\;\;\; \lambda > 1/2
\end{eqnarray}
(for $\lambda < -1/2$ cf.~app.~(\ref{negative lambda})).
Then sheaf cohomology groups on ${\cal E}$ reduce,
by (\ref{vanishing R^1}), (\ref{reduction}),
to the corresponding push-forwards
(direct images) on $b$, cf.~remark after (\ref{lambda assumption}).

Following criterion (\ref{further criterion})
a sufficient condition for $W_b=0$ is by (\ref{vanishing crit})
\begin{eqnarray}
\label{beta crit}
\beta \geq 0\;\;\; , \;\;\; \beta - r < 0
\;\;\;\; \Longrightarrow  \;\;\;\; W_b=0
\end{eqnarray}
whereas an equivalent formulation of the
condition (\ref{criterion}) (necessary for $W_b\neq 0$) is $\beta < 0$
\begin{eqnarray}
\label{dim equality}
h^0\Big({\cal E}, \, l(-F-c)\Big) \; = h^0\Big({\cal E}, \, l(-F)\Big)
\;\;\;\; \stackrel{\mbox{\normalsize{$\Longleftarrow$}}}{\Longrightarrow} \;\;\;\; \beta < 0
\end{eqnarray}
$\beta <0$ is sufficient as then both $h^0$ vanish by (\ref{vanishing crit}).
Remarkably, the converse holds: the $h^0$ in (\ref{dim equality})
can be equal {\em only} if both are zero (cf.~app.~\ref{Lemmata}, Lemma 3).
So one gets

\noindent
\underline{{\bf Third, necessary criterion for $W_b\neq 0$}
(case $\lambda > 1/2$)}
\begin{eqnarray}
\label{third criterion}
W_b\neq 0 \;\;\;\; \Longrightarrow \;\;\;\; \beta < 0 \;\;\;\; \Longleftrightarrow \;\;\;\;
H^0\Big({\cal E}, \, l(-F)\Big) = 0
\end{eqnarray}

\noindent
{\em Remarks:} 1) Note that $\beta <0$ is automatically
fulfilled for $\chi=0$.\\
2) (\ref{third criterion}) can be formulated as a stronger bound on $\eta|_b$
((\ref{irreducibility}) gave already $r \geq n \chi$)
\begin{eqnarray}
\label{sharper bound}
W_b\neq 0 \Longrightarrow
(\lambda - \frac{1}{2})\, r \geq (\frac{1}{2}+n\lambda)\, \chi
\end{eqnarray}
This is indeed sharper:
$\frac{1/2 + n \lambda }{\lambda - 1/2} = \frac{ (1+n)/2}{\lambda - 1/2} + n > n$.
So one gets as necessary condition
\begin{eqnarray}
\label{sharper bound made explicit}
r-n\chi & \geq & \frac{ (1+n)/2}{\lambda - 1/2}\chi
\end{eqnarray}

Let us now come back to the process of making the criteria
(\ref{criterion}) and (\ref{criterion 2}) more explicit.
If one has $\beta <0$, as we will assume, then
(\ref{criterion}) is fulfilled by (\ref{dim equality})
and one has, by (\ref{criterion 2}), to consider the map
(which can be further explicated using
(\ref{l(-F)})-(\ref{beta eval}))
\begin{eqnarray}
\label{the critical map}
H^1\Big({\cal E}, l(-F-c) \Big) \stackrel{\iota_1}{\longrightarrow} H^1\Big({\cal E}, l(-F)\Big)
\end{eqnarray}
This map is induced from multiplication with an element
$\tilde{\iota}  \in  H^0\Big({\cal E}, {\cal O}_{{\cal E}}(ns+rF)\Big)$
and so depends $m\in {\cal M}_{{\cal E}}(c)$ (so actually $c=c_m$).
What (\ref{criterion 2}) says is that
\begin{eqnarray}
\label{extrinsic determinant criterion}
Pfaff\, (m)\; = \; 0 & \Longleftrightarrow & \det \; \iota_1\; (m) \; = \; 0
\end{eqnarray}
(such that
$Pfaff$ equals, up to a constant factor,
$(\det \iota_1)^m$; actually $m=1$ [\ref{BDO}]).
The moduli space ${\cal M}_{{\cal E}}(c)$ has dimension
$h^0\big({\cal E}, {\cal O}_{{\cal E}}(c)\big)-1$ which is by the index theorem
\begin{eqnarray}
\dim {\cal M}_{{\cal E}}(c)\, = \,
h^0\big({\cal E}, {\cal O}_{{\cal E}}(c)\big)-1& = & n(r+1)-\Big(\frac{n(n+1)}{2}-1\Big)\chi
-1\\
&=&n\Big(r-\frac{n}{2}\chi \Big)-(\frac{n}{2}-1)\chi+n-1
\end{eqnarray}
in case $c$ is positive (for $b$ isolated we had $\chi=1$).
Finally the degree of the determinant in
(\ref{extrinsic determinant criterion}) is given
(using (\ref{h^1 reduction})) by
(where explicitly $\frac{\alpha(\alpha-n)}{n}=(\lambda^2-\frac{1}{4})n$)
\begin{eqnarray}
\label{h^1 rewriting}
h^1\Big({\cal E}, l(-F)\Big) & = &
\alpha\Big(-(\beta+1)\Big)+\frac{\alpha(\alpha+1)}{2}\chi -\chi
\; =\; \frac{\alpha(\alpha-n)}{n}(r-\frac{n}{2}\chi)-\chi\;\;\;\;\;\;
\end{eqnarray}

Another way to express the precise criterion (\ref{prec criterion}) is
(with $\sim$ linear equivalence)
\begin{eqnarray}
\label{effectivity equivalence}
Pfaff(m)=0 & \Longleftrightarrow & h^0\Big(c, l(-F)|_c\Big) > 0 \;
\Longleftrightarrow \;\alpha s + \beta F|_c \sim \mbox{effective}
\end{eqnarray}
So one has not only
$R_i=0, \forall i \Longrightarrow f_{\Lambda}=0$ by (\ref{Ri = 0 implication})
but also
$f_{\Lambda}=0 \Longrightarrow  Pfaff =0$ by the remarks after (\ref{specialisation});
if actually codim $\Sigma_{\Lambda}=1$
(cf.~the remark in footn.~\ref{vector-valued fct})
one gets\footnote{\label{reduced factor}generally $f=0\Rightarrow g=0$
gives immediately only $f^{red}|g$ where $f=\prod f_i^{k_i}$
and $f^{red}=\prod f_i$,
but in our actual
cases (cf.~sect.~\ref{Examples} where this remark applies
in various places) $f$ will be irreducible}
\begin{eqnarray}
\label{general Pfaff factor}
f_{\Lambda}\, | \, Pfaff
\end{eqnarray}

\section{\label{section reduction}The idea of reduction}

\resetcounter

The question of contribution of the world-sheet instanton
supported on $b$ to the superpotential amounts
to decide whether $h^0(c, l(-F)|_c)=0$ or not
in dependence on the moduli of $c$ (concretely, in the case of $SU(3)$,
the coefficients of the polynomials $C, B, A$ in the equation of $c$).
The idea of reduction is to translate the
question about $h^0(c, l(-F)|_c)$ to a simpler case. For this one
introduces another ('smaller') line bundle $\bar{l}$
\begin{itemize}
\item leading to a map $\bar{\iota}_1$ between spaces,
now of lower dimension (arguing as for $l(-F)$);
now one has two interesting cases to consider
\begin{itemize}
\item under a certain
{\em equality condition} these spaces have equal dimension and one is
led again to the consideration of a (moduli dependent)
determinantal function $f=\det \bar{\iota}_1$
\item under a {\em vanishing condition} one has (universally in the moduli)
$\ker \bar{\iota}_1\not =0$, i.e.~$f\equiv 0$
\end{itemize}
\item under a {\em reduction condition}
$\bar{l}(-F)$ is related to the original $l(-F)$ such that
if one has $\ker \bar{\iota}_1\not =0$
one gets $\ker \iota_1\not =0$
(both either for certain moduli or universally),
i.e.~the vanishing of $Pfaff$ (at a special locus or universally);
that is one gets
\begin{itemize}
\item $f|Pfaff$
in case the equality condition applies (as $f=\det \bar{\iota}_1$
has lower degree this gives a
'reduction' in the problem of finding zeroes of $Pfaff$)
\item $Pfaff\equiv 0$
in case the vanishing condition applies
\end{itemize}
\end{itemize}
(properly taking into account footn.~\ref{reduced factor}).
In the present section we will treat the reduction condition,
in section \ref{Equality condition} the equality condition
and in section \ref{The vanishing condition} the vanishing condition.

\subsection{The reduction condition: precise version (on $c$)}

Concerning $l(-F)={\cal O}_{{\cal E}}(\alpha s+\beta F)$
we will assume in the following
$\alpha >0$ (actually even $\alpha > n$ by (\ref{technical assumption}))
and the necessary condition (for contribution to the superpotential)
$\beta <0$, cf.~(\ref{third criterion}).
We consider a second line bundle
$\bar{l}(-F):={\cal O}_{{\cal E}}(\bar{\alpha} s+\bar{\beta} F)$
and denote the restrictions to $c$ by
$l(-F)|_c={\cal O}_c(D)$ and $\bar{l}(-F)|_c={\cal O}_c(\bar{D})$.
Now assume
\begin{eqnarray}
\label{precise reduction condition}
\Big(\bar{l}(-F)\Big)^p|_c & \hookrightarrow & l(-F)|_c
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\hspace{2cm}
\mbox{(precise reduction condition)}
\end{eqnarray}
for a positive integer $p$. So the relevant condition is
\begin{eqnarray}
\label{divisor condition}
p\bar{D}  & \leq & D \;\;\;\;\;\; , \;\;\;\;\;\;
\mbox{i.e.}\;\; \tilde{D}=D-p\bar{D} \;\; \mbox{is effective}
\end{eqnarray}
such that $t\in H^0(c, {\cal O}_c(D-p\bar{D})), t \neq 0$ exists.
This gives then
\begin{eqnarray}
\label{reduction phenomenon}
s\in H^0\Big(c, \bar{l}(-F)|_c\Big) & \Longrightarrow &
s^p\in H^0\Big(c, {\cal O}_{{\cal E}}(p\bar{\alpha} s+p\bar{\beta} F)|_c\Big)
\nonumber\\
& \Longrightarrow &
s^p t\in H^0\Big(c, {\cal O}_{{\cal E}}(\alpha s+\beta F)|_c\Big)
\end{eqnarray}
as implication for the existence of nontrivial sections. Therefore one has
\begin{eqnarray}
\label{implication}
h^0\Big(c, \bar{l}(-F)|_c\Big)>0 & \Longrightarrow & h^0\Big(c, l(-F)|_c\Big)>0
\end{eqnarray}
Here the section $t$ has just an auxiliary status:
as the difference of $D$ and $p\bar{D}$ is effective
(this is just the assumption of reduction)
a nontrivial section like $t$ will exist in any case
and the problem of existence of a nontrivial section of $l(-F)|_c$
is just reduced to the corresponding problem for $\bar{l}(-F)$
(note that neither $D$ nor $\bar{D}$ will be effective).\\
\noindent
{\em \underline{Remark:}} One can rephrase the procedure as follows. If one has
$i)$ the condition for a nontrivial section
(on the lhs of (\ref{implication})) fulfilled
and $ii)$ that the reduction condition holds
\begin{eqnarray}
i) \;\;\; h^0(c, \bar{l}(-F)|_c)>0 &\Longleftrightarrow& \bar{D}\sim \bar{D}' \geq 0
\; \Longleftrightarrow \; \bar{D}+(f)\geq 0\\
ii) \;\;\; \;\;\; \;\;\; \;\;\; \;\;\; reduction &\Longleftrightarrow& D\geq p\bar{D}
\end{eqnarray}
(here $\sim$ is linear equivalence)
then one gets a nontrivial section for the original bundle
\begin{eqnarray}
\label{reduction in remark}
h^0(c, l(-F)|_c)>0 &\Longleftrightarrow& D\sim D' \geq 0
\; \Longleftarrow \; D+(f^p)\geq p(\bar{D}+(f))\geq 0
\end{eqnarray}

Precise reduction amounts according to (\ref{divisor condition})
to the effectivity of $\tilde{D}$; this implies that the
complete linear system $|\tilde{D}|$
(of effective divisors, linearly equivalent to $\tilde{D}$)
is nonempty, what just signals the existence of
a nonzero section $t$ of $\tilde{{\cal F}}$
where $\tilde{{\cal F}}={\cal O}_c(\tilde{D})$ is the line
bundle on $c$ such that $\bar{l}(-F)^p|_c\otimes \tilde{{\cal F}}\cong l(-F)|_c$.
So reduction implies $h^0(c, \tilde{{\cal F}})>0$.

\subsection{Strong version of the reduction condition (on ${\cal E}$)}

Actually we will usually assume the following sharper condition on ${\cal E}$
\begin{eqnarray}
\label{sufficient reduction condition}
\Big(\bar{l}(-F)\Big)^p & \hookrightarrow & l(-F)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\hspace{2cm}
\mbox{(strong reduction condition)}
\end{eqnarray}
This condition will imply (\ref{precise reduction condition})
and is easy to check; however is is unnecessarily sharp,
i.e. it is only a sufficient condition.
(\ref{sufficient reduction condition}) amounts to the effectiveness of
$(\alpha - p\bar{\alpha})s$ and $(\beta - p\bar{\beta} )F$, in other words
\begin{eqnarray}
\label{nec cond for suff reduc}
p\bar{\alpha} \leq  \alpha \;\; , \;\;\;\; p\bar{\beta}  \leq  \beta
\end{eqnarray}
As $\beta < 0$ one therefore needs to have $\bar{\beta} <0$;
we will also assume $\bar{\alpha}>0$.

\section{\label{Equality condition}The equality condition}

\resetcounter

The equality condition is the condition which will connect
the existence of a nontrivial section of $\bar{l}(-F)|_c$ with
the vanishing of a corresponding determinantal expression, in precise analogy
to the corresponding phenomenon for $l(-F)|_c$,
cf.~(\ref{extrinsic determinant criterion}).

If one finds a line bundle $\bar{l}(-F)$ fulfilling
(\ref{precise reduction condition})
and wants to use this to find a factor of $Pfaff$
one still has to make sure some things.
The first is the possibility to control $h^0(c, \bar{l}(-F)|_c)>0$ in
(\ref{implication}) again by a determinantal function like in
(\ref{extrinsic determinant criterion}).
For this one needs an equality condition
(\ref{equality of h^1 dim's}) and when this holds one will have
$\det \bar{\iota}_1 \, | \, \det \iota_1$
(modulo the remark before (\ref{general Pfaff factor}))
which reduces the problem of finding a zero of the polynomial $\det \iota_1$
of degree $h^1({\cal E}, l(-F))$ to the polynomial $\det \bar{\iota}_1$ of
degree $h^1({\cal E}, \bar{l}(-F))$ (the degree is the sum of all degrees
in the individual moduli, cf.~for example (\ref{degree 3 pol})).

In this connection we note that one has to check that the lhs of
(\ref{equality of h^1 dim's}) should be indeed nonvanishing. In this question
one has {\em with the assumptions}
$\bar{\alpha}>0, \bar{\beta}<0$ and (\ref{h^1 reduction}) that
$h^1({\cal E}, \bar{l}(-F))>0$ if not either i) $\bar{l}(-F)={\cal O}_{{\cal E}}(s-F)$
or ii) $\bar{l}(-F)={\cal O}_{{\cal E}}(\bar{\alpha}s-F)$ with $\chi=0$;
i) is excluded by $\bar{\alpha}\geq n$
(cf.~remark after (\ref{equality of h^1 dim's})),
ii) by (\ref{betabar fixing}) and (\ref{last expression}).

\subsection{\label{Numerical evaluation}
Numerical evaluation of the equality condition}

{}For $\bar{l}$ most of the arguments
in subsect.~\ref{contribution criteria} are not applicable: one has neither
$h^0(c, \bar{\l}(-F)|_c)=h^1(c, \bar{\l}(-F)|_c)$, because now
$c_1(\bar{V}|_b)\neq 0$,
nor\footnote{as $\alpha-n>0$ from $\alpha=n(\lambda+1/2), \lambda > 1/2$
does not give necessarily $\bar{\alpha}-n>0$}
the vanishing of the $H^2$-terms in the long exact
sequence (\ref{long exact sequence}).
Nevertheless, from the assumption $\bar{\alpha}>0$
and from $\bar{\beta}<0$, one has the
vanishing of the first two terms such that still
$H^0(c, \bar{\l}(-F)|_c) \cong \mbox{Ker}\;\bar{\iota}_1$:
\begin{eqnarray}
0 \rightarrow H^0\Big(c, \bar{\l}(-F)|_c\Big)
\rightarrow H^1\Big({\cal E}, \bar{l}(-F-c)\Big) \stackrel{\bar{\iota}_1}{\rightarrow}
H^1\Big({\cal E}, \bar{l}(-F)\Big) \rightarrow
H^1\Big(c, \bar{\l}(-F)|_c\Big) \rightarrow \dots \;\;\;
\end{eqnarray}
One now wants to find cases where $\bar{\iota}_1$ is a map between spaces of
equal dimension\footnote{Note that the evaluation by (\ref{reduction})
also of the lhs of (\ref{equality of h^1 dim's}),
which has with $\bar{\alpha}-n$ a smaller $s$-coefficient in
$\bar{l}(-F)={\cal O}_{{\cal E}}((\bar{\alpha}-n)s+(\bar{\beta}-r)F)$
which could be potentially $\leq 0$,
will be appropriate as the condition
$\bar{\beta}\leq (\bar{\alpha}-n+1)\chi+r$ from (\ref{H^1 reduction conditions})
is fulfilled
(the rhs is $> 0$ by (\ref{r chi relation}) and $\bar{\alpha}\geq 0$).}
\begin{eqnarray}
\label{equality of h^1 dim's}
h^1\Big({\cal E}, \bar{l}(-F-c)\Big)&=&h^1\Big({\cal E}, \bar{l}(-F)\Big)
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mbox{(equality condition)}
\end{eqnarray}
We proceed now by distinguishing
the cases\footnote{\label{al geq n footn}note that $\bar{\alpha}<n$ where
$h^1({\cal E}, \bar{l}(-F-c))=0$
is not interesting for us as $h^0(c, \bar{l}(-F)|_c)=0$ would mean that
one has no nontrivial section to start with
in the procedure (\ref{reduction phenomenon}).}
$\bar{\alpha}>n$ and $\bar{\alpha}=n$.

\subsubsection{\label{equ cond al > n}
The equality condition in the case $\bar{\alpha}>n$}

To compute the difference of the sides of
(\ref{equality of h^1 dim's}), one can use formula (\ref{h^1 reduction})
for $\bar{\alpha}, \bar{\beta}$ {\em and}, in this case of $\bar{\alpha}>n$,
{\em also for} $\bar{\alpha}-n, \bar{\beta}-r$
and gets
\begin{eqnarray}
h^1\Big({\cal E}, \bar{l}(-F-c)\Big)-h^1\Big({\cal E}, \bar{l}(-F)\Big)
&=& (\bar{\alpha}-\frac{n}{2})(r-n\chi)+(\beta+1-\frac{r+\chi}{2})n
\end{eqnarray}
Equivalently, with $h^2({\cal E}, \bar{l}(-F-c))=0$, one can
use the index formula to compute
\begin{eqnarray}
\label{index computation}
h^1\Big({\cal E}, \bar{l}(-F-c)\Big)-h^1\Big({\cal E}, \bar{l}(-F)\Big)
&=&
h^0(c, \bar{l}(-F)|_c)-h^1(c, \bar{l}(-F)|_c)\nonumber\\
&=&\deg \bar{l}(-F)|_c - \deg K_c^{1/2}\nonumber\\
&=&\Big(\bar{\alpha}s+\bar{\beta}F-\frac{1}{2}(ns+rF+(\chi-2)F)\Big)(ns+rF)
\nonumber\\
\label{final expression}
&=&(\bar{\alpha}-\frac{n}{2})(r-n\chi)+(\bar{\beta}+1-\frac{r+\chi}{2})n\;\;\;
\end{eqnarray}
Thus one gets vanishing just for
\begin{eqnarray}
\label{betabar fixing}
-(\bar{\beta}+1)&=&\frac{\bar{\alpha}-n}{n}(r-n\chi)-\frac{n+1}{2}\chi
\end{eqnarray}
Note that this is just again the condition (\ref{beta rewriting})
(i.e., one has (\ref{equality of h^1 dim's}) if {\em and only if}
$c_1(\bar{V}|_{{\cal E}})=0$, expressed by the relation (\ref{beta rewriting})
between $\bar{\alpha}$ and $\bar{\beta}$, in this case).
The integrality requirement means for $n$ odd
(or equally for $n$ even and $\chi$ even) that $n|\bar{\alpha} r$
while for $n=2m$ and $\chi$ odd that $\bar{\alpha}r/m$ must be an odd integer.

\subsubsection{The equality condition in the case $\bar{\alpha}=n$}

Here one computes $h^1({\cal E}, {\cal O}(0s+(\bar{\beta}-r)F))=-(\bar{\beta}+1)+r$
and gets
\begin{eqnarray}
h^1\Big({\cal E}, \bar{l}(-F-c)\Big)-h^1\Big({\cal E}, \bar{l}(-F)\Big)
&=& -(\bar{\beta}+1)+r+\bar{\alpha}(\bar{\beta}+1)
-\Big(\frac{\bar{\alpha}(\bar{\alpha}+1)}{2}-1\Big)\chi\;\;\;\;\;\;\;\;\;
\end{eqnarray}
Alternatively one gets this by computing the index on $c$
(using (\ref{final expression})) and adding
\begin{eqnarray}
h^2\Big({\cal E}, \bar{l}(-F-c)\Big)
&=&h^0\Big({\cal E}, K_{{\cal E}}\otimes {\cal O}_{{\cal E}}(c)\otimes (\bar{l}(-F))^{-1}\Big)
=h^0\Big(b, {\cal O}_b(\chi-2+r-\bar{\beta})\Big)\nonumber\\
&=&-(\bar{\beta}+1)+r+\chi
\end{eqnarray}

Thus one gets vanishing just for
\begin{eqnarray}
\label{last expression}
-(\bar{\beta}+1) & = & \frac{r}{n-1} - (\frac{n}{2}+1)\chi
\end{eqnarray}
As this expression has to be integral one notes
that for $n$ or $\chi$ even one needs to have $r=(n-1)\rho$ with
$\rho$ a positive integer; if $n=2m+1$ and $\chi$ are odd
one needs to have $\frac{r}{2m}-\frac{1}{2}\in {\bf Z}$, i.e.~that
$r/m$ is an odd integer
(for example $r$ must be odd for $n=3$ with $\chi$ odd).

\subsection{\label{interpretation}
Interpretation of the equality and reduction condition}

We rephrase the condition (\ref{equality of h^1 dim's}) for reduction,
given in the numerical parameters above,
in a more geometric form by restricting the discussion of the line bundles
from ${\cal E}$ to $c$.

\subsubsection{\label{interpret al > n}
The conditions in the case $\bar{\alpha}>n$}

The index computation (\ref{index computation}) shows that
the {\em equality condition} amounts to
\begin{eqnarray}
\label{interpretation equality condition}
\bar{l}(-F)|_c & = & K_c^{1/2}\otimes \bar{{\cal F}}
\end{eqnarray}
with $\bar{{\cal F}}$ a flat bundle on $c$ playing the same role for
$\bar{l}(-F)|_c$ as does ${\cal F}$ for $l(-F)|_c$, cf.~(\ref{degree equalities}).
$\bar{{\cal F}}$ being a flat bundle this is just an equation of degrees:
$\deg \bar{l}(-F)=\deg K_c^{1/2}=\deg l(-F)$.

Now the precise {\em reduction condition} (\ref{precise reduction condition})
implies, together with (\ref{interpretation equality condition})
and (\ref{degree equalities}),
the necessary condition $p\, \deg K_c^{1/2}\leq \deg K_c^{1/2}$.
So for $p\geq 2$ one gets the condition $\deg K_c \leq 0$
which in view of (\ref{canonical degree})
comes down to $r - \frac{n-1}{2}\chi -1\leq 0$ (compare the reasoning
in app.~\ref{Reduction cases}). One gets, in view of
(\ref{r chi relation sharp}), that $r=1, \chi=0$, i.e. the spectral
curve $c=C\cap {\cal E}$ is an elliptic curve.

\subsubsection{The conditions in the case $\bar{\alpha}=n$}

Here the {\em equality condition} amounts to
\begin{eqnarray}
\label{interpretation2 equality condition}
\bar{l}(-F)|_c & = & K_c^{1/2}\otimes ( \omega_{c/b}^{1/2})^{-\frac{1}{n-1}}
\otimes \bar{{\cal F}}
\end{eqnarray}
where $\omega_{c/b}=K_c\otimes \pi_c^* K_b^{-1}$ is the relative dualizing sheaf
with $\deg \omega_{c/b}^{1/2}=n(r - \frac{n-1}{2}\chi)$.
Although written, for analogy with (\ref{interpretation equality condition}),
with the $(n-1)^{th}$ root of the line bundle $\omega_{c/b}^{1/2}$
this is essentially just meant as an equation of degrees
(cf.~also the remarks after (\ref{last expression})).

Now the precise {\em reduction condition} (\ref{precise reduction condition})
gives, together with (\ref{interpretation2 equality condition})
and (\ref{degree equalities}),
the necessary condition
$p \, \deg K_c^{1/2}-\frac{p}{n-1}\deg \omega_{c/b}^{1/2}\leq
\deg K_c^{1/2}$ and so
\begin{eqnarray}
(p-1)(r - \frac{n-1}{2}\chi -1)& \leq &\frac{p}{n-1}(r - \frac{n-1}{2}\chi)
\end{eqnarray}
or
\begin{eqnarray}
\label{final form}
\Big(p(n-2)-(n-1)\Big) (r - \frac{n-1}{2}\chi -1)& \leq & p
\end{eqnarray}
This is usefully applicable for $n=3, p>2$ and $n=4, p\geq 2$,
giving all cases in app.~\ref{list subsection}.
Actually we will proceed slightly differently in
the concrete derivation in app.~\ref{list subsection}
(using $\lambda \geq p - \frac{1}{2}$ in (\ref{first bound})
one would get back (\ref{final form})).

\subsection{All (strong) reduction cases with equality condition}

One has always $\lambda > \frac{1}{2}$ and
$\lambda \in \frac{1}{2}+{\bf Z}$ for $n$ odd, while for $n$ even
the case $\lambda \in \frac{1}{2}+{\bf Z}$ needs $\chi$ even
and $\lambda \in {\bf Z}$ needs $r-\chi$ even.
The condition $\beta <0$ can, according to (\ref{sharper bound}),
be rephrased in the parameters as
$r-n\chi\geq \frac{(1+n)/2}{\lambda - 1/2}\chi$; furthermore $r-n\chi>0$
even for $\chi =0$ by (\ref{r chi relation sharp}).
We are usually interested mainly (physically) in $n=2,3,4,5$ and
$\chi =0$ (for illustration) or $1$ ($b$ isolated).
Now the detailed study in app.~\ref{Reduction cases} gives the following.

\subsubsection{\label{red cases for bar al > n}Cases with $\bar{\alpha}>n$}

For $SU(n)$ bundles one has for
$\chi=0, r=1, \lambda +\frac{1}{2}\in p{\bf Z}^{>1}$
just the $p$-case
\begin{eqnarray}
\l(-F)&=&{\cal O}_{{\cal E}}\Big(n(\lambda+\frac{1}{2})s-(\lambda + \frac{1}{2})F\Big)\\
\label{r=1 general case}
\bar{l}(-F)&=&
{\cal O}_{{\cal E}}\Big(n\frac{\lambda+\frac{1}{2}}{p}s-\frac{\lambda + \frac{1}{2}}{p}F\Big)
\end{eqnarray}
Now $\chi=0, r=1$ were just the conditions for
an elliptic $c$, cf.~(\ref{canonical degree}). One gets
\begin{eqnarray}
K_c&=& {\cal O}_{{\cal E}}(ns+(r+\chi-2)F)|_c={\cal O}_{{\cal E}}(ns-F)|_c\\
l(-F)&=&{\cal O}_{{\cal E}}(ns-F)^{\otimes (\lambda + \frac{1}{2})}|_c
\end{eqnarray}
So here $K_c\cong {\cal O}_c\cong l(-F)|_c$,
$h^0(c, l(-F)|_c)=1>0$ and $Pfaff \equiv 0$, cf.~Ex.~3 below.

\subsubsection{\label{red cases for bar al = n}Cases with $\bar{\alpha}=n$}

Here one has
\begin{eqnarray}
\l(-F)&=&{\cal O}_{{\cal E}}\Big(n(\lambda+\frac{1}{2})s+\beta F\Big)
\; , \;\;\;\;\;\;
\beta =  -\Bigg( (\lambda - \frac{1}{2})r-(n\lambda+\frac{1}{2})\chi+1\Bigg)\\
\label{bar l(-F)}
\bar{l}(-F)&=&{\cal O}_{{\cal E}}\Big(ns + \bar{\beta}F\Big)\; , \;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\,
\bar{\beta}=-\Bigg( \frac{1}{n-1}r-(\frac{n}{2}+1)\chi+1\Bigg)
\end{eqnarray}
The $(\chi, r, \lambda, p)$-list
of occurring cases for $SU(n)$ bundles is given in app.~\ref{list subsection}.
Furthermore
\begin{eqnarray}
\deg \det \iota_1 & = & h^1({\cal E}, l(-F))=-\alpha(\beta+1)
+\Big(\frac{\alpha(\alpha+1)}{2}-1\Big)\chi\nonumber\\
&=&n(\lambda^2-\frac{1}{4})(r-\frac{n}{2}\chi)
-\chi\\
\deg \det \bar{\iota}_1&=&h^1({\cal E}, \bar{l}(-F))=
-(\bar{\beta}+1)+r = \frac{n}{n-1}r-(\frac{n}{2}+1)\chi\nonumber\\
&=&\frac{r}{n-1}+(r-\frac{n}{2}\chi)-\chi
\end{eqnarray}

\section{\label{The vanishing condition}The vanishing condition
(including Example 3)}

\resetcounter

When considering in sect.~\ref{red cases for bar al > n} the reduction case
with $\bar{\alpha}>n$ we saw that for $\chi=0, r=1, \lambda \in \frac{1}{2}+{\bf Z}$
with $c$ elliptic one gets $Pfaff \equiv 0$ by directly computing
$h^0(c, l(-F)|_c)=1>0$. In that case both, $l(-F)|_c$ and $\bar{l}(-F)|_c$,
were powers of the trivial bundle $K_c$.

This type of argument can be applied more widely.
Note that always $\dim \ker \bar{\iota}_1\leq h^0(c, \bar{l}(-F)|_c)$
by the analog of the long exact sequence (\ref{long exact sequence})
for $\bar{l}$ (with equality if $h^0({\cal E}, \bar{l}(-F))=0$). Therefore,
even is one is not in an equality case where (\ref{equality of h^1 dim's})
holds (when vanishing of $h^0(c, \bar{l}(-F)|_c)=\dim \ker \bar{\iota}_1$
will be controlled by $\det \bar{\iota}_1$) one gets
\begin{eqnarray}
\mbox{(vanishing condition)}\;\;
h^1\Big({\cal E}, \bar{l}(-F-c)\Big)-h^1\Big({\cal E}, \bar{l}(-F)\Big)>0& \Longrightarrow &
h^0\Big(c, \bar{l}(-F)|_c\Big)>0\;\;\;\;\;\;\;\;\;
\end{eqnarray}
If the reduction condition holds for $p$ then the argument will be completed
in the usual way by noting that the existence of $s\in H^0(c, \bar{l}(-F)|_c),
s\not = 0$ implies $s^p t\in H^0(c, l(-F)|_c)$; therefore if the
(moduli-independent) vanishing condition holds then $Pfaff \equiv 0$.

Let us see how this condition can be applied. If one is in the case
$\bar{\alpha}>n$ one has $h^1({\cal E}, \bar{l}(-F-c))-h^1({\cal E}, \bar{l}(-F))
=\deg \bar{l}(-F)|_c-\deg K_c^{1/2}$ which can not be positive if we want at
the same time the necessary condition for reduction
$\deg \bar{l}(-F)|_c\leq \deg l(-F)|_c=\deg K_c^{1/2}$ to be fulfilled.
Therefore
$\bar{\alpha}=n$, i.e.~$\bar{l}(-F)={\cal O}_{{\cal E}}(ns+\bar{\beta}F)$ and we have
\begin{eqnarray}
\label{positivity condition}
h^1\Big({\cal E}, \bar{l}(-F-c)\Big)-h^1\Big({\cal E}, \bar{l}(-F)\Big)&=&
(n-1)(\bar{\beta}+1)+r-\Big(\frac{n(n+1)}{2}-1\Big)\chi\nonumber\\
&=&(n-1)\Big( \bar{\beta}+1-\frac{n+2}{2}\chi\Big)+r
\end{eqnarray}

For $SU(3)$ or $SU(4)$ bundles, the only new case (besides the above mentioned
$\chi=0, r=1$ case) where this is strictly positive is $n=3, \lambda = 3/2$ with
$r\not\equiv \chi(2)$, cf.~app.~\ref{vanishing n=3 cases}.

Let us take a closer look on the case of $SU(3)$ bundles with $\lambda = 3/2$ as
this will be an important case in sect.~\ref{Examples}. Here one has $l(-F)=
{\cal O}_{{\cal E}}(6s-(r-5\chi+1)F)$ where $\beta<0$ means $r\geq 5\chi$. Let us take
\begin{eqnarray}
\bar{l}(-F)&=&\left\{ \begin{array}{lll}
{\cal O}_{{\cal E}}\Big(3s-(\frac{r-5\chi}{2}+1)F\Big) & \;\;\;\mbox{if} &
r\equiv \chi(2)\\
{\cal O}_{{\cal E}}\Big(3s-\frac{r-5\chi+1}{2}F\Big) & \;\;\;\mbox{if}&r\not\equiv\chi(2)
\end{array} \right.
\end{eqnarray}

If $r\equiv \chi(2)$ one gets a moduli-dependent statement,
cf.~sect.~\ref{n=3, la =3/2}.

By contrast, if $r\not\equiv\chi(2)$ one gets the moduli-independent
statement $Pfaff\equiv 0$ as here (\ref{positivity condition}) becomes
just $1$ (alternatively one can also argue directly
that $h^0(c, \bar{l}(-F)|_c)=\frac{3}{2}(r-\chi-1)-3(r-\chi-1)
+h^0(c, {\cal O}_{{\cal E}}((r+\chi-2+\frac{r-5\chi+1}{2})F)|_c)\geq 1$).
For $\chi=0, r=1$ cf.~Example $3$.
For an algebraic vanishing argument
in the somewhat special $\chi=0$ case cf.~app.~\ref{chi = 0 app}.

\section{\label{difference subsection}On the difference between
the strong and the precise reduction condition}

\resetcounter

Let us now come back to the precise version of the reduction condition
on $c$ instead of the strong version on ${\cal E}$.
The latter had with (\ref{nec cond for suff reduc})
a precise numerical expession whereas (\ref{precise reduction condition})
implies only the necessary condition for the degrees
\begin{eqnarray}
\label{nec cond for prec reduc}
p(\bar{\alpha}s+\bar{\beta}F)(ns+rF) & \leq & (\alpha s+\beta F)(ns+rF)
\end{eqnarray}
or explicitly\footnote{\label{barbeta geq 0}
Note that we always work under the assumption
$\bar{\alpha}\geq n$, cf.~footn.~\ref{al geq n footn}; here the case
$\bar{\alpha}>n$, which implies that the rhs of
(\ref{nec cond for prec reduc explic}) is $\leq 0$
(cf.~sect.~\ref{interpret al > n}),
still gives $\bar{\beta}\leq 0$ and even $<0$ by (\ref{r chi relation sharp});
however a case $\bar{\alpha}=n, \bar{\beta}\geq 0$ is possible and will
become relevant in sect.~\ref{Exceptional}
and in the Example 1 in sect.~\ref{Ex 1 sect}.}
\begin{eqnarray}
\label{nec cond for prec reduc explic}
p\Big( \bar{\alpha}(r-n\chi)+n\bar{\beta}\Big) & \leq &
n \Big(r-\frac{n-1}{2}\chi -1\Big)
\end{eqnarray}
So the logical relations are
\begin{eqnarray}
strong\; reduction \;(equ.~(\ref{nec cond for suff reduc}))\;
\buildrel i) \over \Longrightarrow \;
precise\; reduction \;(equ.~(\ref{precise reduction condition}))\;
\buildrel ii) \over \Longrightarrow \;equ.~(\ref{nec cond for prec reduc})\;\;\;\;\;
\end{eqnarray}
From the onesided implications two questions arise.

i) How much widened is the domain of possible $\bar{l}(-F)$'s
by considering the precise condition (\ref{precise reduction condition})
instead of the sharpened condition (\ref{nec cond for suff reduc}) ?

ii) The question about sufficiency of the condition
(\ref{nec cond for prec reduc}) for the precise reduction
(\ref{precise reduction condition}):
when is $D-p\bar{D}$ on $c$ (cf.~(\ref{divisor condition})),
which has degree $\geq 0$ by (\ref{nec cond for prec reduc}),
actually even effective ?

Also in connection with these questions we notice the following.

ad i) Note that if the strong reduction condition is violated
we {\em need actually to check explicitely}
that one has $h^0(c, \bar{l}(-F)|_c)>0$
for which we argued only under certain assumptions
in the remark before sect.~\ref{Numerical evaluation}; for example
$\bar{\beta}<0$ followed only because
(\ref{nec cond for suff reduc}).$^{\ref{barbeta geq 0}}$

ad ii)
{\em If the necessary condition
(\ref{nec cond for prec reduc}) for precise reduction is actually saturated}
then $t\in H^0(c, \tilde{{\cal F}})$, cf.~remark after (\ref{reduction in remark}),
is constant ($\tilde{{\cal F}}$ is flat and then even trivial;
a degree zero effective divisor on $c$ is zero);
so, then reduction holds just if $l(-F)|_c\cong \bar{l}(-F)^{\otimes p}|_c$.

\subsection{\label{Exceptional}The exceptional case $\bar{\beta}=0$}

We remarked above that $\bar{\beta}<0$ is necessary to fulfill
(\ref{sufficient reduction condition}),
i.e. concretely $p\bar{\beta}\leq \beta$.
However, as pointed out, this condition is unnecessarily sharp.
Actually one wants only (\ref{precise reduction condition}).
For this (\ref{nec cond for suff reduc}) is not necessary (so $\bar{\beta}$
might be $\geq 0$; cf.~footn.~\ref{barbeta geq 0});
necessary is (\ref{nec cond for prec reduc}).

Let us try to allow $\bar{\beta}=0$
(this will be relevant for Example 1 in sect.~\ref{Ex 1 sect});
consider the first two terms\footnote{Concerning the $H^2$-terms
the second one vanishes
by (\ref{H^2 vanishing conditions}) as $\bar{\alpha}>0$
but $h^2({\cal E}, {\cal O}_{{\cal E}}((\bar{\alpha}-n)s-rF))=0$ only if $\bar{\alpha}>n$
whereas one gets for $\bar{\alpha}=n$ that $h^0(b, {\cal O}_b(r+\chi-2))=r+\chi-1$
if not $\chi=0, r=1$.}
in the long exact sequence
(\ref{long exact sequence}) for $\bar{l}(-F)$:
although still $h^0({\cal E}, {\cal O}_{{\cal E}}((\bar{\alpha}-n)s-rF))=0$ for the
first term, one has $h^0({\cal E}, {\cal O}_{{\cal E}}(\bar{\alpha}s))=1$ for the
second one if $\chi >0$ and $=\bar{\alpha}$ if $\chi=0$.
Thus $\dim \ker \bar{\iota}_1$ equals $h^0(c, \bar{l}(-F))$ {\em minus}
$1$ or {\em minus} $\bar{\alpha}$ if $\chi>0$ or $=0$, resp.; so the vanishing
of $\det \bar{\iota}_1$ (at a point in the moduli space ${\cal M}_{{\cal E}}(c)$)
still indicates the existence of a nontrivial
section of $\bar{l}(-F)|_c$ and one still has
$\det \bar{\iota}_1|\det \iota_1$.

Let us give the equality condition for the case $\bar{\beta}=0$. Then one has
$h^1({\cal E}, \bar{l}(-F))=-(\bar{\alpha}-1)(\bar{\beta}+1)
+(\frac{\bar{\alpha}(\bar{\alpha}+1)}{2}-1)\chi$
with the remark after (\ref{h^1 reduction}) and gets
$-(\bar{\beta}+1)=\frac{r}{n-2}-\frac{(n+2)(n-1)}{2(n-2)}\chi$
which determines $r$
(where (\ref{r chi relation sharp}) excludes $n=2$
and (\ref{r chi relation}) $n>2, \chi=0$)
\begin{eqnarray}
\label{exceptional r fix}
r-n\chi & = & (n-2)\Big(\frac{n+1}{2}\chi-1\Big)
\end{eqnarray}
Here, with $r$ assumed fixed by (\ref{exceptional r fix}),
the interpretation as in sect.~\ref{interpretation}
amounts to
\begin{eqnarray}
\bar{l}(-F)|_c & = & K_c^{1/2}\otimes (K_c^{1/2})^{-\frac{1}{n-1}}\otimes \bar{{\cal F}}
\end{eqnarray}
For $n=3$, where previously $\beta<0\Leftrightarrow r\geq 5\chi$,
we get $\bar{\beta}=0\Leftrightarrow r=5\chi-1$, cf.~Example 1.

This exceptional bundle has the degree on $c$
(using in both cases the $r$-fix (\ref{exceptional r fix}))
\begin{eqnarray}
\deg (\bar{l}(-F))|_c=\deg {\cal O}_{{\cal E}}(ns-0F)|_c&=&n(r-n\chi)
=n(n-2)\Big(\frac{n+1}{2}\chi-1\Big)\\
\deg K_c^{1/2}&=&n(n-1)\Big(\frac{n+1}{2}\chi-1\Big)
\end{eqnarray}
Therefore,
although the conditions (\ref{nec cond for suff reduc}) for
strong reduction (\ref{sufficient reduction condition})
are violated here as $p0\not\leq \beta <0$,
the necessary condition (\ref{nec cond for prec reduc})
for the precise reduction condition (\ref{precise reduction condition})
is fulfilled if $p(n-2)\leq n-1$, i.e.~for $n\leq 1+ \frac{p}{p-1}$. This
holds for $n=2$ or for $n= 3, \, p=2$
where (\ref{nec cond for prec reduc}) is even saturated
(and of course trivially for $p=1$).
When is not only the necessary condition
(\ref{nec cond for prec reduc}) for precise reduction but even the
precise reduction condition (\ref{precise reduction condition}) itself
fulfilled ? Consider the case $n=3, p=2$ where
\begin{eqnarray}
l(-F)&=&{\cal O}_{{\cal E}}\Big(3(\lambda + \frac{1}{2})s-(\lambda -\frac{3}{2})(2\chi -1)F\Big)\\
\label{n=3 exceptional}
\bar{l}(-F)&=&{\cal O}_{{\cal E}}(3s-0F)
\end{eqnarray}
Note that $l_{\lambda = 3/2}(-F)=\bar{l}(-F)^{\otimes 2}$,
and so a fortiori $l_{\lambda = 3/2}(-F)|_c=\bar{l}(-F)^{\otimes 2}|_c$,
but the necessary condition $\beta<0$ from (\ref{third criterion})
requires here $\lambda > 3/2$, cf.~Example 1 in sect.~\ref{Examples}.
But according to the remark
before sect.~\ref{Exceptional} here reduction does hold
precisely if $l_{\lambda}(-F)|_c=\bar{l}(-F)^{\otimes 2}|_c$;
so reduction holds just if $l_{\lambda}(-F)|_c=l_{\lambda = 3/2}(-F)|_c$
and one needs to know the sublocus in the moduli space ${\cal M}_{{\cal E}}(c)$
where (the isomorphism class of) $l(-F)|_c$ is actually independent of $\lambda$,
i.e.~where $\Lambda={\cal O}_{{\cal E}}(ns-(r-n\chi)F)|_c$ becomes trivial;
now cf.~sect.~\ref{Ex 1 sect}.

\section{\label{Examples}Some examples of $SU(3)$ bundles}

\resetcounter

\subsection{Overview over the three main examples}

We illustrate the theory with some examples
(for Example 3 of $Pfaff=0$ cf.~sect.\ref{The vanishing condition})
\begin{center}
\begin{tabular}{|c||c|c|c|}
\hline
                & Ex $1$      & Ex $2$        & Ex $4$ \\
\hline\hline
$\chi$          & $1$         & $1$           & $0$ \\
\hline
$\lambda$       & $5/2$       & $3/2$         & $3/2$ \\
\hline
$c$             & $3s+4F$     & $3s+5F$       & $3s+2F$ \\
\hline
$K_c$           &${\cal O}_{{\cal E}}(3s+3F)|_c$&${\cal O}_{{\cal E}}(3s+4F)|_c$&${\cal O}_{{\cal E}}(3s)|_c$ \\
\hline
$\Lambda$       & ${\cal O}(3s-F)$ & ${\cal O}(3s-2F)$  & ${\cal O}(3s-2F)$  \\
\hline
codim $\Sigma_{\Lambda}$
                &      1      &      2        &    1          \\
\hline
$l(-F)$         & ${\cal O}(9s-F)$ & ${\cal O}(6s-F)$   & ${\cal O}(6s-3F)$\\
\hline
$Pfaff=\det \iota_1$
                &$f^{11}Q_{11}$    & $f^4$         & $f^4$ \\
\hline
$\bar{l}(-F)$   & ${\cal O}(3s)$   & ${\cal O}(3s-F)$   & ${\cal O}(3s-2F)$ \\
\hline
$f=\det \bar{\iota}_1$
                &$f_{\Lambda}$&   $Res(B, A)$ & $f_{\Lambda}$\\
\hline
\end{tabular}
\end{center}
For these Examples we find that
$f:=\det \bar{\iota}_1$ equals $f_{\Lambda}$
if that relation is possible at all, that is if the
codimension of the $f_{\Lambda}=0$ locus $\Sigma_{\Lambda}$ is $1$,
as in Examples $1$ and $4$:
in Example 4 the naive codimension $r-n\chi$ (cf.~sect.~\ref{Ri section})
of $\Sigma_{\Lambda}$ is $2$
but $\bar{l}(-F)=\Lambda$ leads to $f=f_{\Lambda}$;
in Example 2 the codimension is $2$:
one has to demand $R_1=0$ and $R_2=0$,
i.e.~$(R_1, R_2)=f_{\Lambda}=(Res(B, A_i))_i$
where $A=\prod_{i=1}^{r-n\chi}A_i$
(cf.~footn.~\ref{vector-valued fct})
and one has then that codim $\Sigma_{\Lambda}>1$ (the general case).
Although $\Sigma_{\Lambda}$ is a sublocus of $Pfaff=0$
(cf.~the remark after (\ref{effectivity equivalence}))
the 'scalar function' dividing $Pfaff$ is $Res(B, A)$,
cf.~footn.~\ref{multiple resultant zero}.

\subsection{\label{n=3, la =3/2}Bundles with $\lambda =3/2$}

Here one has $l(-F)={\cal O}_{{\cal E}}(6s-(r-5\chi+1)F)$
where $\beta<0$ means $r\geq 5\chi$. Let us take
\begin{eqnarray}
\bar{l}(-F)&=&\left\{ \begin{array}{lll}
{\cal O}_{{\cal E}}\Big(3s-(\frac{r-5\chi}{2}+1)F\Big) & \;\;\;\mbox{if} &
r\equiv \chi(2)\\
{\cal O}_{{\cal E}}\Big(3s-\frac{r-5\chi+1}{2}F\Big) & \;\;\;\mbox{if}&r\not\equiv\chi(2)
\end{array} \right.
\end{eqnarray}

For $r\not\equiv\chi(2)$ we found $Pfaff\equiv 0$
in sect.~\ref{The vanishing condition}.
For $r\equiv \chi(2)$ one gets a moduli-dependent statement:
if an $\rho\in H^0(c, \bar{l}(-F)|_c), \rho\not =0$ exists
(what is described by $\det \bar{\iota}_1=0$)
then $\rho^2u$ and $\rho^2v$ are non-trivial elements of $H^0(c, l(-F)|_c)$,
what suggests for this case that $Pfaff=f^2g$.
Note that at most $Pfaff=f^4$ as
$\deg Pfaff=6r-10\chi, \deg f = \frac{3r-5\chi}{2}$.

\subsubsection{\label{Exa 2}The case $\chi\geq 1$
(including Example 2 as minimal $r$-value)}

For $r=5\chi$, where
$\deg Pfaff=4r, \deg f = r$ and $\chi\geq 1$ by (\ref{r chi relation sharp}),
one gets
\begin{eqnarray}
l(-F)&=&{\cal O}_{{\cal E}}(6s-F)\\
\bar{l}(-F)&=&{\cal O}_{{\cal E}}(3s-F)
\end{eqnarray}
One then has also $\rho z\in H^0(c, l(-F)|_c)$
such that possibly $Pfaff=f^3h$ in this case.

For $\chi=1$, where $c=3s+5F$, this is realized in Example $2$. Note
that there the matrix induced by $\bar{\iota}$ is $m_5={\cal D}_5$ in (\ref{D_5}),
such that $f=\det \bar{\iota}_1=Res(A_2,B_3)$, cf.~sect.~\ref{resultant}.

This is case $3^{\chi=1}$ of app.~\ref{list subsection}.
According to (\ref{the critical map}) one has then to consider
the map $\iota_1: H^1\Big({\cal E}, {\cal O}_{{\cal E}}(3s+(\beta -r)F)\Big)
\longrightarrow H^1\Big({\cal E}, {\cal O}_{{\cal E}}(6s+\beta F)\Big)$ between spaces
of dimension $6r-10\chi$
(for concrete matrix representations
cf.~app.~\ref{concrete matrix representations})
This map is induced by multiplication with an element
$\tilde{\iota}\in H^0\big( {\cal E}, {\cal O}_{{\cal E}}(3s+rF)\big)$.
The determinant of the $\iota_1$ for $r\equiv \chi (2)$ has as factor
the determinant of a corresponding
map $i_r: i_r:  H^1\Big({\cal E}, {\cal O}_{{\cal E}}\big(0s+(\bar{\beta}-r)F\big)\Big) \longrightarrow
H^1\Big({\cal E}, {\cal O}_{{\cal E}}\big(3s+ \bar{\beta}F\big)\Big)$
(again induced by multiplication with
$\tilde{\iota}\in H^0\Big({\cal E}, {\cal O}_{{\cal E}}(3s+rF)\Big)$,
so depending on the same moduli)
between two spaces of dimension $\frac{1}{4}(6r-10\chi)$.

Now, what was generically
$s|_c=\{p_0, t_1\}+\{p_0, t_2\}\, , \;
F_{t_i}|_c=c_{t_i}=\{p_0+p_i^++p_i^-, t_i\}$
(with $A_2(t_i)=0$, $i=1,2$, cf.~(\ref{s|_c generically}))
changes at the specialisation locus
$f=\det \bar{\iota}_1=\det {\cal D}_5 =Res(A_2,B_3)=0$ (where $B_3(t_1)=0$;
the non-generic case for (\ref{s|_c generically})):
the $(w)$ line becomes, in the ${\bf P}^{\bf 2}_{t_1}$-fibre of ${\cal W}_b$,
the $(z)$ line and $p_0$ becomes a three-fold point of $c_{t_1}$
\begin{eqnarray}
(z)|_c=3s|_c=\{3p_0, t_1\}+\{3p_0, t_2\} = F_{t_1}|_c + \{3p_0, t_2\}
\neq  F_{t_1}|_c +F_{t_2}|_c
\end{eqnarray}
In other words, in the specialisation locus $f=0$
one gets $(3s-F)|_c \sim$ effective
or equivalently $h^0(c, {\cal O}(3s-F)|_c)\geq 1$ (the existence of $\rho\neq 0$);
therefore also $(6s-F)|_c \sim$ effective or $h^0(c, l(-F)|_c)\geq 1$; in
other words $f\, | \, Pfaff$.

To demand that even $(3s-2F)|_c \sim$ effective
is a more restrictive condition,
i.e.~$R_1=0=R_2$ is a proper sublocus of $f=0$
(so here $f_{\Lambda}=(R_1, R_2)$, cf.~footn.~\ref{vector-valued fct}).
Note that $(3s-2F)|_c\sim $ effective $\Longrightarrow (3s-F)|_c\sim $ effective;
so, if\footnote{cf.~sect.~\ref{Ri section}, we call here $R_2=R_0$;
cf.~also app.~\ref{more than one root}}
$R_1=0=R_0$
at certain points in moduli space, then $f=0$ there, as
becomes manifest in the representation
\begin{eqnarray}
\label{f in the R_i}
f&=&\frac{1}{a_2^2}\det \left( {\begin{array}{ccc}
a_0  & a_1  & a_2 \\
R_1  & R_0  &   0 \\
  0  & R_1  & R_0
\end{array}} \right)
\end{eqnarray}

\subsubsection{\label{Ex 4 case}The case $\chi=0$
(including Example 4 as minimal $r$-value)}

For $\chi=0$ one has $r$ even and
$\deg Pfaff=12\frac{r}{2}$ and $ \deg f = 3\frac{r}{2}$.
The minimal $r$-value (which must be $> 0$)
is $2$, cf.~Example $4$, where $c=3s+2F$ and
\begin{eqnarray}
l(-F)&=&{\cal O}_{{\cal E}}(6s-3F)\\
\bar{l}(-F)&=&{\cal O}_{{\cal E}}(3s-2F)=\Lambda
\end{eqnarray}
Precisely on the specialisation locus $f=\det \bar{\iota}_1
=\det {\cal D}_3^{\chi=0}=0$, cf.~(\ref{D_3 chi=0}),
the degree $0$ line bundle $\bar{l}(-F)|_c$
gets a nonzero section and so becomes trivial,
in other words
\begin{eqnarray}
f_{\Lambda}&=&f
\end{eqnarray}
{\em \underline{Remark:} i)} Following strictly the procedure of example $2$ one
would expect codim $\Sigma_{\Lambda}=2$
and $f_{\Lambda}$ a two-component expression:
the condition $A_2|B_2$ is here\footnote{with minors
w.r.t.~a development of (\ref{D_3 chi=0}) w.r.t.~the $c$-row,
and in (\ref{f in the M_i}) also w.r.t.~the $b$-row}
$M^{c}_0=0=M^{c}_1$ while
\begin{eqnarray}
\label{Ex 4 resultant}
Res(B_2, A_2)&=&\det \left( {\begin{array}{cccc}
b_2  &  b_1 & b_0 & 0  \\
0    &  b_2 & b_1 & b_0\\
a_2  &  a_1 & a_0 & 0  \\
0    &  a_2 & a_1 & a_0
\end{array}} \right)
= \det \left( {\begin{array}{cc}
M_1^c  &  M_0^c  \\
M_2^c  &  M_0^c
\end{array}} \right)
\end{eqnarray}
(note that $0=a_0M_0^c-a_1M_1^c+a_2M_2^c$ allows to eliminate $M_2^c$).
Now one has
\begin{eqnarray}
\label{f in the M_i}
f&=&\frac{1}{a_2}\det \left( {\begin{array}{cc}
M_1^c  &  M_0^c  \\
M_1^b  &  M_0^b
\end{array}} \right)
\end{eqnarray}
such that the locus where $M^{c}_0=0=M^{c}_1$ is certainly a subset
of the $f=0$ locus, cf.~the remark after (\ref{Ri = 0 implication}).
But $A_2|B_2$ is only a sufficient, not a necessary
condition for $\Lambda|_c\cong {\cal O}_c$,
cf.~(\ref{Ri = 0 implication}).
Here, in this case of symmetry between $C, B$ and $A$ because of $\chi=0$,
the precise condition for $\Lambda|_c\cong {\cal O}_c$ turns out to be
just linear dependence among $C, B$ and $A$
which represents the {\em single} condition $f=0$;
this comes as here $\Lambda = \bar{l}(-F)$ and
demanding a nontrivial section over $c$ leads to the
determinantal condition $f=0$.\\
{\em ii)} That
$3s|_c\sim 2F|_c$ on the locus $f=0$ can also be seen
directly from $\det \bar{\iota}_1=\det {\cal D}_3^{\chi=0}=0$,
cf.~(\ref{D_3 chi=0}), instead of arguing via the long exact sequence.
The equation for the fibre of $c$ over $t_1=u/v\in b$, say, is
$C_2(t_1)z +B_2(t_1)x+A_2(t_1)y=0$ and these
three points lie also in the divisor $\alpha z + \beta x +\gamma y=0$,
understood as $3s|_c$,
where $\alpha = C_2(t_1), \beta=B_2(t_1), \gamma=A_2(t_1)$;
this divisor of degree $6$ contains three further points; we want
to see that these can arise as a further fibre triplet over $t_2$, say.
So we ask whether $t_1, t_2(\neq t_1) \in b$
exist such that the 3-vectors $(D_2(t_1))$ and $(D_2(t_2))$
where $D_2=C_2, B_2, A_2$ are linearly dependent, i.e.~whether
$k\in {\bf C^*}$ exists such that $D_2(t_2)=kD_2(t_1)$
for $D_2=C_2, B_2, A_2$ which indeed just amounts to the nontrivial solvability
of $ {\cal D}_3^{\chi=0}\cdot (kt_1^2-t_2^2, kt_1-t_2, k-1)^t=0$.

\subsection{\label{Ex 1 sect}Example 1 with $\lambda=5/2$}

An $SU(3)$ bundle, $\chi=1$ with $c=3s+4F$,
so\footnote{this is reflected in the sense of
app.~\ref{negative lambda}
from an Example $1$ with $\lambda=-5/2$}
$l(-F)={\cal O}_{{\cal E}}\big( 9s-F\big)$, gives a map
\begin{eqnarray}
\label{ex 1tilde map}
\iota_1: \;
H^1\Big( {\cal E}, {\cal O}_{{\cal E}}\big( 6s-5F \big) \Big)
\longrightarrow
H^1\Big( {\cal E}, {\cal O}_{{\cal E}}\big( 9s-F\big) \Big)
\end{eqnarray}
between $44$-dimensional spaces by (\ref{h^1 reduction}).
$\iota_1$ is induced from multiplication with
$\tilde{\iota}  =  Cz+Bx+Ay$ where
(for the notation cf.~app.~\ref{concrete matrix representations})
\begin{eqnarray}
\tilde{\iota} & \in & H^0\Big( {\cal E}, {\cal O}_{{\cal E}}(3s+rF)\Big)
=\bigoplus_{w'\in StB(\Sigma)}\, w' \, H^0\Big(b, r- [w'] \, \chi\Big)
=\bigoplus_{w'\in StB(\Sigma)}\, w'\, S^{r- [w'] \, \chi}\, V \;\;\;\;\;\;\;\;\;
\end{eqnarray}
(i.e. $w'\in H^0({\cal E}, {\cal O}_{{\cal E}}(3s+[w']\chi F))$,
cf.~(\ref{standard base elements}))
with the accompanying coefficients
\begin{eqnarray}
\label{coefficient expressions}
C=c_4u^4+c_3u^3v+c_2u^2v^2+c_1uv^3+c_0v^4
&\in &S^r\, V \;\;\;\;\, = S^4\, V = Hom(S^4\, V^*, {\bf C})\;\;\;\;\;\;\\
B=b_2u^2+b_1uv+b_0v^2 \;\;\;\;\;\;\;\;\;\;\;
&\in &S^{r-2\chi}\, V
= S^2\, V = Hom(S^2\, V^*, {\bf C})\;\;\;\;\;\;\;\;\\
A=a_1u+a_0v \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
& \in &S^{r-3\chi}\, V= \;\;\;\;\, V= Hom(V^*, {\bf C})\;\;\;\;
\end{eqnarray}
With $l(-F)$ being not among the strong reduction cases we
adopt the more general approach of sect.~\ref{Exceptional}:
we use $\bar{l}(-F)={\cal O}_{{\cal E}}(3s-0F)$ and get the map
\begin{eqnarray}
\bar{\iota}_1:
H^1\Big({\cal E}, {\cal O}_{{\cal E}}(0s-4F)\Big)\longrightarrow H^1\Big({\cal E}, {\cal O}_{{\cal E}}(3s-0F)\Big)
\end{eqnarray}
between $3$-dimensinal spaces.
Or, in matrix form,
(having $\bar{\beta}=0$ the first line is absent)
\begin{eqnarray}
m_r\, : \;\;\;\;\;\;\;\;\;\;\;\;\;\;
\left( \begin{array}{c}
\, C \odot S^{-\bar{\beta}-2}\;\;\;\;\;\, V^*    \\
B \odot S^{-\bar{\beta}-2+2\chi}\, V^*     \\
A \odot S^{-\bar{\beta}-2+3\chi}\, V^*
\end{array} \right)
\end{eqnarray}

Concretely this means one builds the map
\begin{eqnarray}
{\cal D}_3 := B \; \oplus \; A\; \underline{\odot}\; V^*  =
\left( {\begin{array}{ccc}
b_2 & b_1  & b_0  \\
a_1 & a_0  & 0   \\
0   & a_1  & a_0
\end{array}} \right) \; \in \; Hom(S^2\, V^*, {\bf C} \oplus V^*)
\end{eqnarray}
One finds (where $\det \, {\cal D} = Res(A_1, B_2)$, cf.~sect.~\ref{intro Ex 1} and
\ref{resultant})
\begin{eqnarray}
\det\, \iota_1 = -4^{10}\; \Big( \det \, {\cal D}_3 \Big)^{11} \; Q_{11}
\end{eqnarray}

Let us discuss the pair
\begin{eqnarray}
l(-F) & = & {\cal O}_{{\cal E}}(9s-F)\\
\bar{l}(-F)&=&{\cal O}_{{\cal E}}(3s-0F)
\end{eqnarray}
from the point of view of the reduction philosophy described in
sect.~\ref{Exceptional}. Concerning $\bar{l}(-F)$ we are here in the
exceptional case $\bar{\alpha}=n, \bar{\beta}=0$; so the equality condition leads
to the $r$-fix $r=4$ from (\ref{exceptional r fix}) and the discussion after
(\ref{n=3 exceptional}) shows that, although the $\lambda=3/2$ analogue of $l(-F)$
would have given $l(-F)=\bar{l}(-F)^{\otimes 2}$, here $\lambda \geq 5/2$ is required.

Now the long exact sequence for $\bar{l}$
\begin{eqnarray}
 0\;\;\;\;\;\;\;\;\;\;\;\;\; &\longrightarrow &\;
H^0\Big({\cal E}, {\cal O}_{{\cal E}}(3s)\Big)\; \buildrel \rho \over \longrightarrow \;
H^0\Big(c, {\cal O}_{{\cal E}}(3s)|_c\Big)\nonumber\\
\buildrel \delta \over \longrightarrow
H^1\Big({\cal E}, {\cal O}_{{\cal E}}(-4F)\Big)\; &\buildrel \bar{\iota}_1 \over \longrightarrow &
\; H^1\Big({\cal E}, {\cal O}_{{\cal E}}(3s)\Big)
\end{eqnarray}
gives $\ker \bar{\iota}_1\cong
H^0(c, {\cal O}_{{\cal E}}(3s)|_c)/\rho H^0({\cal E}, {\cal O}_{{\cal E}}(3s))$.
So generically, for $f:=\det \bar{\iota}_1\neq 0$,
one has $h^0(c, {\cal O}_{{\cal E}}(3s)|_c)=1$ from the section given by $z$.
The necessary condition  (\ref{nec cond for prec reduc})
for precise reduction $p\deg \bar{l}(-F)|_c\leq \deg l(-F)|_c$
gives $p=1$ or $2$; we consider here the case $p=2$.
Then the bundle $\tilde{{\cal F}}={\cal O}_c(D-2\bar{D})$
from sect.~\ref{difference subsection},
which needs to have a nontrivial section to carry through
the reduction procedure,
is the bundle $\Lambda|_c={\cal O}_{{\cal E}}(3s-F)|_c$, cf.~(\ref{Lambda})
(this is also in line with the remarks after (\ref{n=3 exceptional}));
this flat bundle has $h^0(c, \Lambda|_c)$ either $0$ or $1$.
So, if both,
$\bar{l}(-F)|_c={\cal O}_{{\cal E}}(3s)|_c$ and $\Lambda|_c={\cal O}_{{\cal E}}(3s-F)|_c$,
have a nontrivial section (the first line bundle
has $z$) then so does $l(-F)|_c$.
So (cf.~(\ref{general Pfaff factor}))
\begin{eqnarray}
\label{f_L Pfaff relation}
f_{\Lambda}\, | \, Pfaff
\end{eqnarray}
To find the result $f|Pfaff$ we now show that here actually $f=f_{\Lambda}$.

First we argue for $f_{\Lambda} | f$.
Now, with (\ref{can bund of c}), one has
\begin{eqnarray}
\label{P equ}
h^0(c, {\cal O}_{{\cal E}}(3s)|_c)&=&3-6+h^0(c, {\cal O}_{{\cal E}}(3F)|_c)=: -3+P\\
\label{Q equ}
h^0(c, {\cal O}_{{\cal E}}(3s-F)|_c)&=&0-6+h^0(c, {\cal O}_{{\cal E}}(4F)|_c)=: -6+Q
\end{eqnarray}
such that always $P\geq 3, Q\geq 6$.
Generically one has $P=4$, from the section $z$; note that
$P=h^0(c, {\cal O}_{{\cal E}}(3F)|_c)\geq h^0({\cal E}, {\cal O}_{{\cal E}}(3F))=h^0(b, {\cal O}_b(3))=4$.
As remarked above actually $Q=6$ or $7$.
By (\ref{general Pfaff factor}) the latter is
sufficient for reduction (giving (\ref{f_L Pfaff relation})):
$Q=7\Longleftrightarrow h^0(c,\Lambda|_c)\geq 1\Longleftrightarrow
\Lambda|_c\cong {\cal O}_c \Longleftrightarrow f_{\Lambda}=0 \Longrightarrow l(-f)|_c\cong {\cal O}_{{\cal E}}(2F)|_c$.
Note that $Q\geq 7\Longrightarrow P\geq 5$ as then ${\cal O}(3s)|_c\cong {\cal O}(F)|_c$
with the two sections $\pi^*_cu$ and $\pi^*_cv$; so one gets
another (linearly independent) section $z'$ of ${\cal O}_{{\cal E}}(3s)|_c$, such that
\begin{eqnarray}
\label{f_Lambda to f relation}
f_{\Lambda}\, | \, f
\end{eqnarray}
\hspace{.3cm}Now we argue for $f| f_{\Lambda}$: the generic result
$s|_c=\{p_0, t_0\}\, , \; F_{t_0}|_c=c_{t_0}=\{p_0+p_++p_-, t_0\}$
(with $A_1(t_0)=0$, cf.~(\ref{s|_c generically}))
changes at the locus
$f=\det \bar{\iota}_1=\det {\cal D}_3 =Res(A_1, B_2)=0$
(where $B_2(t_0)=0$): $p_0$ becomes a three-fold point of $c_{t_0}$.
So one has in ${\cal M}_{{\cal E}}(c)$
\begin{eqnarray}
f=0 & \Longleftrightarrow &
(z)|_c=3s|_c=\{3p_0, t_0\} = F_{t_0}|_c
\end{eqnarray}
Therefore $f\, | \, f_{\Lambda}$ and one gets here that actually even
$f=f_{\Lambda}$ and so $f|Pfaff$ by (\ref{f_L Pfaff relation}).

redpajama_set_name: RedPajamaArXiv