%% A manuscript in PLAIN TeX
\def\author{E. ESTEVES, M. GAGN\'E, AND S. KLEIMAN}
\def\title{AUTODUALITY OF THE COMPACTIFIED JACOBIAN}
\def\date{December 18, 2000}
\def\abstract{
We prove the following {\it autoduality theorem} for an integral
projective curve $C$ in any characteristic. Given an invertible sheaf
$\cL$ of degree 1, form the corresponding Abel map
$A_\cL\:C\to\CJ$,
which maps $C$ into its compactified Jacobian, and form its pullback
map $A_\cL^*\:\Pic^0_\CJ\to J$, which carries the connected
component of 0 in the Picard scheme back to the Jacobian. If $C$ has,
at worst, points of multiplicity 2, then $A_\cL^*$ is an isomorphism,
and forming it commutes with specializing $C$.
Much of our work is valid, more generally, for a family of curves with,
at worst, points of embedding dimension 2. In this case, we use the
determinant of cohomology to construct a right inverse to $A_\cL^*$.
Then we prove a scheme-theoretic version of the theorem of the cube,
generalizing Mumford's, and use it to prove that $A_\cL^*$ is
independent of the choice of $\cL$. Finally, we prove our autoduality
theorem: we use the presentation scheme to achieve an induction on the
difference between the arithmetic and geometric genera; here, we use
special properties of points of multiplicity 2.
}
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%%%%%%%%%%%%%%%%% REFERENCE KEYS
\RefKeys
AIK76 AK76 AK79 AK79II AK80 AK90 E95 E99 EGK FGA EGA K87 KM76 L59 M60 M65
M70 R67
\endRefKeys
%%%%%%%%%%%%%%%%% TOPMATTER
\centerline{\twelverm \title}
\medskip\bigskip
%% Subject classification and acknowledgements
\footnote{}{\noindent %
MSC-class: 14H40 (Primary) 14K30, 14H20 (Secondary).}
%% Authors' names and support
\centerline{EDUARDO ESTEVES\footnote{$^{1}$}{%
Supported in part by
PRONEX, Conv\^enio 41/96/0883/00,
and CNPq Proc. 300004/95-8 (NV).},
MATHIEU GAGN\'E,
{\it and}
STEVEN KLEIMAN\footnote{$^{2}$}{%
Supported in part by NSF grant 9400918-DMS.}
}
\bigskip\bigskip
%% Abstract
{\eightpoint \centerline{\smc Abstract}\par \abstract \par}
%%\end of Topmatter
%% Body
\sct1 Introduction
Let $C$ be an integral projective curve, defined over an algebraically
closed field of any characteristic, and $\cL$ an invertible sheaf of
degree 1. Form the (generalized) Jacobian, the connected component of
the identity of the Picard scheme, $J:=\Pic^0_C$. If $C$ is smooth,
then $J$ is an Abelian variety, and the Abel map
$A_\cL\:C\to J$
is defined by $P\mapsto\cL(-P)$. Also, the corresponding pullback is an
isomorphism, $A_\cL^*\:\Pic^0_J \risom J$, which is independent of
the choice of $\cL$; thus $J$ is ``autodual,'' or canonically isomorphic
to its own dual Abelian variety $\Pic^0_J$. (See Theorem 3 on \p.156 in
\cite{L59} or Proposition 6.9 on \p.118 in \cite{M65}.) Our main
result is the autoduality theorem of (2.1); it
asserts that, more generally, if $C$ has, at worst, double points
(arbitrary points of multiplicity 2), then a similar pullback is an
isomorphism, and forming it commutes with specializing $C$.
Suppose first that $C$ has arbitrary singularities. Recall (see
\cite{AK76}, \cite{AK79II}, \cite{AK80}) that $J$ has a natural
compactification $\CJ$, the (fine) moduli space of torsion-free sheaves
of rank 1 and degree 0. Also, the Abel map
$A_\cL\:C\to \CJ$ is
defined by
$P\mapsto\cI_P\ox\cL$
where $\cI_P$ is the ideal of $P$; it is a closed embedding if $C$ is
not of genus 0. Furthermore, the Picard scheme $\Pic_{\CJ}$ exists and
is a union of quasi-projective, open and closed
subschemes\UThin---\thinspace including $\Pic^0_{\CJ}$ and
$\Pic^\tau_{\CJ}$, which are the connected component of $0$ and the
subscheme of points with multiples in $\Pic^0_{\CJ}$.
Suppose now that all the singularities of $C$ are {\it surficial}, that
is, of embedding dimension 2. Recall (see \cite{AIK76}) that $\CJ$ is
rather nice; it is a local complete intersection, and is integral and
projective. So forming $\Pic_{\CJ}$ commutes with specializing $C$, but
conceivably forming $\Pic^0_{\CJ}$ does not. Nevertheless, we prove two
general results, Propositions (2.2) and (3.7). The former asserts that
$A_\cL^*\:\Pic^0_\CJ\to J$ has a natural right inverse $\beta$,
which is independent of the choice of $\cL$. The latter is much deeper,
and asserts that $A_\cL^*$ is itself independent of the choice of
$\cL$.
Suppose finally that all the singularities of $C$ are double points.
Then $A_\cL^*$ is an isomorphism and
$\Pic^0_{\CJ}=\Pic^\tau_{\CJ}$ by
our autoduality
theorem;
this is our
deepest result, and rests on everything preceding it.
All three of our results are compatible with specializing $C$. More
precisely, we prove relative versions of them for flat, projective
families of geometrically integral curves over an arbitrary locally
Noetherian base scheme. Some fibers may be smooth, others not. A node
may degenerate into a cusp; two nodes may coalesce into a tac.
To prove the autoduality theorem, we put together
the Abel maps $A_\cL$, as $\cL$ varies, to form the Abel map of
bidegree (1,1):
$$A\:C\x J^1\to \CJ.$$
Here $J^1:=\Pic^1_C$, the component of the Picard scheme that
parameterizes the invertible sheaves of degree 1. This map was first
studied in the authors' paper \cite{EGK} (where, however, the two
factors are taken in the opposite order; that is, $A$ maps $J^1\x C$
into $\CJ$). It was proved in \cite{EGK} that $A$ is rather
nice. Indeed, if $C$ is Gorenstein, then $A$ is smooth by Corollary
(2.6) of \cite{EGK}; so its image $V$ is open. Furthermore, if $g$
denotes the arithmetic genus of $C$, then the complement $\CJ-V$ is of
dimension at most $g-2$ if and only if all the singularities of $C$ are
double points by Corollary~(6.8) of \cite{EGK}.
These two properties of $A$ are central to our proof of autoduality.
(Indeed, they were discovered when we developed this proof, as were many
other results in \cite{EGK}.) We use these properties to show that
$A_\cL^*$ is injective. Here's how. Let $\cN$ be an invertible sheaf
on $\CJ$.
Suppose that all the singularities of $C$ are double points. Then
$\CJ-V$ is small. Moreover, $\CJ$ is a local complete intersection. So
$\cN$ is trivial if its restriction $\cN|V$ is trivial. In turn, to
show that $\cN|V$ is trivial, we may use descent theory since $A$ is
smooth, so flat.
%
Suppose that $\cN$ corresponds to a point of $\Pic^0_\CJ$. Then there
are invertible sheaves $\cN_1$ on $C$ and $\cN_2$ on $J^1$ such that
$$A^*\cN=p_1^*\cN_1\ox p_2^*\cN_2,$$
where the $p_i$ are the projections; the existence of the $\cN_i$
results from our general theory of the theorem of the cube, especially
Part (2) of Lemma~(3.6). Consequently, $A_\cL^*\cN$ is equal to
$\cN_1$, so is independent of the choice of $\cL$, as was asserted
above.
Suppose also that $A_\cL^*\cN$ is trivial for some $\cL$.
We have to prove that $\cN$ is trivial too. To begin, note that $\cN_1$ is
trivial, so $A^*\cN$ is equal to $p_2^*\cN_2$. We proceed by
induction on the arithmetic genus $g$. Choose a double point $Q$ on
$C$, and blow $Q$ up, getting $\vf\:\dr C\to C$. Then $\dr C$ too has
only double points by Lemma~(6.4) of \cite{EGK}, and its
arithmetic genus is $g-1$ by Proposition (6.1) of \cite{EGK}.
To relate the compactified Jacobians $\CJ_C:=\CJ$ and $\CJ_{\dr C}$ of
$C$ and $\dr C$, we use the presentation scheme $P$ and the maps
$\kappa\:P\to\CJ_C$ and $\pi\:P\to\CJ_{\dr C}$; they are studied in
\cite{EGK}. By Theorem~(6.3) of \cite{EGK}, because $Q$ is a double
point, $\pi$ is a locally trivial $\IP^1$-bundle. On each $\IP^1$, the
restriction of $\kappa^*\cN$ is trivial because $\cN$ corresponds to a
point of $\Pic^0_{\CJ_C}$. Hence, $\kappa^*\cN$ is the pullback of a
sheaf $\dr\cN$ on $\CJ_{\dr C}$.
Because $C$ and $\dr C$ are Gorenstein, there are two natural
commutative diagrams
$$\CD
\dr C\x J_{C}^1 @>\Lambda>> P \\
@V 1\x\vf^*VV @V\pi VV \\
\dr C\x J_{\dr C}^1 @>A_{\dr C}>> \CJ_{\dr C}
\endCD\qquad\CD
\dr C\x J_{C}^1 @>\Lambda>> P \\
@V\vf\x1VV @V\kappa VV \\
C\x J_{C}^1 @>A_C>> \CJ_{C}
\endCD$$
by Corollary (5.5) in \cite{EGK}; here $A_C$ and $A_{\dr C}$ are the
Abel maps of $C$ and $\dr C$. The diagrams imply that, if
$\dr\cL:=\vf^*\cL$, then $A_{\dr\cL}^*\dr\cN$ is equal to
$\vf^*A_\cL^*\cN$, which is trivial by hypothesis.
Since autoduality holds for $\dr C$ by induction, $\dr\cN$ is trivial.
Since the pullback of $\dr\cN$ to $P$
is equal to $\kappa^*\cN$, the latter is
trivial. So thanks to the commutativity of the second diagram above,
$(\vf\x1)^*A_C^*\cN$ is trivial. Since $A_C^*\cN$ is
equal to $p_2^*\cN_2$, it follows that $\cN_2$ is trivial.
Since $C$ and $\dr C$ are Gorenstein, by Corollary (5.5) in \cite{EGK},
the second diagram above is Cartesian. Consider the descent data on
$(\vf\x1)^*A_C^*\cN$ with respect to $\Lambda$; since $\kappa^*\cN$ is
trivial, this data is trivial. Hence, so is that on $A_C^*\cN$ with
respect to $A_C$ because $\kappa$ is birational. Therefore $\cN$ is
trivial. Thus $A_\cL^*$ is injective.
We construct the right inverse $\beta\:J\to\Pic^0_\CJ$ to $A_\cL^*$ by
using the determinant of cohomology $\cD$ along the projection
$q_2\:C\x\CJ\to\CJ$. We proceed as follows. Fix a universal sheaf
$\cI$ on $C\x\CJ$. Then, given any invertible sheaf $\cM$ on $C$ of
degree~0, set
$$\beta(\cM):=(\cD(\cI\ox q_1^*\cM))^{-1}\ox\cD(\cI),$$
where $q_1\:C\x\CJ\to C$ is the projection.
This construction was suggested by Breen [pvt.\ comm., 1985]. It is a
modern formulation of an older construction using the theta divisor.
Namely,
$$\beta(\cM)=\Theta_\cL-\tau^*_\cM\Theta_\cL$$
where $\tau_\cM\:\CJ\to\CJ$ is the translation, given by tensoring
with $\cM$, and where $\Theta_\cL$ is the divisor obtained by pulling
back the canonical theta divisor along the isomorphism
$\CJ\risom\CJ^{g-1}$ given by tensoring with $\cL^{g-1}$. We consider
the equivalence of the two formulations in more detail in Remark (2.4).
Since $A_\cL^*\beta=1$, the map $\beta$ is a closed embedding.
Since $A_\cL^*$ is injective, we could conclude that it is an
isomorphism if we knew, a priori, that $\Pic^0_\CJ$ is reduced. We
don't. So we must prove that $A_\cL^*$ is a monomorphism, that is,
injective on $T$-points; we take care to do so in (4.1).
Is $A_\cL^*$ an isomorphism when all the singularities of $C$ are
assumed only {\it surficial}? The evidence is mixed. On the
one hand, Propositions (2.2) and (3.7) suggest so, as they assert that
$A_\cL^*$ has a right inverse $\beta$, and both maps are
independent of the choice of $\cL$. On the other hand, our proof of
the autoduality theorem suggests not; it doesn't simply fail when $C$ has
singularities of higher multiplicity, rather it suggests that then there
may be a counterexample.
Indeed, if $C$ has higher surficial singularities, then $\CJ$ is
irreducible, and $\CJ-V$ contains a set of codimension 1. This set
could support a Cartier divisor $D$. If $D$ exists, then
$A_\cL^*\cO(D)$ is trivial for any $\cL$. Furthermore, $D$ could vary
in an algebraic family with support on $\CJ-V$ and with two linearly
inequivalent members. If so, then their difference would correspond to
a point of $\Pic^0_\CJ$, other than $0$. Thus $A_\cL^*$ would not be
injective, and we'd have a counterexample.
What happens when $C$ is reducible? The theory of the compactified
Jacobian for reducible curves is not as mature as that for integral
curves. Fine moduli spaces of torsion-free, rank-1 sheaves were only
recently constructed, see \cite{E99}. Abel maps too were considered
there, but not thoroughly treated. Once this theory is more mature, our
methods may well yield an autoduality theorem for $C$ if all its
singularities are double points.
In short, in Section 2, we formulate the autoduality theorem, our main
result: if the curves in a family have double points at worst, then the
Abel map $A_\cL^*$ is an isomorphism. Then we treat $\beta$, which
is the canonical right inverse to $A_\cL^*$. In Section 3, we
generalize Mumford's scheme-theoretic theorem of the cube, and conclude
that $A_\cL^*$ is independent of the choice of $\cL$. Finally, in
Section 4, we prove our autoduality
theorem.
\sct2 Autoduality
\dsc1 Statement. Consider a {\it flat projective family of integral
curves} $p\:C\to S$; that is, $S$ is a locally Noetherian scheme, and
$p$ is a flat and projective map with geometrically integral fibers of
dimension 1. Recall (see \cite{AK76}, \cite{AK79II}, \cite{AK80}) that,
given an integer $n$, there exists a projective $S$-scheme $\CJ^n_{C/S}$ that
parameterizes the torsion-free rank-1 sheaves of degree $n$ on the
fibers of $C/S$. Furthermore, there exists an open subscheme $J^n_{C/S}$
parameterizing those sheaves that are invertible. Also, forming
$\CJ^n_{C/S}$ and $J^n_{C/S}$ commutes with changing the base $S$.
As is customary, call $J^n_{C/S}$ the (relative generalized) {\it Jacobian}
of $C/S$, and $\CJ^n_{C/S}$ the {\it compactified Jacobian}. We will
often abbreviate $J^n_{C/S}$ by $J^n$ and $\CJ^n_{C/S}$ by $\CJ^n$. Set
$$J_{C/S}:=J^0_{C/S}\and\CJ_{C/S}:=\CJ^0_{C/S}.$$
We will also abbreviate $J_{C/S}$ by $J$ and $\CJ_{C/S}$ by $\CJ$.
More precisely, a (relative) {\it torsion-free rank-1 sheaf $\cI$} on
$C/S$ is an $S$-flat coherent $\cO_C$-module $\cI$ such that, for each
point $s$ of $S$, the fiber $\cI(s)$ is a torsion-free rank-1 sheaf on
the fiber $C(s)$. Moreover, $\cI$ is {\it of degree} $n$ if $\cI(s)$
satisfies the relation,
$$\chi(\cI(s))-\chi(\cO_{C(s)})=n.$$
Given a locally Noetherian $S$-scheme $T$, a torsion-free rank-1 sheaf
of degree $n$ on $C\x T/T$ defines an $S$-map $T\to \CJ^n$. Conversely,
every such $S$-map arises from such a sheaf, which is determined up to
tensor product with the pullback of an invertible sheaf on $T$, at least
if the smooth locus of $C/S$ admits a section. If so, then in
particular the identity map $1_{\CJ^n}$ arises from such a sheaf on $C\x
\CJ^n/\CJ^n$; the latter sheaf is known as a {\it universal\/} (or
Poincar\'e) sheaf, as any $T\to\CJ^n$ arises from the sheaf on $C\x
T/T$ obtained by pulling back a universal sheaf.
In general, an $S$-map $T\to\CJ^n$ arises rather from a pair $(T'/T,\,
\cI')$ where $T'/T$ is an \'etale covering (that is, the map $T'\to T$
is \'etale, surjective, and of finite type) and where $\cI'$ is a
torsion-free rank-1 sheaf of degree $n$ on $C\x T'/T'$. Such a pair
defines such an $S$-map if and only if there is an \'etale covering
$T''\big/\, T'\x_TT'$ such that the two pullbacks of $\cI'$ to $C\x T''$
are equal. A second such pair $(T'_1/T,\,\cI'_1)$ defines the same
$S$-map if and only if there is an \'etale covering $T''\big/\,
T'\x_TT'_1$ such that the pullbacks of $\cI'$ and $\cI'_1$ to $C\x T''$
are equal. In sum, $\CJ^n$ represents the \'etale sheaf associated to
the functor of torsion-free rank-1 sheaves.
Given an invertible sheaf $\cL$ of degree 1 on $C/S$, define the {\it
Abel map,}
$$A_\cL\:C\to \CJ,$$
as follows. Let $\cI_{\Delta}$ be the ideal of the diagonal $\Delta$
of $C\x C$, and $p_1\:C\x C\to C$ be the first projection. Then
$\cI_{\Delta}$ is a torsion-free rank-1 sheaf of degree $-1$ on $C\x
C/C$, and the tensor product $\cI_{\Delta}\ox p_1^*\cL$ defines
$A_\cL$. Forming $A_\cL$ commutes with changing the base $S$,
and if the fibers of $C/S$ are not of arithmetic genus 0,
then $A_\cL$ is a closed embedding by \cite{AK80, (8.8), \p.108}.
Assume now that the geometric fibers of $C/S$ have only surficial
singularities (ones with embedding dimension 2), for example, double
points. Then the projective $S$-scheme $\CJ^n$ is flat, and its
geometric fibers are integral local complete intersections; see
\cite{AIK76, (9), \p.8}. Hence, the Picard scheme $\Pic_{\CJ^n/S}$
exists and is a disjoint union of quasi-projective $S$-schemes; see
Th\'eor\`eme~3.1, \p.232-06, in \cite{FGA}, and Corollary~(6.7)(ii),
\p.96, in \cite{AK80}. So the Abel map induces an $S$-map,
$$\textstyle A_\cL^*\:\Pic_{\CJ/S}\to \coprod_nJ^n.$$
As is customary \cite{FGA, \p.236-03}, let $\Pic^0_{\CJ/S}$ denote the
set-theoretic union of the connected components of the identity 0 in the
fibers of $\Pic_{\CJ/S}$, and let $\Pic^\tau_{\CJ/S}$ denote the set of
points of $\Pic_{\CJ/S}$ that have a multiple in $\Pic^0_{\CJ/S}$. The
set $\Pic^\tau_{\CJ/S}$ is open; give it the induced scheme structure.
The following theorem asserts that, if the geometric fibers of $C/S$
only have double points (of arbitrary order) as singularities, then
$\Pic^0_{\CJ/S}$ and $\Pic^\tau_{\CJ/S}$ are equal, and under
$A_\cL^*$, they are isomorphic to $J$. This is our main result,
and its proof occupies the rest of the paper.
\medbreak{\smc Theorem} \(Autoduality). {\it
Let $C/S$ be a flat projective family of integral curves. Assume
its geometric fibers have double points at worst.
Then\/ $\Pic^0_{\CJ/S}=\Pic^\tau_{\CJ/S}$.
Furthermore,
the Abel map induces an isomorphism,
$$A_\cL^*\:\Pic^\tau_{\CJ/S}\risom J,$$
which is independent of the choice of the invertible sheaf $\cL$ of
degree 1 on $C/S$; in fact, the isomorphism exists whether or not any
sheaf $\cL$ does.}
\medbreak{\smc Proposition (\number\sctno.2)} \(Right inverse).
\bgroup\it Let $C/S$ be a flat projective family of integral curves.
Assume its geometric fibers only have surficial singularities.
Then there exists a natural map,
$$\beta\:J\to\Pic_{\CJ/S},$$
whose formation commutes with base change, and whose image lies in the
subset $\Pic^0_{\CJ/S}$. Furthermore, $A_\cL^*\circ\beta=
1_{J}$ for any $\cL$.
\pf
Choose an \'etale covering $S'/S$ such that the smooth locus of $C\x
S'/S'$ admits a section (such a covering exists by \cite{EGA, IV$_4$
17.16.3(ii), \p.106}). Choose universal sheaves $\cI$ on $C\x\CJ\x
S'$ and $\cM$ on $C\x J\x S'$. Form $C\x\CJ\x J\x S'$, and let
$p_{ijk}$ be the projection onto the product of the indicated factors.
Set
$$\dM:=(\cD_{p_{234}}(p_{124}^*\cI\ox p_{134}^*\cM))^{-1}
\ox\cD_{p_{234}}(p_{124}^*\cI)\on \CJ\x J\x S'$$
where $\cD_{p_{234}}$ denotes the determinant of cohomology; see
Section 6 in \cite{E99}, or \cite{KM76}.
So $\dM$ is an invertible sheaf. It defines the desired
map $\beta$ as we now prove.
The sheaf $\cI$ is determined up to tensor product with the pullback of
an invertible sheaf $\cN$ on $\CJ\x S'$. So the projection formula
for the determinant of cohomology yields
$$\eqalign{\cD_{p_{234}}(p_{124}^*\cI\ox p_{24}^*\cN\ox p_{134}^*\cM)
&= \cD_{p_{234}}(p_{124}^*\cI\ox p_{134}^*\cM)\ox p_{13}^*\cN^{\ox m}\cr
\cD_{p_{234}}(p_{124}^*\cI\ox p_{24}^*\cN)
&= \cD_{p_{234}}(p_{124}^*\cI)\ox p_{13}^*\cN^{\ox n}\cr}$$
where the $p$'s are the indicated projections and where $m$ and $n$ are
the Euler characteristics of $p_{124}^*\cI\ox p_{134}^*\cM$ and
$p_{124}^*\cI$ on the fibers of $p_{234}$ (thus $m$ and $n$ are locally
constant functions on $\CJ\x J\x S'$). Now, $m=n$ because the
fibers of $p_{134}^*\cM$ have degree 0. Therefore $\dM$ does not depend
on the choice of $\cI$.
Similarly, the sheaf $\cM$ is determined up to tensor product with the
pullback of an invertible sheaf $\cP$ on $J\x S'$. Moreover, the
preceding argument shows that, if $\cM$ is replaced by its tensor product
with the pullback of $\cP$, then $\dM$ is replaced by its tensor product
with the pullback of $\cP^{\ox m}$.
Set $S'':=S'\x S'$. There are two pullbacks of $\cI$ to $C\x\CJ\x
S''$, and both are universal sheaves. Similarly, there are two
pullbacks of $\cM$ to $C\x J\x S''$, and both are universal sheaves.
Now, forming the determinant of cohomology commutes with changing the
base. Therefore, by the preceding paragraphs, the two pullbacks of
$\dM$ to $\CJ\x J\x S''$ differ by tensor product with the
pullback of an invertible sheaf on $J\x S''$. Hence $\dM$ defines a
map $\beta\:J\to \Pic_{\CJ/S}$.
Consider another choice of covering $S'_1/S$ and of sheaves $\cI_1$ and
$\cM_1$, and form the corresponding $\dM_1$. Set $S'':=S'\x S'_1$. Then
the pullbacks of $\cI_1$ and $\cI$ to $C\x\CJ\x S''$ are both
universal. Similarly, the pullbacks of $\cM_1$ and $\cM$ to $C\x J\x
S''$ are both universal. Hence, by the preceding argument, the
pullbacks of $\dM$ and $\dM_1$ to $\CJ\x J\x S''$ differ by
tensor product with the pullback of an invertible sheaf on $J\x S''$.
So $\dM_1$ and $\dM$ define the same map $\beta$.
Forming $\beta$ commutes with changing $S$ since forming the determinant
does.
The image of $\beta$ lies in $\Pic^0_{\CJ/S}$. Indeed, we may change
the base to an arbitrary geometric point of $S$, and so work over an
algebraically closed field. Then $J$ is integral. So it suffices to
prove $\beta(0)=0$. Now, we may choose $\cI$ on $C\x\CJ$ and $\cM$ on
$C\x J$. Then the fiber $\cM(0)$ is equal to $\cO_C$. Since forming
the determinant commutes with passing to the fiber, it follows that
$\dM(0)= \cO_{\CJ}$. So $\beta(0)=0$.
Finally, $A_\cL^*\circ\beta=1_{J}$. Indeed, it suffices to check this
equation after changing the base to $S'$; so assume $S'=S$. Then $\cI$
sits on $C\x\CJ$, and $\cM$ sits on $C\x J$. So $A_\cL$ is defined by
$(1_C\x A_\cL)^*\cI$, as well as by $\cI_{\Delta}\ox p_1^*\cL$. Hence
these two sheaves differ by tensor product with the pullback, along the
projection $p_2$, of an invertible sheaf on $C$. It follows as above
from the properties of the determinant of cohomology that
$$(A_\cL\x1_{J})^*\dM=
(\cD_{p_{23}}(p_{12}^*\cI_{\Delta}\ox p_1^*\cL\ox p_{13}^*\cM))^{-1}
\ox\cD_{p_{23}}(p_{12}^*\cI_{\Delta}\ox p_1^*\cL)\eqno\Cs2.1)$$
on $C\x J$. So both sides of this equation define the same map
$J\to\coprod_nJ^n$.
To evaluate the right-hand side of \Cs2.1), consider the natural sequence,
$$0\to\cI_{\Delta}\to\cO_{C\x C}\to\cO_{\Delta}\to0.\eqno\Cs2.2)$$
Pull it back to $C\x C\x J$, then tensor with $p_1^*\cL\ox
p_{13}^*\cM$ and with $p_1^*\cL$. The additivity of the
determinant of cohomology now yields
$$\eqalign{
\cD_{p_{23}}(p_{12}^*\cI_{\Delta}\ox p_1^*\cL\ox p_{13}^*\cM)
&=\cD_{p_{23}}(p_1^*\cL\ox p_{13}^*\cM)\ox(p_1^*\cL\ox\cM)^{-1},\cr
\cD_{p_{23}}(p_{12}^* \cI_{\Delta}\ox p_1^*\cL)
&=\cD_{p_{23}}(p_1^*\cL)\ox(p_1^*\cL)^{-1}
.\cr}$$
Consider the following
Cartesian square:
$$\CD
C\x J @>> \cJ^{n+1} @>>> \cJ^n @>>> \cJ^n/\cJ^{n+1} @>>>0\\
@. @VVV @VVV @VVV \\
0@>>>\beta_*\cO_B(n+1) @>>> \beta_*\cO_B(n)@>>> \beta_*\cO_E(n)
\endCD$$
Since $E=\IP(\cJ/\cJ^2)$, the right vertical map is an isomorphism for
$n\ge-1$ by Serre's computation again. The left vertical map is an
isomorphism for $n\gg0$ by Serre's theorem. Hence, by descending
induction on $n$, the middle vertical map is an isomorphism for
$n\ge-1$. Thus Formulas \Cs4.3) hold, and the proof is complete.
\medbreak{\smc Lemma (\number\sctno.5)} \(Generalized theorem of the
cube). \bgroup\it Let $S$ be a connected and locally Noetherian scheme.
Let $g\:Y\to S$ and $f\:X\to Y$ be flat and proper maps. Let
$\sigma\:S\to Y$ and $\tau\:Y\to X$ be sections of $g$ and $f$.
Assume\smallbreak
\item i that $\cO_S=g_*\cO_Y$ and $\cO_Y=f_*\cO_X$ hold universally, and
\item ii that, for every closed point $s\in S$,
the natural map on the fibers is
injective:
$$w\:H^0(Y(s), R^1f(s)_*\cO_{X(s)}) \into
H^1(f(s)^{-1}\sigma(s),\cO_{f(s)^{-1}\sigma(s)}).$$
\smallbreak\noindent
Let $s_0\in S$ and $\cL$ be an invertible sheaf on $X$. If
the three restrictions,
$$\cL|X(s_0),\ \cL|\tau(Y),\and\cL|f^{-1}\sigma(S),$$
are trivial, then $\cL$ is trivial.
\pf
It is not hard to adapt, mutatis mutandis, Mumford's proof of his
similar theorem on \p.91 in \cite{M70}. It is a straightforward job,
except at the beginning and at the end. At the beginning, Mumford uses
his proposition on \p.89 to obtain the existence of a maximum closed
subscheme $T$ of $S$ carrying an invertible sheaf $\cN$ such that
$h^*\cN=\cL|h^{-1}T$ where $h:=gf$. (In fact, forming $T$ commutes with
base-changing $S$.) Mumford's construction of $T$ is not hard to adapt;
here's the idea.
Since $h$ is flat and proper, by \cite{EGA, III$_2$ 7.7.6, \p.69},
there are coherent sheaves $\cM$ and $\cN$ on $S$ such that, for
every coherent sheaf $\cF$ on $S$, we have
$$h_*(\cL\ox h^*\cF)=\cHom(\cM,\cF)
\and h_*(\cL^{-1}\ox h^*\cF)=\cHom(\cN,\cF).$$
Take $T:=\Supp(\cM)\bigcap\Supp(\cN)$ using the annihilators of $\cM$
and $\cN$ to define the scheme structures on their supports.
Over a point $t\in T$, the restrictions $\cL|X(t)$ and $\cL^{-1}|X(t)$
each have a nonzero section. Compose the first section with the dual of
the second, obtaining a nonzero map $\cO_{X(t)}\to\cO_{X(t)}$. This map
is given by multiplication by a scalar because $h_*\cO_X=\cO_T$ holds
universally. Hence $\cL|X(t)$ is trivial. Therefore, on a neighborhood
of $t$, one element suffices to generate $\cN$. It follows that $\cN|T$
is an invertible $\cO_T$-module and that $h^*\cN|h^{-1}T=\cL|h^{-1}T$.
At the end of the proof of his theorem, Mumford uses the K\"unneth
formula. Instead, we must use a related injection, which we get
as follows. For each closed point $s\in S$,
form the exact sequence of terms of low
degree of the Leray spectral sequence:
$$H^1(Y(s),f(s)_*\cO_{X(s)}) \TO u H^1(X(s),\cO_{X(s)}) \TO v
H^0(Y(s),R^1f(s)_*\cO_{X(s)}). $$
Since $f(s)_*\cO_{X(s)}=\cO_{Y(s)}$ by Assumption (i), the section
$\tau(s)\:Y(s)\to X(s)$ of $f(s)$ yields a map,
$$u'\: H^1(X(s),\cO_{X(s)}) \to H^1(Y(s),\cO_{Y(s)}),$$
splitting $u$. Hence, by Assumption (ii), the map $(u',w\circ v)$ is
an injection,
$$H^1(X(s),\cO_{X(s)}) \into H^1(Y(s),\cO_{Y(s)})\textstyle
\bigoplus H^1(f(s)^{-1}\sigma(s),\cO_{f(s)^{-1}\sigma(s)}).$$
This injection works in place of the K\"unneth formula.
\lem6 Let $C/S$ be a flat projective family of integral curves with
surficial singularities, and let $\cP$ be an invertible sheaf on $\CJ$.
Assume that $S$ is connected, and that $S$ contains a point $s_0$ such
that the fiber $\cP(s_0)$ is trivial. Form the Abel pullback
$A^*\cP$ on $C\x J^1$. Then the following two assertions
hold.
\part 1 The pullback $A^*\cP$ defines a map $a\:J^1\to J$,
and $a$ factors uniquely through the structure map $J^1\to S$.
\part 2 If the smooth locus $\smC$ of $C/S$ admits a section, then
there are invertible sheaves $\cM_1$ on $C$ and $\cM_2$ on $J^1$ such
that
$$A^*\cP=p_1^*\cM_1\ox p_2^*\cM_2,$$
where the $p_i$ are the projections.
\pf
Consider Part (1). A priori, $A^*\cP$ defines a map from
$J^1$ to $\Pic_{C/S}$. However, the image lies in the open subscheme
$J$ because $A^*\cP(s_0)$ is trivial and $S$ is connected.
If there is an $S$-factorization $a\:J^1\TO bS\TO cJ$, then $b$ must
be the structure map. So $b$ is faithfully flat. Hence, by descent
theory, $c$ is uniquely determined.
Assume for a moment that the smooth locus $\smC$ of $C/S$ admits a
section, and that Part (2) holds. Then $\cM_1$ defines an $S$-map
$c\:S\to J$ such that $a=cb$, where $b$ is the structure map.
In general, there is an \'etale covering $S'/S$ such that $\smC\x
S'/S'$ admits a section. So, by the reasoning above, $a\x S'$ factors
through a unique map $c'\:S'\to J\x S'$. Set $S'':=S'\x S'$. Then $a\x
S''$ factors through both $c'\x S'$ and $S'\x c'$. So the latter two
maps are equal. Hence $c'$ descends to a suitable map $c\:S\to J$.
Thus Part (1) follows from Part (2).
To prove Part (2), assume from now on that $\smC$ admits a section.
Then there exists a universal sheaf $\cI$ on $C\x\CJ^1$. Set
$P:=\IP(\cI)$. Let $\rho\:P\to C\x\CJ^1$ denote the structure map, and
$$\rho_1\:P\to C\and\rho_2\:P\to\CJ^1$$
the natural maps. Now, $A\:C\x J^2\to \CJ^1$ is smooth; see
(3.1). So its image $V$ is open. Also $V\supset J^1$. Set
$$X:=\rho^{-1}(C\x V)\and Z:=\rho^{-1}(C\x J^1).$$
Then $\rho$ induces an isomorphism $Z\risom C\x J^1$; so the $\rho_i$
extend the projections $p_i$.
Say $\theta\:S\to C$ is the section, let $Q$ be its image, and set
$\cL:=\cO_C(Q)$. Then $\cO_C(Q)$ defines a section $\phi\:S\to \CJ^1$,
whose image lies in $J^1$. So $\phi$ defines a section $\xi_1\:C\to
P$ of $\rho_1$ because $\rho$ is an isomorphism over $C\x J^1$.
Moreover, $\theta$ defines a section $\xi_2\:\CJ^1\to P$ of $\rho_2$
because $\rho$ is an isomorphism over $\smC\x\CJ^1$.
Consider the map $\zeta\:P\to\CJ$ of \Cs2) extending
$A\:C\x J^1\to\CJ$. Set
$$\vcenter{\openup1\jot
\halign{$\hfil#$&${}#\hfil$&$#$\hfil\cr
\cQ_2&:=\zeta^*\cP, &\cN_2:=\xi_2^*\cQ_2;\cr
\cQ_1&:=\cQ_2\ox \rho_2^*\cN_2^{-1},&\cN_1:=\xi_1^*\cQ_1;\cr
\cQ_0&:=\cQ_1\ox \rho_1^*\cN_1^{-1}=\cQ_2\ox\rho_2^*&\cN_2^{-1}
\ox \rho_1^*\cN_1^{-1}.\cr}}$$
Notice that, by hypothesis and by construction, the restrictions,
$$\cQ_0|P(s_0)\and\cQ_0|\xi_2(\CJ^1)
\and\cQ_0|\rho_2^{-1}\phi(S),\eqno\Cs6.1)$$
are trivial. It suffices to prove that $\cQ_0|X$ is trivial.
Set $q:=\rho_2|X$, so $q\:X\to V$. Then $q$ is flat. Indeed, let
$\Delta$ be the diagonal subscheme of $C\x C$, and $\cI_\Delta$ its
ideal. Set $W:=\IP(\cI_\Delta)$. Then there is a
Cartesian square
$$\CD
X @<<< W\x J^2 \\
@VVV @(\fiberbox) @VVV \\
C\x V @<1\x A<< C\x C\x J^2
\endCD$$
because, on $C\x C\x J^2$, the pullback of $\cI|C\x V$ is equal to the
tensor product of the pullback of $\cI_\Delta$ with an invertible sheaf
(namely, the pullback of a universal sheaf on $C\x J^2$). So there is a
Cartesian square
$$\CD
X @<<< W\x J^2 \\
@VVqV @(\fiberbox) @VVV \\
V @\Lambda>> P \\
@V 1\x\vf^*VV @V\pi VV \\
\dr C\x J_{\dr C}^1 @>A_{\dr C}>> \CJ_{\dr C}
\endCD\qquad\CD
\dr C\x J_C^1 @>\Lambda>> P \\
@V\vf\x1VV @V\kappa VV \\
C\x J_C^1 @>A_C>> \CJ_C
\endCD\eqno\Cs1.2)$$
Those diagrams yield these equations:
$$(A_{\dr\cL}\x1)^*\dr\cR=(\Lambda_\cL\x1)^*(\kappa\x1)^*\cN
=(\vf\x1)^*(A_\cL\x1)^*\cN=p_2^*\cR[\cL],$$
where $p_2\:\dr C\x T\to T$ is the projection, and $\Lambda_\cL$ is the
composition of $\Lambda$ with the map $\dr C\to\dr C\x J^1_C$ defined by
$\cL$.
By induction, autoduality holds for $\dr C$. Hence $\dr\cR$ is the
pullback of an invertible sheaf on $T$, whence so is $(\kappa\x1)^*\cN$.
Invert the latter sheaf on $T$, pull it back to $\CJ_C\x T$, tensor with
$\cN$, and use the product to replace $\cN$. Thus we may assume that
$(\kappa\x1)^*\cN$ is trivial. So $(\Lambda\x1)^*(\kappa\x1)^*\cN$ is
trivial too. So, since the second diagram above is commutative,
$(\vf\x1\x1)^*(A_C\x1)^*\cN$ is trivial. Hence Equation~\Cs1.1) implies
that $(A_C\x1)^*\cN$ is trivial.
Fix isomorphisms,
$$u\:(A_C\x1)^*\cN\risom\cO_{C\x J^1_C\x T}
\and v\:(\kappa\x1)^*\cN\risom\cO_{P\x T},$$
and set $\dr u:=(\Lambda\x1)^*v$. Since the second diagram above is
commutative, $\dr u$ and
$(\vf\x1\x1)^*u$
differ by multiplication with an invertible (regular) function on $\dr
C\x J^1_C\x T$. Since $\dr C$ is complete and integral, this function
is the pullback of an invertible (regular) function on $J^1_C\x T$.
Modifying $u$ accordingly, we may assume that $\dr u=(\vf\x1\x1)^*u$.
Set $R:=(C\x J^1_C)\x_{\CJ_C}(C\x J^1_C)$ and $\dr R:=R\x_{\CJ_C}P$, and
let $g\:\dr R\to R$ be the projection. By
Corollary~(5.5) in
\cite{EGK},
the second square in \Cs1.2) is Cartesian. Hence
$\dr R=(\dr C\x J^1_C)\x_P(\dr C\x J^1_C)$, and all
three squares in the following diagram are Cartesian:
$$\CD
\dr R\x T @''' \dr C\x J^1_C\x T @>\Lambda\x1>> P\x T \\
@VVg\x1V @(\Fiberbox) @VV\vf\x1\x1V @(\fiberbox) @VV\kappa\x1V\\
R\x T @''' C\x J^1_C\x T @>A_C\x1>> \CJ_C\x T
\endCD$$
Form the two pullbacks $u_1,\ u_2$ of $u$ to $R\x T$ and
correspondingly those $\dr u_1,\ \dr u_2$ of $\dr u$ to $\dr R\x T$.
Now, $\dr u:= (\Lambda\x1)^*v$; so $\dr u_1=\dr u_2$.
The Abel map $A_C$ is smooth; see (3.1). So the two projections from
$R\x T$ to $C\x J^1\x T$ are smooth too. Hence the associated points of
$R\x T$ map to simple points of $C$. Now, $\vf$ is an isomorphism off
the double point $Q$.
Hence $g\:\dr R\to R$ is an isomorphism over the associated points of
$R$. Since
$$(g\x1)^*u_1=\dr u_1=\dr u_2=(g\x1)^*u_2,$$
therefore $u_1=u_2$ holds at every associated point of $R\x T$, so
everywhere.
Since $u_1=u_2$, by descent theory $u$ descends to a trivialization of
$\cN$ on the image $V\x T$ of $A_C\x1$. Now, $\CJ_C$ is a local
complete intersection of dimension $g$ by \cite{AIK76, (9), \p.8}, where
$g$ is the arithmetic genus of $C$. Since each singular point of $C$ is
a double point,
$\cod(\CJ_C-V,\, \CJ_C)\ge2$
by Corollary~(6.8) in \cite{EGK}.
Hence, $\cN$ is the direct image of its restriction to $V\x T$.
Therefore, $\cN$ is trivial. The proof is now complete in the case
where $\phi(T)\subset U^0$. Call this the ``first case.''
Using our work in the first case, we will now establish the general
case. To do so, fix an arbitrary rational point $t_0\in T$. Let
$\cN_0$ be the fiber $\cN(t_0)$ viewed on $\CJ$. We will prove that
$\cN_0$ is trivial. Then, since $T$ is connected, $\phi(T)\subset U^0$,
and so we will have a complete proof of the autoduality theorem of
(2.1).
By hypothesis, $\cN_0$ corresponds to a point of $U$. So some multiple
$\cN_0^n$, for $n>0$, corresponds to a point of $U^0$. Moreover,
$$A_\cL^*(\cN_0^n)=(A_\cL^*\cN_0)^n=\cO_C.$$
Hence, by the preceding case with $S$ for $T$ and $\cN_0^n$ for $\cN$,
we may conclude that $\cN_0^n$ is the pullback of a sheaf on $S$. Since
$S$ is a point, $\cN_0^n$ is trivial on $\CJ$.
Let $K_n$ be the kernel of the $n$th power map $J\to J$; let's prove
that $K_n$ is affine. Let $\eta\:\wt C\to C$ be the normalization map.
Then $\eta$ is a surjective map between two proper schemes over a field;
so the induced map on the Jacobians $\eta^*\:J_C\to J_{\wt C}$ is affine
by Cor.1.5b) on \p.598 of \cite{R67}. Let $\wt K_n$ be the kernel of
the $n$th power map $J_{\wt C}\to J_{\wt C}$. Since $\wt{C}$ is smooth,
$J_{\wt C}$ is an Abelian variety. Hence, by Application 2 on \p.62 of
\cite{M70}, $\wt K_n$ is finite, so affine. Hence $\eta^{*-1}\wt K_n$
is affine too. Since it contains $K_n$ as a closed subscheme, $K_n$ is
therefore affine.
In fact, $K_n$ is finite if $n$ is not a multiple of the characteristic
$p$ of the ground field (for example, if $p=0$), because of the
following standard result: every commutative affine algebraic group has
a descending chain of closed subgroups such that each subquotient is
either the additive group, the multiplicative group, or a finite cyclic
group. (However, if $n$ is a multiple of $p$, then $K_n$ may be
infinite; for example, it is infinite if $C$ is a cuspidal curve.)
Consider $A^*\cN_0$ on $C\x J^1$. It defines a map $\vt\:J^1\to J$ such
that $\vt[\cL]=0$. Moreover, $\cN_0^n$ is trivial; so
$\vt(J^1)\subseteq K_n$. Now, $\vt(J^1)$ is connected and contains 0.
Hence, if $K_n$ is finite, then $\vt(J^1)=\{0\}$.
In fact, we always have $\vt(J^1)=\{0\}$; we'll prove it below. Hence
$\vt=0$ since $J^1$ is reduced. Therefore, $A^*\cN_0$ is equal to the
pullback of some invertible sheaf $\cR$ on $J^1$. We can now prove that
$\cN_0$ is trivial, as asserted above. We proceed by induction on
$\delta$ as in the first case, but with $S$ for $T$ and $\cN_0$ for
$\cN$. If $\delta=0$, then $U^0=U$ as we saw above, and so $\cN_0$ is
trivial by the first case. If $\delta\ge1$, then the argument in the
first case goes through exactly as before since the analogue of
Equation~\Cs1.1) holds. Thus $\cN_0$ is trivial.
It remains to prove that we always have $\vt(J^1)=\{0\}$. Let $V$ be
the image of the Abel map $A\:C\x J^2\to\CJ^1$, and let $\cI$ be the
restriction of a universal sheaf to $C\x V$. Set $X:=\IP(\cI)$, and
denote by $\rho\:X\to C\x V$ the structure map. By Lemma~(3.2), the
Abel map $A\:C\x J^1\to\CJ$ extends to a map $\zeta\:X\to\CJ$. Set
$\cM:=\zeta^*\cN_0$. Then $\cM^n$ is trivial since $\cN_0^n$ is.
We'll prove below that $\rho_*\cM$ is invertible and
$\cM=\rho^*\rho_*\cM$. Hence $\rho_*\cM$ defines a map
$\gamma\:V\to\Pic_C$. This map extends $\vt$ because $\zeta$ extends
$A\:C\x J^1\to\CJ$. Let $\Delta$ be the diagonal subscheme of $C\x C$,
and $\cI_{\Delta}$ its ideal. Set $T':=C\x J^2$ and
$W:=\IP(\cI_{\Delta})\x J^2$. Let $\psi\: W\to C\x T'$ be the induced
map. As in the proof of Lemma~(3.6), there is a Cartesian square
$$\CD
X @<<< W \\
@VV\rho V @(\fiberbox) @V\psi VV \\
C\x V @<1\x A<< C\x T'
\endCD\eqno\Cs1.3)$$
where $A\:T'\to V$ is the Abel map. Since $A$ is faithfully flat,
Lemma~(3.4) implies that $\rho_*\cO_X=\cO_{C\x V}$. Hence, since $\cM^n$
is trivial, we have
$$(\rho_*\cM)^n=(\rho_*\cM)^n\ox\rho_*\cO_X=
\rho_*\rho^*(\rho_*\cM)^n=\rho_*\cM^n=\rho_*\cO_X=\cO_{C\x V}.$$
Therefore, $\gamma(V)\subseteq K_n$.
Let $\gamma'\:T'\to K_n$ be the composition $\gamma\circ A$. Since
$T':=C\x J^2$ and since $K_n$ is affine and $C$ is complete, $\gamma'$
contracts $C\x\{t\}$ to a point for every $t\in J^2$.
Given an invertible sheaf $\cL'$ of degree 1 on $C$, there exist smooth
points of $C$, say $Q_1,\dots,Q_{2m}$, such that
$$\cL'\cong\cL\ox\cO_C(Q_1-Q_2+Q_3-\cdots-Q_{2m}).$$
For $i=1,\dots,m$ let $t_i$ be the point of $J^2$ representing the sheaf,
$$\cL\ox\cO_C(Q_1-Q_2+\cdots+Q_{2i-1}).$$
Then $A(Q_1,t_1)=[\cL]$ and $A(Q_{2m},t_m)=[\cL']$; so
$\gamma'(Q_1,t_1)= \gamma[\cL]=\vt[\cL]=0$ and $\gamma'(Q_{2m},t_m)=
\gamma[\cL']$. Furthermore, $A(Q_{2i},t_i)= A(Q_{2i+1},t_{i+1})$ for
$i=1,\dots,m-1$; so $\gamma'(Q_{2i},t_i)= \gamma'(Q_{2i+1},t_{i+1})$ for
these $i$. On the other hand, $\gamma'$ contracts $C\x\{t\}$ for every
$t$; so $\gamma'(Q_{2i-1},t_i)= \gamma'(Q_{2i},t_i)$ for $i=1,\dots,m$.
It follows that $\gamma[\cL']=0$. Thus $\gamma(J^1)=\{0\}$, and hence
$\vt(J^1)=\{0\}$.
It remains to show that the natural map $w\:\psi_*\cK\to\psi_*(\cK|W_t)$
is surjective. Let $\cA$ be an ample sheaf on $C$, and $pr_1\:C\x T'\to
C$ and $pr_2\:C\x T'\to T'$ be the projections. Then $w$ is surjective
if and only if
$$pr_{2*}(w\ox pr_1^*\cA^m)\:pr_{2*}((\psi_*\cK)\ox pr_1^*\cA^m)
\to pr_{2*}((\psi_*(\cK|W_t))\ox pr_1^*\cA^m)$$
is surjective for $m\gg0$. Set $\cK_m:=\cK\otimes\psi_1^*\cA^m$. It
then follows that $w$ is surjective if and only if the base-change map
$$\psi_{2*}(\cK_m)(t)\to H^0(W_t,\cK_m|W_t)$$
is surjective for $m\gg0$.
Since $\psi_2$ is flat by Lemma~(3.4), it is therefore enough to prove
that $$H^1(W_t,\cK_m|W_t)=0$$ for $m\gg0$. Now, $\cK_m|E$ is trivial
for each $m$; hence $H^1(E,\cK_m|E)=0$. On the other hand, since
$\varphi$ is finite, $H^1(\dr{C},\cJ\otimes\varphi^*\cA^m)=0$ for
$m\gg0$. Therefore, since $\cJ\otimes\varphi^*\cA^m$ is the kernel of
the restriction $\cK_m|W_t\to\cK_m|E$, we conclude that
$H^1(W_t,\cK_m|W_t)=0$ for $m\gg0$, as desired. The proof is now
complete.
\references
\serial{adv}{Adv. Math.}
\serial{ajm}{Amer. J. Math.}
\serial{bams}{Bull. Amer. Math. Soc.}
\serial{bpm}{Birkh\"auser Prog. Math.}
\serial{ca}{Comm. Alg.}
\serial{crasp}{C. R. Acad. Sci. Paris}
\serial{MathScand}{Math. Scand.}
\def\GrothFest#1 #2 {\unskip, {\it
The Grothendieck Festschrift #1} (eds P.~Cartier, L.~Illusie,
N.M.~Katz, G.~Laumon, Y.~Manin, K.A.~Ribet), Progress in Mathematics 86
(Birkh\"auser, 1990), pp.\ #2.}
\def\osloii{\unskip, {\it Real and complex singularities,} Proceedings,
Oslo 1976, (ed P. Holm, Sijthoff \& Noordhoff, 1977), pp.~}
\def\sitgesii{\unskip, {\it Enumerative Geometry} Proceedings,
Sitges, 1987 (eds S. Xamb\'o Descamps), \splnm 1436 1990 }
\def\splnm #1 #2 {Lecture Notes in Mathematics #1
(Springer, Berlin, #2)}
AIK76
{A. Altman, A. Iarrobino, {\rm and} S. Kleiman},
`Irreducibility of the compactified Jacobian'
\osloii 1--12.
AK76
A. Altman {\rm and} S. Kleiman,
`Compactifying the Jacobian',
\bams {(6) 82} 1976 947--49
AK79
A. Altman {\rm and} S. Kleiman,
`Bertini theorems for hypersurface sections containing a subscheme',
\ca {(8) 7} 1979 775--790
AK79II
A. Altman {\rm and} S. Kleiman,
`Compactifying the Picard scheme II',
\ajm 101 1979 10--41
AK80
A. Altman {\rm and} S. Kleiman,
`Compactifying the Picard scheme',
\adv 35 1980 50--112
AK90
A. Altman {\rm and} S. Kleiman,
`The presentation functor and the compactified Jacobian'
\GrothFest I 15--32
E95
D. Eisenbud,
{\it Commutative Algebra,}
(Springer GTM 150, 1995).
E99
E. Esteves,
`Compactifying the relative Jacobian over families of reduced curves',
http://arXiv.org/abs/math/alg-geom/9709009,
to appear in
{\it Trans. Amer. Math. Soc.}
EGK
E. Esteves{,} M. Gagn\'e{,} and S. Kleiman,
`Abel maps and presentation schemes',
http://arXiv.org/abs/math.AG/9911069,
Hartshorne Issue
{\it Comm. Alg.} (12) 28 (2000).
FGA
A. Grothendieck,
`Technique de descente et th\'eor\`emes d'existence en g\'e\-om\'e\-trie
alg\'e\-bri\-que V', {\it S\'eminaire Bourbaki} 232 Feb. 1962, and VI,
{\it S\'eminaire Bourbaki} 236, May 1962.
EGA
A. Grothendieck {\rm and} J. Dieudonn\'e,
{\it El\'ements de G\'eom\'etrie Alg\'ebrique,}
(Publ. Math. IHES, O$_{\rm III}$, Vol. 11, 1961;
III$_2$, Vol. 17, 1963; IV$_3$, Vol. 28, 1966; and
IV$_4$, Vol. 32, 1967).
K87
S. Kleiman,
`Multiple-point formulas II: the Hilbert scheme'
\sitgesii, pp.\ 101--138.
KM76
F. Knudsen {\rm and} D. Mumford,
`The projectivity of the moduli space of curves I',
\MathScand 39 1976 19--55
L59
S. Lang,
{\it Abelian varieties}
Interscience Tract 7 (Interscience, New York, 1959).
M60
A. Micali,
`Alg\`ebre sym\'etrique d'un id\'eal',
\crasp 251 1960 54--56
M65
D. Mumford,
{\it Geometric invariant theory},
Ergebnisse 34 (Springer, Berlin, 1965).
M70
D. Mumford,
{\it Abelian Varieties},
(Oxford Univ. Press, London, 1970).
R67
M. Raynaud,
`Un th\'eor\`eme de representabilit\'e relative sur le foncteur de Picard',
{\it Th\'eorie des intersections et th\'eor\`eme de
Riemann--Roch (SGA 6) Expos\'e XII} , \splnm 225 1971 .
\endreferences
\medskip
{\it \obeylines
\indent Instituto de Matem\'atica Pura e Aplicada
\indent\indent Estrada D. Castorina {\sl110, 22460--320},\ %
Rio de Janeiro RJ, BRAZIL
\indent\indent\indent {\rm E-mail: \tt Esteves\@impa.br}
\medskip
\indent EMC Corporation
\indent\indent 171 South Street, Hopkinton, MA {\sl01748}, USA
\indent\indent\indent {\rm E-mail: \tt MGagne\@emc.com}
\medskip
Department of Mathematics, Room {\sl 2-278} MIT,
\indent\indent {\sl77} Mass Ave, Cambridge, MA {\sl02139-4307}, USA
\indent\indent\indent \rm E-mail: \tt Kleiman\@math.mit.edu
}
\bye