Jean–Marc Drézet

Institut de Mathématiques de Jussieu - Paris Rive Gauche

Case 247

4 place Jussieu

F-75252 Paris, France
Moduli of vector bundles] Moduli of vector bundles on primitive
multiple schemes

A primitive multiple scheme is a Cohen-Macaulay scheme $Y$ such that the
associated reduced scheme $X=Y_{red}$ is smooth, irreducible, and that $Y$ can
be locally embedded in a smooth variety of dimension (X)+1. If $n$ is
the multiplicity of $Y$, there is a canonical filtration  X=X_1⊂X_2⊂⋯⊂X_n=Y, such that X_i is a primitive multiple
scheme of multiplicity $i$. The simplest example is the trivial primitive
multiple scheme of multiplicity $n$ associated to a line bundle $L$ on $X$: it
is the $n$-th infinitesimal neighborhood of $X$, embedded in the line bundle
$L^*$ by the zero section.

The main subject of this paper is the construction and properties of fine
moduli spaces of vector bundles on primitive multiple schemes. Suppose that
$Y=X_n$ is of multiplicity $n$, and can be extended to $X_{n+1}$ of
multiplicity $n+1$, and let $M_n$ a fine moduli space of vector bundles on
$X_n$. With suitable hypotheses, we construct a fine moduli space $M_{n+1}$ for
the vector bundles on $X_{n+1}$ whose restriction to $X_n$ belongs to $M_n$. It
is an affine bundle over the subvariety $N_n\subset M_n$ of bundles that can be
extended to $X_{n+1}$. In general this affine bundle is not banal. This applies
in particular to Picard groups.

We give also many new examples of primitive multiple schemes $Y$ such that the
dualizing sheaf $\omega_Y$ is trivial (in the case of surfaces they are called

Mathematics Subject Classification : 14D20, 14B20

§ INTRODUCTION

In this paper a scheme is a noetherian separated scheme over $\C$.
A primitive multiple scheme is a Cohen-Macaulay scheme $Y$ over such that:

– $Y_{red}=X$ is a smooth connected variety,

– for every closed point x∈X, there exists a neighbourhood $U$
of $x$ in $Y$, and a smooth variety $S$ of dimension  (X)+1 such that
$U$ is isomorphic to a closed subvariety of $S$.

We call $X$ the support of $Y$. If $Y$ is quasi-projective we say that it
is a primitive multiple variety with support $X$. In this case $Y$ is
projective if $X$ is.

For every closed subscheme Z⊂Y, let _Z (or _Z,Y)
denote the ideal sheaf of $Z$ in $Y$. For every positive integer $i$,
let X_i be the closed subscheme of $Y$ corresponding to the ideal sheaf
_X^i. The smallest integer $n$ such that  X_n=Y  is called the
multiplicity of $Y$. For 1≤i≤n, X_i is a primitive
multiple scheme of multiplicity $i$, L=_X/_X_2 is a line bundle on
$X$, and we have  _X_i/_X_i+1=L^i.
We call $L$ the line bundle on $X$ associated to $Y$. The ideal sheaf
_X can be viewed as a line bundle on X_n-1. If n=2, $Y$ is
called a primitive double scheme.

The simplest case is when $Y$ is contained in a smooth variety $S$ of dimension
d+1. Suppose that $Y$ has multiplicity $n$. Let P∈X and
f∈_S,P  a local equation of $X$. Then we have
_X_i,P=(f^i)  for 1<j≤n in $S$, in particular
_Y,P=(f^n), and  L=_X(-X) .

For any L∈(X), the trivial primitive variety of multiplicity
$n$, with induced smooth variety $X$ and associated line bundle $L$ on $X$ is
the $n$-th infinitesimal neighborhood of $X$, embedded by the zero section in
the dual bundle $L^*$, seen as a smooth variety.

The primitive multiple curves where defined in [19], [2].
Primitive double curves were parametrized and studied in [4] and
[18]. More results on primitive multiple curves can be found in
[9], [10], [11], [12], [13], [14],
[15], [16], [6], [28], [29], [30]. Some
primitive double schemes are studied in [3], [21] and
[23]. The case of varieties of any dimension is studied in [17],
where the following subjects were treated:

– construction and parametrization of primitive multiple schemes,

– obstructions to the extension of a vector bundle on X_m to

– obstructions to the extension of a primitive multiple scheme of
multiplicity $n$ to one of multiplicity n+1.

The main subject of this paper is the construction and properties of moduli
spaces of vector bundles on X_n. If X_n+1 is an extension of X_n
to a primitive multiple scheme of multiplicity n+1, and $\bf M$ is a moduli
space of vector bundles on X_n, we will see how to construct a moduli space
M' for the vector bundles on X_n+1 whose restriction to X_n
belongs to $\bf M$. It is an affine bundle over the closed subvariety N⊂M of bundles that can be extended to X_n+1. This applies
in particular to Picard groups.

Fine moduli spacesF_M

Let $\chi$ be a set of isomorphism classes of vector bundles on a scheme $Z$.
Suppose that $\chi$ is open, i.e. for every scheme $V$ and vector bundle
$\E$ on Z×V, if v∈V is a closed point such that
_v∈χ, then there exists an open neighbourhood $U$ of $v$ such that
_u∈χ for every closed point u∈U.
A fine moduli space for $\chi$ is the data of a scheme $M$ and of

– a bijection

\[\xymatrix@R=5pt{M^0\ar[r] & \chi\\ m\fmaps[r] & E_m

(where M^0 denotes the set of closed points of $M$),

– an open cover (M_i)_i∈I of $M$, and for every i∈I, a
vector bundle _i on X×M_i such that for every m∈M_i,

such that:
for any scheme $S$, any vector bundle $\kf$ on Z×S such that for any
closed point s∈S, _s∈χ, there is a morphism
f_:S→M  such that for every s∈S, if m=f_(s) then
_s≃E_m, and if m∈M_i, then there exists an open
neighbourhood U of $s$ such that  f_(U)⊂M_i and

Extensions of a vector bundle to higher multiplicity
– Let $\E$ be a vector bundle on X_n and  E=_|X. In
[17], 7.1, a class  Δ()∈^2__X(E,EL^n)  is
defined, such that $\E$ can be extended to a vector bundle on X_n+1 if
and only if  Δ()=0.

If $\ke$ is a vector bundle on X_n+1 such that  _|X_n=, the
kernel of the restriction morphism  →  is isomorphic to EL^n, and we have an exact sequence on X_n+1

\begin{equation}\label{equ24}
0\lra E\ot L^n\lra\ke\lra\E\lra 0 \ . \end{equation}

Hence if we want to extend $\E$ to X_n+1, it is natural to study
^1__X_n+1(,EL^n). We have a canonical exact sequence
0[r]     ^1__X(E,EL^n)[r]^-ψ
EL^n)[r]^-ϕ    (E)[r]^-δ_     ^2__X(E,EL^n)
and we have  Δ()=δ_(I_E). We have also the following
description of Δ() from [17]: we have a canonical exact
sequence of sheaves on X_n: 0→L^n→Ω_X_n+1|X_n→Ω_X_n→0, associated to
σ_Ω_X_n+1∈^1__X_n(Ω_X_n,L^n), inducing
σ_,X_n+1∈^1__X_n(Ω_X_n,L^n). We have then

\begin{equation}\label{equ25}\Delta(\E) \ = \
\ov{\sigma}_{\E,X_{n+1}}\nabla_0(\E) \ ,\end{equation}

where  ∇_0()∈^1__X_n(,Ω_X_n)  is the
fundamental class of $\E$.

Let $\E$ be a vector bundle on X_n. Then we have
H^0(_X_n)⊂(). We say that $\E$ is simple if

Let X_n be a primitive multiple scheme of support $X$ and multiplicity $n$,
that can be extended to a primitive multiple scheme X_n+1 of multiplicity
n+1. Suppose that we have a nonempty open set χ_n of isomorphism
classes of simple vector bundles on X_n such that there is a fine moduli
space $M_n$ for $\chi_n$. Then there exists a closed subvariety  N_n⊂M_n  such that for every closed point t∈N_n, the corresponding bundle
can be extended to X_n+1, and that for every scheme $T$ and every vector
bundle $\E$ on X_n+1×T such that for every closed point t∈T,
_t|X_n∈χ_n, then if  =_|X_n×T  and
f_:T→M_n  is the associated morphism, we have  f_(T)⊂N_n.

Let χ_n+1 be the set of isomorphism classes of vector bundles $E$ on
X_n+1 such that  E_|X_n∈χ_n. We will show that with suitable
hypotheses, there is also a fine moduli space M_n+1 for χ_n+1,
and that the restriction morphism  M_n+1→N_n  is an affine bundle.

Construction of fine moduli spaces – Let χ_n be an
open set of isomorphism classes of vector bundles on X_n. Suppose that
there is a fine moduli space for χ_n, defined by a smooth irreducible
variety M_n, and that the subvariety N_n⊂M_n of
bundles that can be extended to X_n+1 is smooth (this variety is defined
in <ref>). We suppose that for every ∈χ_n, if E=_|X,

(i) $\ke$ and $E$ are simple,

(ii) $\dim(\Ext^1_{\ko_X}(E,E\ot L^n))$ and
$\dim(\Ext^1_{\ko_{X_n}}(\ke,\ke\ot\Omega_{X_n}))$ are independent of $\ke$,

(iii) $\dim(\Ext^2_{\ko_X}(E,E\ot L^n))$ is independent of $\ke$.

(iv) $\Delta(\ke)=0$ .

(v) every vector bundle $\kf$ on X_n+1 such that
_|X_n≃  is simple.

Let χ_n+1 be the set of isomorphism classes of vector bundles $\E$ on
X_n+1 such that  _|X_n∈χ_n. We have then (cf. <ref>)

Theorem:  The set χ_n+1 is open,
and there is a fine moduli space M_n+1 for χ_n+1, which is
an affine bundle over N_n.

We have an exact sequence  0→L^n→_X_n+1→_X_n→0,
whence a connecting homomorphism  δ^0:H^0(X_n,_X_n)→H^1(X,L^n) . Let ∈N_n, E=_|X. Since E^*EL^n≃L^n⊕((E)L^n), we can see (δ^0) as a
subspace of ^1__X(E,EL^n). Let $\bf A$ be the vector bundle on
N_n associated to the affine bundle M_n+1→N_n. Then we have a
canonical isomorphism

\[{\bf A}_\ke \ \simeq \ \Ext^1_{\ko_X}(E,E\ot L^n)/\imm(\delta_0) \ . \]

Suppose that for some open subset S∈M_n, there is an universal vector
bundle $\E$ on X_n×S. Let S' be the inverse image of $S$ in
M_n+1. Then we have

\[{\bf A}_{|S} \ = \ R^1p_{S*}(\HHom(\E,E\ot p_X^*(L^n))/(\ko_S\ot
\imm(\delta_0)) \ . \]

Since  Δ(_s)=0  for every s∈S, we will see that

\[\Delta(\E) \ \in \ H^1(S,R^1p_{S*}(\HHom(\E,E\ot p_X^*(L^n)))) \ . \]

Suppose that η∈H^1(A_|S) corresponds to the affine bundle
S'→S. Then (cf. <ref>)

Theorem:  We have  .ϵ=η.

In other words there is a link between the possibility to extend a whole family
in higher multiplicity and the structure of the affine bundle.

The hypothesis (v) is satisfied if $L$ is a non trivial ideal sheaf and if the
restrictions to $X$ of the bundles of χ_n are simple (proposition

The assumption on N_n is satisfied in the the following cases:

– The case of Picard groups (cf. <ref>), i.e. $\chi_n$ consists
of line bundles.

– When $N_n=M_n$, in particular if $X$ is a curve, because in this
case, all the obstructions to extend vector bundles on X_n to X_n+1

Picard groupsintro2

Let ${\bf P}$ be an irreducible component of (X) and P_0 the
component that contains _X. Let _n,P be the set of line
bundles $\L$ on X_n such that  Ł_|X∈P.
We prove by induction on $n$ that if _n,P is not empty, then
there is a fine moduli space ^P(X_n) for _n,P.

Suppose that some ∈_n,P can be extended to X_n+1. Then
the product with $\kb$ defines an isomorphism  η_:^P_0(X_n)→^P(X_n). The
set of line bundles in _n,P that can be extended to X_n+1 is
a smooth closed subvariety

\[\Gamma^{\bf P}(X_{n+1}) \ \subset \ \Pic^{\bf P}(X_n) \ .\]

We have  Γ^P(X_n+1)=η_(Γ^P_0(X_n+1)), and Γ^P_0(X_n+1) is a subgroup of
^P_0(X_n): it is the kernel of the morphism of groups

\[\Delta:\Pic^{{\bf P}_0}(X_n)\to H^2(X,L^n) \ , \]

(the obstruction morphism). The case n=1 is simpler: since the
fundamental class of a line bundle on $X$ is a discrete invariant, we have
either  Γ^P(X_2)=∅  or  Γ^P(X_2)=^P(X). We have (cf. <ref>)

Theorem:  Suppose that _n+1,P is nonempty. Then ^P(X_n+1) is an affine
bundle over Γ^P(X_n+1) with associated vector bundle
_Γ^P(X_n+1)(H^1(X,L^n)/(δ^0)) .

The case of curvesCur

Suppose that $X$ is a curve. In this case the obstructions to extend a
primitive multiple curve or a vector bundle to higher multiplicity vanish (from
[17] they lie in the H^2 of some vector bundles on $X$).

Suppose that (L)<0. Let $r$, $d$ be integers such
that r>0 and $r$, $d$ are coprime. We take for M_n the set of
isomorphism classes of vector bundles $\ke$ on X_n such that _|X
is stable of rank $r$ and degree $d$. The hypotheses of <ref> are
satisfied, and we obtain by induction on $n$

Theorem:  There is a fine moduli space
M_X_n(r,d) for M_n, which is a smooth irreducible variety. It is an
affine bundle over M_X_n-1(r,d).

Of course for n=1, M_X_n(r,d) is the well known moduli space of
stable vector bundles of rank $r$ and degree $d$ on $X$. From [12], the
vector bundles of M_X_n(r,d), n≥2, are also stable. Hence
M_X_n(r,d) is an open subset of a moduli space of stable sheaves on

Let $\E$ be a Poincaré bundle on  X×M_X(r,d). Let  p_X:X×M_X(r,d)→X,p_M:X×M_X(r,d)→M_X(r,d)  be the
projections. Then if n≥2,

\[{\bf A}_{n-1} \ = \ R^1p_{M*}(\E\ot\E^*\ot p_X^*(L^{n-1}))\]

is a vector bundle on M_X(r,d). Let  p_n-1:M_X_n-1(r,d)→M_X(r,d)  be the restriction morphism. Then p_n-1^*(A_n-1) is
the vector bundle on M_X_n-1(r,d) associated to the affine bundle M_X_n(r,d)→M_X_n-1(r,d).

The case of Picard groups – We don't require here that
(L)<0. The irreducible components of (X) are the smooth
projective varieties  P_d=^d(X)  of line bundles of degree $d$.

Then if follows from <ref> that

Theorem: The irreducible components of
(X_n) are the ^P_d(X_n), d∈, and^P_d(X_n+1) is an affine bundle over ^P_d(X_n), with associated vector bundle _^P_d(X_n)H^1(L^n) .

These affine bundles need not to be banal: (cf. <ref>)

Theorem :
^d(X_2) is not a vector bundle over ^d(X) if either $X$
is not hyperelliptic and  (L)≤2-2g, or L=ω_X.

Using some parts of the proof of theorem <ref> we obtain

Theorem : Suppose that $C$ is not
hyperelliptic, X_2 not trivial and  (L)≤2-2g.
Then M_X_2(r,-1) is not a vector bundle over M_X(r,-1).

Primitive double schemes with trivial canonical line bundletriv_can

Since X_n is a locally complete intersection, it has a dualizing
sheaf ω_X_n which is a line bundle on X_n. We call also
ω_X_n the canonical line bundle of X_n. We have

\[\omega_{X_n|X} \ \simeq \ \omega_X\ot L^{1-n} \ . \]

In particular we have  ω_X_2|X≃ω_XL^-1. We have an
exact sequence
0[r]     L[r]     _X_2[r]^-ζ    _X[r]     0  .
We will prove (cf. <ref>)

Theorem:  The map
H^1(ζ):H^1(X_2,_X_2)→H^1(X,_X)  is surjective.

which will imply (cf. <ref>)

Corollary: We have
ω_X_2≃_X_2  if and only if  ω_X≃L .

If $X$ is a surface, and  ω_X_2≃_X_2, X_2 is called a
K3-carpet (cf. [3]).

Let $C$ and $D$ be smooth projective irreducible curves, X=C×D and
π_C:X→C, π_D:X→D  the projections. Let g_C (resp.
g_D) be the genus of $C$ (resp. $D$). Suppose that g_C≥2,
g_D≥2. Let L_C (resp. L_D) be a line bundle on $C$ (resp. $D$)
and  L=π_C^*(L_C)π_D^*(L_D). The non trivial double schemes X_2
with associated line bundle $L$ are parametrized by (H^1(X,T_XL)),
where T_X is the tangent bundle of $X$ (cf. [17]). We have

\[H^1(X,T_X\ot L) \ = \ \big(H^0(\omega_C^*\ot L_C)\ot
\big(H^1(\omega_C^*\ot L_C)\ot H^0(L_D)\big)\oplus\]

\[ \qquad\qquad\qquad\qquad\quad
\big(H^0(L_C)\ot H^1(\omega_D^*\ot L_D)\big)\oplus
\big(H^1(L_C)\ot H^0(\omega_D^*\ot L_D)\big) \ , \]

and using this decomposition, the double schemes $X_2$ are defined by
quadruplets (η_1,η_2,η_3,η_4).

We denote by ϕ_L_C, ϕ_L_D the canonical maps
H^0(ω_C^*L_C)H^1(ω_C)→H^1(L_C)  and H^0(ω_D^*L_D)H^1(ω_D)→H^1(L_D)  respectively.

Using <ref> we will see that

\[\Delta(M) \ = \ (\phi_{L_C}\ot
\phi_{L_D})(\eta_4\ot\nabla_0(M_D)) \ . \]

The canonical class of a line bundle on a smooth projective curve is in fact an
integer. For every line bundle $\kl$ on $C$ we have
∇_0()=()c, where  c=∇_0(_C(P)), for any P∈C.

For example let $\Theta$ be a theta characteristic on $C$ such that
h^0(C,Θ)>0. We take  L_C=Θ, L_D=ω_D. We have
η_1=0. Suppose that  η_4=0. Such non trivial double schemes are
parametrized by  ((H^1(Θ^-1)H^0(ω_D))⊕(H^0(Θ)H^1(_D))) (pairs (η_2,η_3)). Suppose that
η_3≠0. Then every line bundle on $X$ can be extended to X_2,
which is projective, and (using theorem <ref> below) there exists an
extension of X_2 to a primitive multiple scheme X_3 of multiplicity 3,
and X_3 is projective.

The case of K3 carpets – From <ref>, X_2 is a K3 carpet if and only if
L≃ω_X, i.e L_C=ω_C and L_D=ω_D.
In this case, let
Here we have

\[\eta_1\in\C \ , \quad \eta_2\in H^1(C,\ko_C)\ot H^0(D,\omega_D) \ , \quad
\eta_3\in H^0(C,\omega_C)\ot H^1(D,\ko_D) \ , \quad \eta_4\in\C \ . \]

If  M=π_C^*(M_C)π_D^*(M_D), then

\[\Delta(M) \ = \ \eta_1\nabla_0(M_C)+\eta_4\nabla_0(M_D) \ , \]

hence $M$ can be extended to a line bundle on X_2 if and only if

It follows that
the scheme X_2 is projective if and only if  η_1=η_4=0  or
η_1η_4<0  and η_1/η_4 is rational.
We will see that X_2 can be extended to a primitive multiple scheme of
multiplicity 3 if and only ω_X can be extended to a line bundle on
X_2. This is the case if and only  (g_C-1)η_1+(g_D-1)η_4=0, and
then X_2 is projective.

We will see also that there may exist components $\bf P$ of (X) such
that the affine bundle ^P(X_2) is not empty, and not a vector
bundle over $\bf P$.

Improvements of results of [17]Impr

Extensions of X_n to multiplicity n+1 –
If X_n can be extended to a primitive multiple scheme X_n+1 of
multiplicity n+1, then _X,X_n, which is a line bundle on
X_n-1, can be extended to a line bundle on X_n, namely
_X,X_n+1. Conversely, given an extension of _X,X_n to a
line bundle $\L$ on X_n, an invariant  Δ”_n(X_n,Ł)∈H^2(X,T_XL^n)  is defined in [17], such that X_n can be
extended to X_n+1 with  _X,X_n+1≃Ł  if and only if

By <ref>, another extension $\L'$ of _X,X_n to X_n+1 comes
from some  η∈H^1(X,L^n-1). Let  ζ∈H^1(X,T_XL)  correspond to the exact sequence  0→L→Ω_X_2|X→Ω_X→0.
We have a canonical product

\[H^1(X,T_X\ot L)\ot H^1(X,L^{n-1})\lra H^2(X,T_X\ot L^n)\]

and (cf. <ref>)

Theorem:  We have
Δ”_n(X_n,Ł')-Δ”_n(X_n,Ł)=ζη .

This implies that if the product is a surjective map, then there exists an
extension of X_n in multiplicity n+1.

Extensions of a line bundle on X_n to
X_n+1 – Let  Ł=_X,X_n+1|X_2, which is a line bundle on X_2. Let
δ^1_Ł^n-1 be the connecting morphism  H^1(X,L^n-1)→H^2(X,L^n)  coming from the exact sequence  0→L^n→Ł^n-1→L^n-1→0.

Suppose that X_n can be extended to X_n+1, and let $\D$ be a line
bundle on X_n. By <ref>, $\D$ can be extended to a line bundle on
X_n+1 if and only if  Δ()=0. Let '∈(X_n) be such
that  '_|X_n-1≃_|X_n-1. If  =_|X_n-1,
E=_|X, we have exact sequences

\[0\to E\ot L^{n-1}\to\D\to\E\to 0 \ , \qquad 0\to E\ot L^{n-1}\to\D'\to\E\to 0
\ , \]

corresponding to  η,η'∈^1__X_n(,EL^n-1)=H^1(X,L^n-1)  respectively. Then (cf. <ref>)

Theorem: We have Δ(')-Δ()=nδ^1_Ł^n-1(η'-η) .

Theorem <ref> implies for Picard groups (cf. <ref>, <ref>)

Proposition: Let $Z$ be the image of he
restriction morphism  Γ^P(X_n+1)→Γ^P(X_n). Then
$Z$ is smooth and Γ^P(X_n+1) is an affine bundle over $Z$ with
associated vector bundle  _Z(δ^1_Ł^n-1) .

Outline of the paperOutl

In chapter 2, we give several ways to use Čech cohomology, and recall some
basic definitions about fine moduli spaces and affine bundles.

In the very technical chapter 3, we give a description of Leray's spectral
sequence and the exact sequence of Ext's in terms on Čech cohomology (which
is our main tool).

The chapter 4 is devoted to the definitions and properties of primitive
multiple schemes, with the improvement <ref> of [17]. It contains
also supplementary results on vector bundles on primitive multiple schemes,
that are used in the following chapters.

In chapter the definition and properties of the fundamental class of a vector
bundles are recalled. We also give here the results of <ref>.

Chapter 6 contains the proofs of the main results (the construction and
properties of fine moduli spaces of vector bundles on primitive multiple

In chapter 7 we treat the special case of Picard groups (cf. <ref>), and
prove the improvement <ref> of [17].

In chapter 8 we give the examples <ref>, when $X$ is a product of curves.

The chapter 9 is devoted to moduli spaces of vector bundles on primitive
multiple curves.

Notations and terminology: – An algebraic variety is a
quasi-projective scheme over $\C$. A vector bundle on a scheme is an
algebraic vector bundle.

– If $X$ is a scheme and P∈X is a closed point, we denote
by m_X,P (or m_P) the maximal ideal of $P$ in _X,P.

– If $X$ is a scheme and Z⊂X is a closed subscheme,
_Z (or _Z,X) denotes the ideal sheaf of $Z$ in $X$.

– If V is a finite dimensional complex vector space, (V) denotes the
projective space of lines in $V$, and we use a similar notation for projective

– Let $X$ be a set and (X_i)_i∈I a family of subsets of $X$. If
i,j∈I, X_ij denotes the intersection X_i∩X_j, and
similarly for more indices.

– If $X$ and $Y$ are two schemes and f:X→Y is a morphism, d(f):f^*(Ω_Y)→Ω_X  is the corresponding canonical morphism.

– If $R$ is a commutative ring, $n$ a positive integer, a∈R[t]/(t^n)  and $i$ an integer such that 0≤i<n, let a^(i)
denote the coefficient of t^i on $\bf a$, so that we have  a=_i=0^n-1a^(i)t^i.

§ PRELIMINARIES

Čech cohomologycech

Let $X$ be a scheme over $\C$ and $E$, $V$ vector bundles on $X$ of rank
$r$. Let (U_i)_i∈I be an open cover of $X$ such that
we have isomorphisms:
α_i:E_|U_i[r]^-≃    V_|U_i .
V_|U_ij ,
so that we have the relation  α_ijα_jk=α_ik  on

Let $n$ be a positive integer, and for every sequence (i_0,…,i_n) of
distinct elements of $I$, σ_i_0⋯i_n∈H^0(U_i_0⋯i_n,V). Let

\[\theta_{i_0\cdots i_n} \ = \ \alpha_{i_0}^{-1}\sigma_{i_0\cdots i_n} \ \in \
H^0(U_{i_0\cdots i_n},E) \ . \]

The family (θ_i_0⋯i_n) represents an element of H^n(X,E)
if the cocycle relations are satisfied: for every sequence
(i_0,…,i_n+1) of distinct elements of $I$,

\[\sigg_{k=0}^n(-1)^k\theta_{i_0\cdots\widehat{i_k}\cdots i_{n+1}} \ = \ 0 \ ,
\]

which is equivalent to

\[\alpha_{i_0i_1}\sigma_{i_1\cdots i_{n+1}}+\sigg_{k=1}^{n+1}
(-1)^k\sigma_{i_0\cdots\widehat{i_k}\cdots i_{n+1}} \ = \ 0 \ . \]

For n=1, this gives that elements of H^1(X,E) are represented by
families (σ_ij),σ_ij∈H^0(U_ij,V),
such that

\[\alpha_{ij}\sigma_{jk}+\sigma_{ij}-\sigma_{ik} \ = \ 0 \ . \]

In Čech cohomology, it is generally assumed that
θ_ji=-θ_ij. This implies that

Let $F$ be a vector bundle on $X$.
Similarly, an element of H^n(X,EF) is
represented by a family (μ_i_0⋯i_n), with  μ_i_0⋯i_n∈H^0(U_i_0⋯i_n,VF), satisfying the relations

\[(\alpha_{i_0i_1}\ot I_F)(\mu_{i_1\cdots i_{n+1}})+\sigg_{k=1}^{n+1}
(-1)^k\mu_{i_0\cdots\widehat{i_k}\cdots i_{n+1}} \ = \ 0 \ . \]

The corresponding element of H^0(U_i_0⋯i_n,EF) is

Coboundaries – In the situation of
<ref>, the family (θ_i_0⋯i_n) represents $0$ in
H^n(X,EF) if and only if there exists a family (τ_j_0⋯j_n-1), with  τ_j_0⋯j_n-1∈H^0(U_j_0⋯j_n-1,EF), such that

\[\theta_{i_0\cdots i_n} \ = \
\sigg_{k=0}^n(-1)^k\tau_{i_0\cdots\widehat{i_k}\cdots i_n} \ . \]

Similarly, the family (μ_i_0⋯i_n) represents $0$ if and only
there exists a family  (ν_j_0⋯j_n-1), with
ν_j_0⋯j_n-1∈H^0(U_j_0⋯j_n-1,VF), such that

\[\mu_{i_0\cdots i_n} \ = \
(\alpha_{i_0i_1}\ot I_F)(\nu_{i_1\cdots i_n})+\sigg_{k=1}^{n}
(-1)^k\nu_{i_0\cdots\widehat{i_k}\cdots i_{n}} \ . \]

Representation of morphisms – Suppose that we
have also local trivializations of $F$:
β_i:F_|U_i[r]^-≃    _U_i^s .
We have then local trivializations of (E,F)

\[\xymatrix@R=5pt{\Delta_i:\HHom(E,F)_{|U_i}\ar[r]^-\simeq &
\ko_{U_i}\ot L(\C^r,\C^s)\\ \phi\fmaps[r] & \beta_i\circ\phi\circ\alpha_i^{-1}

such that, for every  λ∈H^0(_U_ijL(^r,^s)), we have

\[\Delta_{ij}(\lambda) \ = \ \Delta_i\Delta_j^{-1}(\lambda) \ = \
\beta_{ij}\lambda\alpha_{ij}^{-1} \ . \]

Products – Let  χ_E,F:H^1(X,E)H^1(X,F)→H^2(X,EF)  be the canonical map. Let  ϵ∈H^1(X,E), ϕ∈H^1(X,F), represented by cocycles (ϵ_ij),
(ϕ_ij), ϵ_ij∈H^0(U_ij,E), ϕ_ij∈H^0(U_ij,F). Then χ_E,F(ϵϕ) is represented by the
cocycle (ϵ_ijϕ_jk).

With respect to the canonical isomorphism  FE≃EF  we have
χ_F,E(ϕϵ)=-χ_E,F(ϵϕ) .

Suppose that we have vector bundles $V$, $W$ on $X$ with isomorphisms

\[\alpha_i:E_{|U_i}\lra V_{|U_i} \ , \qquad \beta_i:F_{|U_i}\lra W_{|U_i} \ .
\]

Let  e_ij=α_iϵ_ij, f_ij=β_iϕ_ij . We have

\[\epsilon_{ij}\ot\phi_{jk} \ = \ (\alpha_i^{-1}\ot\beta_j^{-1})(e_{ij}\ot
f_{jk}) \ , \]

\[(\alpha_i\ot\beta_i)(\epsilon_{ij}\ot\phi_{jk}) \ = \ (I\ot\beta_{ij})
(e_{ij}\ot f_{jk}) \ . \]

It follows that the family (e_ijβ_ijf_jk) represents
χ_E,F(ϵϕ) in the sense of <ref>, with respect to
the local isomorphisms  α_iβ_i:(EF)_|U_i(VW)_|U_i.

Construction of vector bundles via local
isomorphisms – Le $Z$ be a scheme over $\C$, (Z_i)_i∈I an open
cover of $Z$, and for every i∈I, a scheme U_i, with an isomorphism
δ_i:Z_i→U_i. Let
which is an isomorphism.
Then a vector bundle of $X$ can be constructed (in an obvious way) using vector
bundles E_i on U_i and isomorphisms

\[\Theta_{ij}:\delta_{ij}^*(E_{j|\delta_j(Z_{ij})})\lra

such that  Θ_ij∘δ_ij^*(Θ_jk)=Θ_ik  on

Representation of extensions – Let $E$, $F$ be
coherent sheaves on $X$, andσ∈H^1(X,(F,E))⊂^1__X(F,E). Let

\[0\lra E\lra\ke\lra F\lra 0\]

be the corresponding exact sequence. Suppose that $\sigma$ is represented by a
cocycle (σ_ij), σ_ij:F_|U_ij→E_|U_ij. Then
$\ke$ can be constructed by gluing the sheaves (E⊕F)_|U_i
using the automorphisms of (E⊕F)_|U_ij defined by the matrices
I_E     σ_ij
0     I_F.

Moduli spaces of sheavesFine

(cf. [8])

Let $X$ be a scheme over $\C$.

Families and sets of vector bundles – Let $S$ be a scheme over $\C$.

A family of vector bundles on $X$ parametrized by $S$ is a vector bundle
on X×S. If f:T→S is a morphism of schemes and $\ke$
a family of vector bundles on $X$ parametrized by $S$, we note

\[f^\sharp(\ke) \ = \ (I_X\times f)^*(\ke) \ , \]

which is a family of vector bundles on $X$ parametrized by $T$.

Let $\chi$ be a nonempty set of isomorphism classes of vector bundles on $X$.
A family of vector bundles of $\chi$ parametrized by $\chi$ is a family
of vector bundles $\ke$ on $X$ parametrized by $S$ such that for every closed
point s∈S, _s belongs to $\chi$.

We say that $\chi$ is open if for every $S$ and every family of vector
bundles $\ke$ on $X$ parametrized by $S$, if s∈S is a closed point such
that _s∈χ, then there exists a neighbourhood $U$ of $s$ such that
_u∈χ for every closed point u∈U.

Fine moduli spaces – Let $\chi$ be a nonempty
open set of isomorphism classes of vector bundles on $X$. A fine moduli
space for $\chi$ is the data of a scheme $M$ and of

– a bijection

\[\xymatrix@R=5pt{M^0\ar[r] & \chi\\ m\fmaps[r] & E_m

(where M^0 denotes the set of closed points of $M$),

– An open cover (M_i)_i∈I of $M$, and for every i∈I, a
vector bundle _i on X×M_i such that for every m∈M_i,

such that for any scheme $S$, any family $\kf$ of vector bundles of $\chi$
parametrized by $S$, there exists a morphism  f_:S→M such that:
for every closed point s∈S, if _s≃E_m,
with m∈M_i, then there exists an open neighbourhood $U$ of $s$ such that
f_(U)⊂M_i  and f^♯_|U(_i)≃_|X×U.

If $\chi$ is an open set, then
for every closed point m∈M, if m∈M_i, then _i is a
semi-universal deformation of _i,m, hence the Kodaïra-Spencer map T_m M_i→^1__X(_i,m,_i,m)  is an isomorphism.

Fine moduli spaces are unique.

Affine bundlesaff

Let  f:→S  be a morphism of schemes, and r≥0 an integer. We
say that $f$ (or $\ka$) is an affine bundle of rank $r$ over $S$ if there
exists an open cover (S_i)_i∈I of $S$ such that for every i∈I
there is an isomorphism  τ_i:f^-1(S_i)→S_i×^r  over
S_i such that for every distinct i,j∈I, f_j∘f_i^-1:S_ij×^r→S_ij×^r  is of the form

\[(x,u)\longmapsto (x,A_{ij}(x)u+b_{ij}(x)) \ , \]

where A_ij is a r×r-matrix of elements of _S(S_ij) and
b_ij is a morphism from S_ij to ^r. We have then the cocycle

\[A_{ij}A_{jk} \ = \ A_{ik} \ , \quad b_{ik} \ = \ A_{ij}b_{jk}+b_{ij} \ . \]

The first relation shows that the family (A_ij) defines a vector bundle
$\A$ on $S$, and the second that (b_ij) defines  λ∈H^1(S,) (according to [17], 2.1.1).

The vector bundle $\A$ is uniquely defined, as well as

We say that $\ka$ is a vector bundle if $f$ has a section. This is the
case if and only λ=0, and then a section of $f$ induces an
isomorphism ≃ over $S$.

For every closed point s∈S there is a canonical action of the additive
group _s on _s

\[\xymatrix@R=5pt{\A_s\times\ka_s\ar[r] & \ka_s\\ (u,a)\fmaps[r] & a+u}\]

such that for every a∈_s, $u\mapsto a+u$ is an isomorphism

Quotients – Let  ⊂  be a
subbundle of $\A$, q:→/  the quotient morphism and
R:H^1(S,)→H^1(S,/)  the induced morphism. Let  λ∈H^1(S,)  and  λ”=R(λ). Let  f”:”→S  be the
affine bundle corresponding to λ”. We will also use the notation
”=/. Then there is an obvious
$S$-morphism  p:→”, such that for every closed point s∈S,
a∈_s and u∈_s, we have  p_s(a+u)=p_s(a)+q_s(u). Moreover
$p$ is an affine bundle with associated vector bundle f”^*().

§ CANONICAL EXACT SEQUENCES IN ČECH COHOMOLOGY

The Ext spectral sequence allows to compute Ext groups in terms of cohomology
groups and Leray's spectral sequence to compute the cohomology groups of a
sheaf in terms if the cohomology groups of its direct images by a suitable
morphism. We will describe simple examples of consequences of these spectral
sequences in terms of Čech cohomology (this provides also elementary proofs
of these consequences).

Leray's spectral sequenceEx_Ler

Let  π:X→S  be a proper flat morphism of sheaves, and $\ke$ a
coherent sheaf on $X$ such that  π_*()=0. Leray's spectral sequence
implies that we have an exact sequence
0[r]     H^1(S,R^1π_*())[r]^-λ    H^2(X,)[r]^-η
H^0(S,R^2π_*())[r]     0  .
We will describe $\lambda$ and $\eta$ in terms of Čech cohomology.

Let (S_m)_m∈M (resp. (X_i)_i∈I) be an affine cover of $S$
(resp. $X$), such that for every i∈I, m∈M, π^-1(S_m)∩X_i is affine.

Description of $\eta$ – Let Σ∈H^2(X,), represented by the cocycle (B_ijk)_i,j,k∈I, B_ijk∈H^0(X_ijk,). For every m∈M,
(B_ijk|π^-1(S_m)∩X_ijk)_i,j,k∈I defines  Σ_m∈H^2(π^-1(S_m),), and (Σ_m)_m∈M defines

Description of $\lambda$ – Let  θ∈H^1(S,R^1π_*()), represented by the cocycle
(A_mn)_m,n∈M, A_mn∈H^1(π^-1(S_mn),). Suppose that
A_mn is represented by the cocycle (α_ij^mn)_i,j∈I,α_ij^mn∈H^0(X_ij∩π^-1(S_mn),). The
cocycle condition for (A_mn) implies that (α_ij^mn+α_ij^np-α_ij^mp)_i,j∈I represents
0 in H^1(π^-1(S_mnp),). Hence we can write

\[\alpha_{ij}^{mn}+\alpha_{ij}^{np}-\alpha_{ij}^{mp} \ = \
\beta_i^{mnp}-\beta_j^{mnp} \ , \]

where  β_k^mnp∈H^0(π^-1(S_mnp)∩X_k,)  for every
k∈I. We have, for every distinct m,n,p,q∈M, i,j∈I

\[\beta_i^{mnp}-\beta_i^{mnq}+\beta_i^{mpq}-\beta_i^{npq} \ = \
\beta_j^{mnp}-\beta_j^{mnq}+\beta_j^{mpq}-\beta_j^{npq} \ .\]

Hence there exists  Δ^mnpq∈H^0(π^-1(S_mnpq),)  such

\[\beta_i^{mnp}-\beta_i^{mnq}+\beta_i^{mpq}-\beta_i^{npq} \ = \
\Delta^{mnpq}_{|\pi^{-1}(S_{mnpq})\cap V_i}\]

for every i∈I. Since π_*()=0, we have  Δ^mnpq=0.
Hence for every i∈I, (β_i^mnp)_m,n,p∈M satisfies the
cocycle condition, and defines an element of H^2(X_i,)=0 (because
X_i is affine). So we can write, for every distinct m,n,p∈M, i∈I,
\[\beta_i^{mnp} \ = \ \Gamma_i^{mn}+\Gamma_i^{np}-\Gamma_i^{mp} \ , \]

with  Γ_i^mn∈H^0(π^-1(S_mn)∩X_i,). Let

\[a^{mn}_{ij} \ = \ \alpha_{ij}^{mn}-\Gamma_i^{mn}+\Gamma_j^{mn} \ . \]

Then A_mn is also represented by (a^mn_ij)_i,j∈I. For every
distinct i,j∈I, we have a^mn_ij+a^np_ij-a^mp_ij=0,
hence (a^mn_ij)_m,n∈M represents an element of H^1(X_ij,),
which is zero (because X_ij is affine). So we can write

\[a_{ij}^{mn} \ = \ \mu_{ij}^m-\mu_{ij}^n \ , \]

μ_ij^m∈H^0(π^-1(S_m)∩X_ij). We have, for every distinct
i,j,k∈I, m,n∈M,

\[\mu_{ij}^m+\mu_{jk}^m-\mu_{ik}^m \ = \ \mu_{ij}^n+\mu_{jk}^n-\mu_{ik}^n \]

on π^-1(S_mn)∩X_ijk. Hence there exists  B_ijk∈H^0(X_ijk,) such that μ_ij^m+μ_jk^m-μ_ik^m=B_ijk|
π^-1(S_m)∩X_ijk. Then (B_ijk)_i,j,k∈I verifies the
cocycle condition, and defines  λ(θ)∈H^2(X,).

With these definitions, it is easy to see that $\lambda$ is injective, $\eta$
surjective and (η)=(λ).

The exact sequence of Ext'sEx_Ext

Let $X$ be a scheme and $E$, $F$ be coherent sheaves on $X$. From [22],
7.3 (Ext spectral sequence), we have an exact sequence Γ_E,F
H^0(^1__X(E,F))[r]^-δ_X     H^2(X,(E,F))
We will give a simple description of this exact sequence using Čech

Description of α_X – Let (X_i)_i∈I be an open affine cover of $X$, and u∈H^1(X,(E,F)), represented by the cocycle (ϕ_ij) with
respect to the cover (X_i), with ϕ_ij:E_|X_ij→F_|X_ij.
Let $\ke$ be the coherent sheaf on $X$ obtained by gluing the sheaves
(E⊕F)_|X_i using the automorphisms of (E⊕F)_|X_ij
defined by the matrices I_E     0
I_F. Then we have an obvious exact sequence  0→F→→E→0, whose associated element in ^1__X(E,F) is α_X(u).

Let σ∈^1__X(E,F), corresponding to the extension
0[r]     F[r]^-λ    [r]^-μ    E[r]     0.
Let u∈H^1(X,(E,F)), represented by the cocycle
(ϕ_ij). Let σ'=σ+α_X(u), corresponding to the
extension  0→F→'→E→0. Then ' is obtained by gluing
the sheaves _|X_i using the automorphisms

Description of β_X – Let σ∈^1__X(E,F), corresponding to an exact sequence 0→F→→E→0, whose restriction to X_i defines σ_i∈^1__X_i(E_|X_i,F_|X_i)=^1__X(E,F)(X_i).
then β_X(σ) is the section $s$ of ^1__X(E,F) such
that for every i∈I, s_|X_i=σ_i.

Description of δ_X – Let  s∈H^0(^1__X(E,F)). For every i∈I, we have s_|X_i∈^1__X_i(E_|X_i,F_|X_i), which defines an exact
Σ_i:0[r]     F_|X_i[r]^-γ_i     _i[r]^-p_i
E_|X_i[r]     0  .
The restrictions of Σ_i and Σ_j to X_ij define the same
element of ^1__X_ij(E_|X_ij,F_|X_ij), i.e. we have a
commutative diagram
Σ_i:  0[r]     F_|X_ij[r]^-γ_i[d]
_i|X_ij[r]^-p_i[d]^ψ_ij     E_|X_ij[r][d]     0

Σ_j:  0[r]     F_|X_ij[r]^-γ_j
_j|X_ij[r]^-p_j     E_|X_ij[r]     0
We have then
ψ_jkψ_ij-ψ_ik:_i|X_ijk→_k|X_ijk, and there
exists  σ_ijk:E_|X_ijk→F_|X_ijk  such that
ψ_jkψ_ij-ψ_ik=γ_kσ_ijk p_i. It is easily checked
that the family (σ_ijk) satisfies the cocycle relation, and then
defines an element of H^2(X,(E,F)), which is δ_X(s).

Local extensions – Let  s∈H^0(^1__X(E,F)). Let (U_m)_m∈M be an open cover of $X$ such
that for every i∈I and m∈M, the intersection X_i∩U_m is
affine. For every i,j∈I and m,n∈M, let  Z_i^m=X_i∩U_m,
Z_ij^m=X_ij∩U_m, etc.

Given two extensions
Σ:0[r]     F[r]^-λ    [r]^-μ    E[r]     0,       Σ':0[r]     F[r]^-λ'    '[r]^-μ'     E[r]     0  ,
an isomorphism ϕ:→' is called a I-isomorphism (with respect to $\Sigma$, Σ') if the following diagram
0[r]     F[r]^-λ[d]     [r]^-μ[d]^ϕ
E[r][d]     0

0[r]     F[r]^-λ'     '[r]^-μ'     E[r]     0
is commutative. If ϕ' is another I-isomorphism →', there
exists a morphism  τ:E→F  such that

Suppose that for every m∈M there exists
σ_m∈^1__U_m(E,F)  such that
δ_U_m(σ_m)=s_|U_m. Let
0[r]     F_|U_m[r]^-c_m     _m[r]^-π_m[r]
E_|U_m[r]     0
be the extension corresponding to σ_m.
Then for every i∈I, m∈M,
there exists an I-isomorphism  α_i^m:_i|Z^m_i→_m|Z^m_i.
Let j∈I. Then (α_j^m)^-1α_i^m is an I-isomorphism
_i|Z^m_ij→_j|Z^m_ij. So we can write

\[\psi_{ij}-(\alpha_j^m)^{-1}\alpha_i^m \ = \ \gamma_j\mu^m_{ij}p_i \ , \]

with  μ_ij^m:E_|Z_ij^m→F_|Z_ij^m. It follows that

\[\big(\psi_{jk}\psi_{ij}-\psi_{ik}\big)_{|Z_{ijk}^m} \ = \
\gamma_k(\mu_{jk}^m+\mu_{ij}^m-\mu_{ik}^m)_{|Z_{ijk}^m}p_i \ , \]

so we have

\[\sigma_{ijk|Z_{ijk}^m} \ = \
(\mu_{jk}^m+\mu_{ij}^m-\mu_{ik}^m)_{|Z_{ijk}^m} \ . \]

Let  a_ij^mn=μ_ij^m-μ_ij^n. Then
for every m,n∈M, (a^mn_ij)_i,j∈I is a cocycle and thus
defines  A_mn∈H^1(U_mn,(E,F)).

The case of families – We suppose now that $X$ is a product : X=Y×S, and that
p_S*((E,F))=0, where  p_S:X→S  is the projection. We have
from <ref> a canonical injection

\[\lambda : H^1(S,R^1p_{S*}(\HHom(E,F)))\lra H^2(X,\HHom(E,F)) \ . \]

Let (S_m)_m∈M be an affine open cover of $S$. We suppose that
U_m=p_S^-1(S_m)  for every m∈M. Then (A_mn)_m,n∈M
defines  ϵ∈H^1(S,R^1p_S*((E,F))). It follows from
<ref> that

Proposition:  We have

§ PRIMITIVE MULTIPLE SCHEMES

Definition and constructionPMS_def

Let $X$ be a smooth connected variety, and  d=(X). A multiple
scheme with support $X$ is a Cohen-Macaulay scheme $Y$ such that
Y_red=X. If $Y$ is quasi-projective we say that it is a multiple
variety with support $X$. In this case $Y$ is projective if $X$ is.

Let $n$ be the smallest integer such that Y=X^(n-1), X^(k-1)
being the $k$-th infinitesimal neighborhood of $X$, i.e. _X^(k-1)=_X^k . We have a filtration  X=X_1⊂X_2⊂⋯⊂X_n=Y  where $X_i$ is the biggest Cohen-Macaulay
subscheme contained in Y∩X^(i-1). We call $n$ the multiplicity
of $Y$.

We say that $Y$ is primitive if, for every closed point $x$ of $X$,
there exists a smooth variety $S$ of dimension d+1, containing a
neighborhood of $x$ in $Y$ as a locally closed subvariety. In this case,
L=_X/_X_2 is a line bundle on $X$, X_j is a primitive multiple
scheme of multiplicity $j$ and we have
_X_j=_X^j, _X_j/_X_j+1=L^j  for 1≤j<n.
We call $L$ the line bundle on $X$ associated to $Y$. The ideal sheaf
_X,Y can be viewed as a line bundle on X_n-1.

Let P∈X.
Then there exist elements y_1,…,y_d, $t$ of m_S,P whose images
in m_S,P/m_S,P^2 form a basis, and such that for 1≤i<n we have
_X_i,P=(t^i). In this case the images of y_1,…,y_d in
m_X,P/m_X,P^2 form a basis of this vector space.

A multiple scheme with support $X$ is primitive if and only if
_X/_X^2 is zero or a line bundle on $X$ (cf. [14], proposition

Even if $X$ is projective, we do not assume that $Y$ is projective. In fact we
will see examples of non quasi-projective $Y$.

The simplest case is when $Y$ is contained in a smooth variety $S$ of dimension
d+1. Suppose that $Y$ has multiplicity $n$. Let P∈X and
f∈_S,P  a local equation of $X$. Then we have
_X_i,P=(f^i)  for 1<j≤n in $S$, in particular
_Y,P=(f^n), and  L=_X(-X) .

For any L∈(X), the trivial primitive variety of multiplicity
$n$, with induced smooth variety $X$ and associated line bundle $L$ on $X$ is
the $n$-th infinitesimal neighborhood of $X$, embedded by the zero section in
the dual bundle $L^*$, seen as a smooth variety.

Construction of primitive multiple schemes – Let $Y$ be a primitive multiple scheme of multiplicity $n$, X=Y_red.
Let  Z_n=([t]/(t^n)).
Then for every closed point P∈X, there exists an open neighborhood $U$
of $P$ in $X$, such that if U^(n) is the corresponding neighborhood of
$P$ in $Y$, there exists a commutative diagram

U^(n)[rr]^-≃        U×𝐙_n
i.e. $Y$ is locally trivial ([10], théorème 5.2.1, corollaire 5.2.2).

It follows that we can construct a primitive multiple scheme of multiplicity
$n$ by taking an open cover (U_i)_i∈I of $X$ and gluing the varieties
U_i×Z_n (with automorphisms of the  U_ij×Z_n
leaving U_ij invariant).

Let (U_i)_i∈I be an affine open cover of $X$ such that we have
δ_i:U_i^(n)[r]^-≃    U_i×Z_n ,
and  δ_i^*:_U_i×Z_n→_U_i^(n)  the
corresponding isomorphism. Let
U_ij×Z_n  .
Then  δ_ij^*=δ_i^*-1δ_j^*  is an automorphism of
_U_i×Z_n=_X(U_ij)[t]/(t^n), such that for every
ϕ∈_X(U_ij), seen as a polynomial in $t$ with coefficients in
_X(U_ij), the term of degree zero of δ_ij^*(ϕ) is the
same as the term of degree zero of $\phi$.

The ideal sheaf of $X$ – There exists  α_ij∈_X(U_ij)_[t]/(t^n-1)  such that
δ_ij^*(t)=α_ijt. Let
For every i∈I, δ_i^*(t) is a generator of
_X,Y|U^(n). So we have local trivializations

\[\xymatrix@R=5pt{\lambda_i:\ki_{X,Y|{U_i^{(n-1)}}}\ar[r] &
\ko_{U_i^{(n-1)}}\\
\delta_i^*(t)\fmaps[r] & 1}\]

Hence  λ_ij=λ_iλ_j^-1:
_U_ij^(n-1)→_U_ij^(n-1)  is the multiplication by
δ_j^*(α_ij). It follows that
(δ_j^*(α_ij)) (resp.
(α^(0)_ij))  is a cocycle representing the line bundle
_X,Y (resp. L) on X_n-1 (resp. $X$).

The associated sheaves of non abelian groups and
obstructions to extension in higher multiplicity – For every open subset $U$ of $X$, we have
_X×Z_n(U)=_X(U)[t]/(t^n). Let  _n be
the sheaf of (non abelian) groups on $X$ defined by: for every open subset $U$
of $X$, _n(U) is the group of automorphisms $\theta$ of the $\C$-algebra
_X(U)[t]/(t^n) such that for every α∈_X(U)[t]/(t^n),
θ(α)_|U=α_|U. We have _1=_X^*. Then
(δ_ij^*) is a cocycle of _n, which describes completely

In this way we see that there is a canonical bijection between the cohomology
set H^1(X,_n) and the set of isomorphism classes of primitive multiple
schemes X_n such that X=(X_n)_red. There is an obvious surjective
morphism  ρ_n+1:_n+1→_n, such that if n≥2,

\[H^1(\rho_{n+1}):H^1(X,\kg_{n+1})\lra H^1(X,\kg_n)\]

sends a primitive multiple scheme of multiplicity n+1 to the underlying
scheme of multiplicity $n$, whereas

\[H^1(\rho_2):H^1(X,\kg_2)\lra H^1(X,\ko_X^*)=\Pic(X)\]

sends X_2 to $L$.

We have (ρ_2)≃T_X  and
(ρ_n+1)≃T_X⊕_X  if n≥2.
The fact that they are sheaves of abelian groups allows to compute obstructions. Let g_n∈H^1(X,_n), corresponding to the primitive
multiple scheme X_n, and for 1≤i<n, g_i the image of g_n in
H^1(X,_i). Let  (ρ_n+1)^g_n be the associated sheaf of
groups (cf. [17], 2.2, [20]). We have then

\[\ker(\rho_2)^{g_1} \ \simeq \ T_X\ot L  \]

By cohomology theory, we find that there is a
canonical surjective map

\[H^1(X,T_X\ot L)\lra H^1(\rho_2)^{-1}(L)\]

sending 0 to the trivial primitive scheme, and whose fibers are the orbits of
the action of $\C^*$ on H^1(X,T_XL) by multiplication. Hence there is a
bijection between the set of non trivial double schemes with associated line
bundle $L$, and (H^1(X,T_XL)). For a non trivial double scheme
X_2, we have an exact sequence

\[0\lra L\lra\Omega_{X_2|X}\lra\Omega_X\lra 0 \ , \]

corresponding to  σ∈^1__X(Ω_X,L)=H^1(T_XL), and
σ is the element of (H^1(T_XL)) corresponding to X_2.
We can write for n=2

\[(\delta_{ij}^*)_{|\ko_X(U_{ij})} \ = \
I_{|\ko_X(U_{ij})}+tD_{ij} \ , \]

where D_ij is a derivation of _X(U_ij). Then the family
(D_ij) represents $\sigma$ in the sense of <ref>.

If n>2 we have

\[\ker(\rho_{n+1})^{g_n} \ \simeq \ (\Omega_{X_2|X})^*\ot L^n\]

We have then (by the theory of cohomology of
sheaves of groups) an obstruction map

\[\Delta_n:H^1(X,\kg_n)\lra H^2((\Omega_{X_2|X})^*\ot L^n)\]

such that  g_n∈(H^1(ρ_n+1))  if and only  Δ_n(g_n)=0.

Extensions of the ideal sheaf of $X$ – The
ideal sheaf
_X,X_n is a line bundle on X_n-1. A necessary condition to
extend X_n to a primitive multiple scheme X_n+1 of multiplicity
n+1 is that _X,X_n can be extended to a line bundle on X_n
(namely _X,X_n+1). This is why we can consider pairs (X_n,Ł),
where $\L$ is a line bundle on X_n such that

The corresponding sheaf of groups on $X$ is defined as follows: for every open
subset U⊂X, _n(U) is the set of pairs (ϕ,u), where
ϕ∈_n(U), and u∈_X(U)[t]/(t^n) is such that
ϕ(t)=ut (cf. [17], 4.5). The set of isomorphism classes of the
above pairs (X_n,Ł)
can then be identified with the cohomology set H^1(X,_n).

There is an obvious morphism  τ_n:_n+1→_n, such that

\[H^1(\tau_n):H^1(X,\kg_{n+1})\lra H^1(X,\kh_n)\]

sends X_n+1 to (X_n,_X,X_n+1).
Let g∈H^1(X,_n). Then
(τ_n)≃T_X  and

\[\ker(\tau_n)^g \ \simeq \ T_X\ot L^n \ . \]

Consequently there is again, by cohomology theory, an obstruction map

\[\Delta''_n:H^1(X,\kh_n)\lra H^2(T_X\ot L^n)\]

such that, if (X_n,Ł) corresponds to $g$, there is an extension of X_n
to a primitive multiple scheme X_n+1 of multiplicity n+1 with
_X,X_n+1≃Ł  if and only if  Δ”_n(g)=0.

Descriptions using the open cover (U_i)const_sh

(i)  Construction of sheaves and morphisms –  (cf. [17],
2.1.3). Let $\ke$ be a
coherent sheaf on X_n. We can define it in the usual way, starting with
sheaves _i on the open sets U_i^(n) and gluing them. We take
these sheaves of the form  _i=δ_i^*(_i), where _i is a
sheaf on  U_i×Z_n. To glue the sheaves _i on the
U_ij^(n) we use isomorphisms
ρ_ij:_j|U_ij^(n)→_i|U_ij^(n), with the relations
ρ_ik=ρ_jkρ_ij. Let

\[\theta_{ij} =
Z}_n})\lra\ke_{i|U_{ij}\times{\bf Z}_n} \ .
\]

We have then the relations
θ_ik=θ_ij∘δ_ij^*(θ_jk). Conversely,
starting with sheaves _i and isomorphisms θ_ij satisfying the
preceding relations, one obtains a coherent sheaf on X_n.

This applies to trivializations, i.e when  _i=_U_i^(n)^r.
We have then θ_ij:_U_ij×Z_n^r→_U_ij×Z_n^r. In particular,
_X,X_n is represented by

Suppose that we have another sheaf ' on X_n, defined by sheaves
'_i on  U_i×Z_n  and isomorphisms θ'_ij.
One can see easily that a morphism  Ψ:→'  is defined by
morphisms  Ψ_i:_i→'_i  such that

(ii)  Other construction of sheaves –  Let $E$ be a vector bundle
on $X$. For every open subset i∈I, let  p_i:U_i×Z_n→U_i  be the projection. Then we have
We construct a vector bundle $\E$ on X_n by gluing the
vector bundles δ_i^*(p_i^*(E_|U_i)) on U_i^(n). For this we
take isomorphisms

\[\epsilon_{ij}:\delta_j^*(p_j^*(E_{|U_{ij}}))_{U_{ij}^{(n)}}\lra
\delta_i^*(p_i^*(E_{|U_{ij}}))_{U_{ij}^{(n)}}\]

satisfying  ϵ_ijϵ_jk=ϵ_ik. Let

\[\tau_{ij}={\delta_i^*}^{-1}(\epsilon_{ij}):\delta_{ij}^*(p_j^*(D))=p_i^*(D)\lra
p_i^*(D) \ . \]

Then we have  τ_ij∘δ_ij^*(τ_jk)=τ_ik.
Conversely, given automorphisms τ_ij satisfying the preceding
we can define ϵ_ij and the vector bundle $\E$ on X_n.

Suppose that the τ_ij are homotheties (multiplication by ν_ij
∈(U_ij×Z_n)^*). Then the family (ν_ij) defines a
line bundle $\L$ on X_n (by (i)), and we have  _|X≃EŁ_|X.

(iii)  Cohomology of vector bundles –  Suppose that a vector
bundle $E$ on X_n is built using isomorphisms
θ_ij:_U_ij×Z_n^r→_U_ij×Z_n^r  as in (i).
Any  β∈H^1(X_n,E)  is represented by a family (β_ij),
β_ij∈H^0(_U_ij×Z_n^r), with the cocycle
relations  β_ik=β_ij+θ_ijδ_ij^*(β_jk).

Connecting morphismscon_hom

Recall that on X_2 we can write
δ_ij|_X(U_ij)^*=I+tD_ij, where D_ij is a derivation
of _X(U_ij) (cf. <ref>).

Let $\E$ be a vector bundle on X_2 of rank $r$, defined as in
<ref>, (i), using isomorphisms θ_ij:_U_ij×Z_2^r→_U_ij×Z_2^r. We can write

\[\theta_{ij} \ = \ \theta_{ij}^{(0)}+\theta_{ij}^{(1)}t \ , \]

where θ_ij^(0), θ_ij^(1) are matrices with
coefficients in _X(U_ij).

Let  E=_|X. We have a canonical exact sequence

\[0\lra E\ot L\lra\E\lra E\lra 0 \ , \]

whence a connecting morphism

\[\delta^1_\E:H^1(X,E)\lra H^2(X,E\ot L) \ . \]

Let  β∈H^1(X,E), represented by a family (β_ij),
β_ij∈H^0(_U_ij^r), in the sense of

Lemma:  δ^1_(β) is represented,
in the sense of <ref>, (iii), by the family (ν_ijk), with

\[\nu_{ijk} \ = \
\theta_{ij}^{(0)}D_{ij}(\beta_{jk})+\theta_{ij}^{(1)}\beta_{jk} \ . \]

We construct δ^1_(β) as follows: we view β_ij,
β_kj, β_ik as elements of H^0(_U_ij×Z_n^r). Then we can take

\[\nu_{ijk} \ = \ \beta_{ij}-\beta_{ik}+\theta_{ij}\delta_{ij}^*(\beta_{jk})
\]

which gives lemma <ref>.

Suppose that X_n can be extended to a primitive multiple scheme X_n+1
of multiplicity n+1.

We have an exact sequence of sheaves on X_n+1

\[0\lra L^n\lra\ko_{X_{n+1}}\lra\ko_{X_n}\lra 0 \ , \]

whence a connecting morphism

\[\delta^0:H^0(\ko_{X_n})\lra H^1(X,L^n) \ . \]

We have  H^0(X,L^n-1)⊂H^0(_X_n). Let  η∈H^0(X,L^n-1), represented by a cocycle (η_i),
η_i∈_X(U_i) (in the sense of <ref>, i.e. the cocycle relation
is  η_i=(α_ij^(0))^n-1η_j). Recall that
_X,X_n+1 is represented by the cocycle (α_ij), where
α_ij is an invertible element of _X(U_ij)[t]/(t^n).

The proof of the following lemma is similar to that of lemma <ref>:

Lemma:  δ^0(η) is
represented by the cocycle (γ_ij), with

\[\gamma_{ij} \ = \ (\alpha^{(0)}_{ij})^{n-1}D_{ij}(\eta_j)+
(n-1)(\alpha^{(0)}_{ij})^{n-2}\alpha^{(1)}_{ij}\eta_j \ . \]

The sheaf of differentialsPMS_Omeg

We use the notations of <ref>. We will give a construction of
Ω_X_n using <ref> : it is defined by the sheaves
_i=Ω_U_i×Z_n, and

\[{\bf d}(\delta_{ij}):\delta_{ij}^*(\Omega_{U_i\times{\bf Z}_n})=
\Omega_{U_i\times{\bf Z}_n}\lra\Omega_{U_i\times{\bf Z}_n}\]

(more precisely  d(δ_ij)(a.db)=δ_ij^*(a)d(δ_ij^*(b)))).

Let σ∈H^1(X_n,Ω_X_n), represented by the cocycle
(σ_ij), σ_ij∈Ω_U_ij^(n). Then σ'_ij=d(δ_i^-1)(σ_ij)∈Γ(Ω_U_ij
×Z_n) , and we have the relations

\[\sigma'_{ik} \ = \ \sigma'_{ij}+{\bf d}(\delta_{ij})(\sigma'_{jk}) \ . \]

Conversely, every family (σ'_ij) satisfying these relations defines
an element of H^1(X_n,Ω_X_n).

The dualizing sheaf – Suppose that $X$ is
projective. Then X_n is a proper Cohen-Macaulay scheme. Hence there exists
a dualizing sheaf ω_X_n for X_n, which is a line bundle.

Extensions of primitive multiple schemesExt_sh

Let X_n a primitive multiple scheme of multiplicity n≥2, with
underlying smooth variety $X$ projective, irreducible, and
associated line bundle $L$ on $X$. For every open subset U⊂X, let
U^(n) denote the corresponding open subset of X_n. Let
ζ∈H^1(X,T_XL), corresponding to the exact
sequence  0→L→Ω_X_2|X→Ω_X→0.

If X_n can be extended to a primitive multiple scheme X_n+1 of
multiplicity n+1, then _X,X_n, which is a line bundle on
X_n-1, can be extended to a line bundle on X_n, namely
_X,X_n+1. We have exact sequences

\begin{equation}\label{equ12}0\lra L^n\lra\ki_{X,X_{n+1}}\lra\ki_{X,X_n}\lra 0
\ ,
\end{equation}

0[r]     H^1(X,L^n-1)[r]
^1__X_n(_X,X_n,L^n)[r]^-ϕ    (L)[r]^-δ
(cf <ref>), and given any exact sequence

\[0\lra L^n\lra\ke\lra\ki_{X,X_n}\lra 0 \ , \]

corresponding to  σ∈^1__X_n(_X,X_n,L^n),
the sheaf $\ke$ is a line bundle on X_n if and only if ϕ(σ)
is an automorphism. In this case we can suppose that  ϕ(σ)=I_L .
In this way we can see that, given $(\ref{equ12})$, the extensions of
_X,X_n to a line bundle on X_n are parametrized by

More precisely, using the notations of <ref> and <ref>, let
Ł_0∈(X_n) an extension of _X,X_n. Suppose that is is
defined  by the family (α_ij), with
α_ij∈[_X(U_ij)[t]/(t^n)]^*, satisfying the relations
α_ik=α_ijδ_ij^*(α_jk), so that $L$ is defined
by the family (α_ij^(0)). Let $\L$ be an extension
of _X,X_n to a line bundle on X_n, corresponding to  η∈H^1(X,L^n-1). Then $\L$ is defined by a cocycle (θ_ij), with

\[\theta_{ij} \ = \ \alpha_{ij}+\beta_{ij}t^{n-1} \ , \]

β_ij∈_X(U_ij). The cocycle relation
θ_ik=θ_ijδ_ij^*(θ_jk)  is equivalent to

\begin{equation}\label{equ14}\frac{\beta_{ik}}{\alpha_{ik}^{(0)}} \ = \
\frac{\beta_{ij}}{\alpha_{ij}^{(0)}}+\big(\alpha_{ij}^{(0)}\big)^{n-1}
\frac{\beta_{jk}}{\alpha_{jk}^{(0)}} \ . \end{equation}

According to <ref>, this means that
(β_ij/α_ij^(0)) defines an element of
H^1(X,L^n-1), which is $\eta$.

Now we use the results of <ref>. The family
(δ_ij^*,α_ij) defines an element $\bf h$ of H^1(X,_n).
The family (δ_ij^*,θ_ij) defines an element $\bf k$
of H^1(X,_n) (corresponding to (X_n,Ł)), and we have
Δ”_n(k)=0  if and only there exists an extension X'_n+1
of X_n in multiplicity n+1 such that  _X,X'_n+1≃Ł. We
will compute Δ”_n(k-h).

We can write

\[(\delta_{ij}^*)_{|\ko_X(U_{ij})} \ = \
I_{|\ko_X(U_{ij})}+tD_{ij}+t^2\Phi_{ij} \ , \]

where D_ij is a derivation of _X(U_ij). The family (D_ij)
represents the element $\boldsymbol{\zeta}$ of H^1(X,T_XL) associated
to X_2 (cf. <ref>) in the sense of <ref> (i.e. the cocycle
relations are  D_ik=D_ij+α^(0)_ijD_jk). We have a canonical

\[H^1(X,T_X\ot L)\ot H^1(X,L^{n-1})\lra H^2(X,T_X\ot L^n)\]

Theorem:  We have  Δ”_n(k-h)=ζη .

To construct Δ”_n(k-h) we start from the exact sequence
0[r]     T_X[r]     _n+1[r]^-τ_n     _n[r]     0
(cf. [17], 2.2, [20]). We take
ψ_ij∈(_X(U_ij)[t]/(t^n))  over
(δ_ij^*,θ_ij) (such that  ψ_ji=ψ_ij^-1).
Then ψ_ijψ_jkψ_ki is of the form

\[\psi_{ij}\psi_{jk}\psi_{ki} \ = \ I+\nu_{ijk}t^n \ , \]

where ν_ijk is a derivation of _X(U_ij). The family
(ν_ijk) represents Δ”_n(k) in the sense of <ref>,
i.e. we have the cocycle relations  ν_ijk-ν_ijl+ν_ikl-
(α^(0)_ij)^nν_jkl=0 .

Similarly we take
ϕ_ij∈(_X(U_ij)[t]/(t^n))  over
(δ_ij^*,α_ij) (such that  ϕ_ji=ϕ_ij^-1).
Then ϕ_ijϕ_jkϕ_ki is of the form

\[\phi_{ij}\phi_{jk}\phi_{ki} \ = \ I+\tau_{ijk}t^n \ , \]

where τ_ijk is a derivation of _X(U_ij). The family
(τ_ijk) represents Δ”_n(h) in the sense of <ref>.

We can take

\[\psi_{ij}-\phi_{ij} \ = \ t^n\Delta_{ij} \ , \]

where, if we want that ψ_ij is a morphism of rings,
Δ_ij:_X(U_ij)[t]/(t^n+1)→_X(U_ij)  is such that there
exist a derivation E_ij of _X(U_ij) and
μ_ij∈_X(U_ij)  such that for every
u∈_X(U_ij)[t]/(t^n+1)  we have

\[\Delta_{ij}(u) \ = \ E_{ij}(u^{(0)})+u^{(1)}\mu_{ij} \ . \]

Since ψ_ij is over (δ_ij^*,θ_ij) we have
μ_ij=β_ij. It is easy to see that if
ϕ_ji=ϕ_ij^-1, the condition
ψ_ji=ψ_ij^-1  is equivalent to

\begin{equation}\label{equ16}\beta_{ji} \ = \

which is already fulfilled by $(\ref{equ14})$ and

\begin{equation}\label{equ15} E_{ji} \ = \
\frac{\beta_{ij}}{(\alpha_{ij}^{(0)})^{n+1}}D_{ij}-
\frac{1}{(\alpha_{ij}^{(0)})^n}E_{ij} \ . \end{equation}

We have then

\[\nu_{ijk}-\tau_{ijk} \ = \
\phi_{ij}\phi_{jk}(t^n\Delta_{ki}) \ . \]

We have, for every  u∈_X(U_ij)[t]/(t^n+1)

\begin{eqnarray*}
t^n\Delta_{ij}\phi_{jk}\phi_{ki}(u) & = &
\big(E_{ij}(u^{(0)})+(D_{ji}(u^{(0)})+\alpha^{(0)}_{ji}u^{(1)})\beta_{ij}
\big)t^n \ ,\\
\phi_{ij}(t^n\Delta_{jk})\phi_{ki}(u) & = &
\big(E_{jk}(u^{(0)})+(D_{ki}(u^{(0)})+
\alpha^{(0)}_{ki}u^{(1)})\beta_{jk}\big)\big(\alpha^{(0)}_{ij}\big)^nt^n \ , \\
\phi_{ij}\phi_{jk}(t^n\Delta_{ki})(u) & = & \big(\alpha^{(0)}_{ik}\big)^n\big(
E_{ki}(u^{(0)})+\beta_{ki}u^{(1)}\big)t^n \ . \\
\end{eqnarray*}

Hence we have

\begin{eqnarray*}
\nu_{ijk}(u)-\tau_{ijk}(u) & = &
\big(E_{ij}+\big(\alpha^{(0)}_{ij}\big)^nE_{jk}+
\big(\alpha^{(0)}_{ik}\big)^nE_{ki}\big)(u^{(0)})+\\
& & \big(\beta_{ij}D_{ji}+\big(\alpha^{(0)}_{ij}\big)^n\beta_{jk}D_{ki}
\big)(u^{(0)})+\\
& & \big(\alpha^{(0)}_{ji}\beta_{ij}+
\big(\alpha^{(0)}_{ij}\big)^n\alpha^{(0)}_{ki}\beta_{jk}+
\big(\alpha^{(0)}_{ik}\big)^n\beta_{ki}\big)u^{(1)} \ . \\
\end{eqnarray*}

By $(\ref{equ14})$ and $(\ref{equ16})$ we have  α^(0)_jiβ_ij+
(α^(0)_ik)^nβ_ki=0. On the other hand, by <ref>
the family ((α^(0)_ij)^nE_jk-E_ik+E_ij) is a
coboundary, and by $(\ref{equ15})$ we have E_ki=
1/(α_ik^(0))^nE_ik. Hence we can assume that

\begin{eqnarray*}\nu_{ijk} & = & \frac{\beta_{ik}}{\alpha^{(0)}_{ik}}D_{ik}
& = & \Big(\frac{\beta_{ik}}{\alpha^{(0)}_{ik}}-
\frac{\big(\alpha^{(0)}_{ij}\big)^n}{\alpha^{(0)}_{ik}}\beta_{jk}\Big)D_{ik}
+\beta_{ij}D_{ji} \ . \\ \end{eqnarray*}

But  β_ik/α^(0)_ik-
β_ij/α^(0)_ij  by $(\ref{equ14})$, and  D_ik=
D_ij+α^(0)_ijD_jk, D_ji=-1/α^(0)_ij
D_ij. Hence

\[\nu_{ijk} \ = \ \beta_{ij}D_{jk} \ = \ \alpha^{(0)}_{ij}\Big(
\frac{\beta_{ij}}{\alpha^{(0)}_{ij}}\Big)D_{jk} \ . \]

Since (β_ij/α^(0)_ij) represents $\eta$
and (D_ij) represents $\boldsymbol{\zeta}$, the result follow from

Extensions of vector bundles to higher multiplicityext_B

Suppose that X_n can be extended to a primitive multiple scheme X_n+1
of multiplicity n+1. Let $\E$ be a vector bundle on X_n, and
E=_|X. If $\E$ can be extended to a vector bundle _n+1 on
X_n+1, then we have an exact sequence

\[0\lra E\ot L^n\lra\E_{n+1}\lra\E=\E_{n+1|X_n}\lra 0 \ . \]

Obstruction to the extension of a vector bundle in
higher multiplicity – In [17], 7.1, a class
Δ()∈^2__X(E,EL^n)  is defined, such that $\E$ can be
extended to a vector bundle on X_n+1 if and only  Δ()=0.

The canonical exact sequence  0→L^n→Ω_X_n+1|X_n→Ω_X_n→0  induces σ_,X_n+1∈^1__X_n(Ω_X_n,L^n).
We have a canonical product

\[\Ext^1_{\ko_{X_n}}(\E\ot\Omega_{X_n},\E\ot L^n)\times
\Ext^1_{\ko_{X_n}}(\E,\E\ot\Omega_{X_n})\lra
\Ext^2_{\ko_{X_n}}(\E,\E\ot L^n)= \Ext^2_{\ko_X}(E,E\ot L^n) \ , \]

\begin{equation}\label{equ3}
\Delta(\E)=\ov{\sigma}_{\E,X_{n+1}}\nabla_0(\E) \ ,
\end{equation}

where ∇_0() is the canonical class of $\E$ (cf. <ref>,
[17], theorem 7.1.2).

Lemma:  We have ^1__X_n+1(,EL^n)≃(E).

Let U⊂X be a nonempty subset and U_n, U_n+1
the corresponding open subsets of X_n, X_n+1 respectively.
Suppose that $U$ is affine. It follows from <ref> that _|U_n
(resp. _U,U_n+1) can be extended to a vector bundle $\F$ (resp.
$\L$) on U_n+1. We have a canonical morphism Ł→_U_n+1:
Ł[r]     Ł_|U_n=_U,U_n+1[r]     _U_n+1  ,
inducing  α:Ł^n+1→Ł^n  and  β:Ł^n→_n+1. We
have then an obvious locally free resolution of _|U_n+1:
⋯[r]     Ł^n+1 [rr]^-I_Fα         Ł^n
[rr]^-I_Fβ         [r] [r]     _|U_n[r]     0  .
Using this resolution it follows that
^1__U_n+1(_U_n+1,E_|UL^n)≃(E_|U). It
is easy to see that these isomorphisms can be glued together to define the
isomorphism of lemma <ref>.

Other description of Δ() and of the
extensions – From <ref> and lemma <ref>, we have the exact sequence
Γ_,EL^n on X_n+1
0[r]     ^1__X(E,EL^n)[r]^-ψ
EL^n)[r]^-ϕ    (E)[r]^-δ_     ^2__X(E,EL^n)
and we have  Δ()=δ_(I_E) (cf. [17]).

Other description of the extensions – Let

\[0\lra E\ot L^n\lra\ke\to\E\lra 0\]

be an extension on X_n+1, with $\ke$ locally free, and
β∈^1__X(E,EL^n). There is an open cover (U_i)_i∈I of $X$ such that there are trivializations

\[\theta_i:\ke_{|U_i^{(n+1)}}\lra\ko_{U_i^{(n+1)}}\ot\C^r \ \]

so that $\ke$ is represented by the family (θ_ij),
Recall that  _X_n,X_n+1=L^n, viewed as a sheaf on X_n+1. Let
β∈^1__X(E,EL^n) . Then (for a suitable cover
(U_i)) $\beta$ is represented by (β_ij), with

\[\beta_{ij} \ \in \
H^0(\ko_{U_{ij}}\ot\End(\C^r)\ot L^n) \ \subset \
H^0(\ko_{U_{ij}^{(n+1)}}\ot\End(\C^r)) \ , \]

in the sense of <ref>, i.e.
the cocycle relation is
β_ik=β_ij+θ_ijβ_jkθ_ij^-1. Let

\[0\lra E\ot L^n\lra\ke'\lra\E\lra 0\]

be the extension, corresponding to σ+Ψ(β). Then '
is represented by the family ((1+β_ij)θ_ij). This description
is similar to that of <ref>.

If  Δ()=0 then $\phi$ is surjective (because
(ϕ) contains all the automorphisms of $E$).

This is also a consequence of

Lemma:  Let
0→EL^n→→0  be an extension, corresponding to
σ∈^1__X_n+1(,EL^n). Then the sheaf $\ke$ is
locally free if and only if ϕ(σ) is an isomorphism.

We need only to prove the analogous following result: let $U$ be
a nonempty opensubset of $X$, U_n, U_n+1 the corresponding
open  subsets of X_n, X_n+1 respectively, such that_U,U_n+1 can
be extended to a line bundle $\L$ on U_n+1. Let $r$ be a positive integer
and 0→L^n_|U^r→→_U_n^r→0  an exact
sequence of sheaves on U_n+1, and ϕ∈(L^n_|U^r)
the associated morphism. Then we have  ≃_U_n+1^r  if
and only $\phi$ is an isomorphism.

The result follows easily from the following fact, using the locally free
resolution of lemma <ref>: $\ke$ is isomorphic to the cokernel of the
(L^n_|U^r)⊕(_U_n+1^r)  ,
where $i$ is the canonical inclusion (cf. [7], 4.2).

It follows that when $\ke$ is locally free we can always assume that

Isomorphism classes – We have an exact
sequence  0→L^n→_X_n+1→_X_n→0, whence a connecting

\[\delta^0:H^0(X_n,\ko_{X_n})\lra H^1(X,L^n) \ . \]

Since  E^*EL^n≃L^n⊕((E)L^n), we can see
(δ^0) as a subspace of^1__X(E,EL^n).

Recall that $\E$ is called simple if the only endomorphisms of $\E$ are
the homotheties, i.e. if the canonical morphism  H^0(_X_n)→()
is an isomorphism.

Proposition:  Suppose that $\E$ is simple.

\[0\lra E\ot L^n\lra\ke\to\E\lra 0\]

be an extension on X_n+1, with $\ke$ locally free, corresponding to
σ∈^1__X_n+1(,EL^n), and

\[0\lra E\ot L^n\lra\ke'\lra\E\lra 0\]

another extension, corresponding to σ+Ψ(β),
β∈^1__X(E,EL^n). Then '≃ if and only if

We use the notations of <ref>. Let
θ'_i:'_|U_i^(n+1)→_U_i^(n+1)^r  be
trivializations, with  θ'_ij=(1+β_ij)θ_ij.

There is an automorphism  f:→'  if and only for every i∈I
there exists an automorphism f_i of _U_i^(n+1)^r such that
the following square is commutative

'_|U_i^(n+1)[r]^-θ'_i     _U_i^(n+1)^r
Suppose that $f$ exists. Then since $\E$ is simple the morphism
_|X_n=→'_|X_n=  induced by $f$ is the multiplication by
μ∈H^0(_X_n)^*. Hence we can write  f_i=μ_i.I+η_i, where
μ_i∈H^0(_X_n+1(U_i^(n+1)))  is such that
μ_i|U_i^(n)=μ  and
η_i:_U_i^(n+1)^r→_X_n,X_n+1|U_i^(n+1). Let
μ_0=μ_|X∈^*. We have on U_ij^(n+1),
θ'_i^-1f_iθ_i=θ'_j^-1f_jθ_j=f, i.e.
f_iθ_ij=θ'_ijf_j, which gives

\[\beta_{ij} \ = \
\frac{\mu_i-\mu_j}{\mu_j}+\frac{\eta_i-\theta_{ij}\eta_j\theta_{ij}^{-1}}{\mu_0}
\ .\]

By <ref>, the family (η_i-θ_ijη_jθ_ij^-1) is a
coboundary, hence we can suppose that  β_ij=
μ_i-μ_j/μ_j. Then, since  μ_i-μ_j∈H^0(U_ij^(n+1),_X_n,X_n+1), we have
μ_i-μ_j/μ_j=μ_i/μ_0-μ_j/μ_0. Hence
β=δ^0(μ/μ_0). The converse is proved in
the same way.

Canonical filtrations, quasi locally free sheaves and Serre dualityQF

Let P∈X be a closed point, z∈_Y,P an equation of $X$ and $M$
a _X_n,P-module of finite type. Let $\ke$ be a coherent sheaf on

The two canonical filtrations are useful tools to study the coherent
sheaves on primitive multiple curves.

First canonical filtration – The first canonical filtration of $M$ is

\[M_n=\nsp\subset M_{n-1}\subset\cdots\subset M_{1}\subset M_0=M\]

where for 0≤i< n, M_i+1 is the kernel of the surjective
canonical morphismM_i→M_i__n,P_X,P . So we

\[M_{i}/M_{i+1} \ = \ M_{i}\ot_{\ko_{n,P}}\ko_{X,P}, \ \ \ \
M/M_i \ \simeq \ M\ot_{\ko_{n,P}}\ko_{X_i,P}, \ \ \ \
M_i \ = \ z^iM .\]

One defines similarly the first canonical filtration of $\ke$: it is the

\[\ke_n=0\subset \ke_{n-1}\subset\cdots\subset \ke_{1}\subset \ke_0=\ke\]

such that for 0≤i< n, _i+1 is the kernel of the canonical
surjective morphism  _i→_i|X. So we have _i/_i+1=_i|X ,
/_i=_|X_i .

Second canonical filtration – One defines similarly the second canonical filtration of $M$: it is the

\[M^{(0)}=\nsp\subset M^{(1)}\subset\cdots\subset M^{(n-1)}\subset M^{(n)}=M\]

with  M^(i)  =  {u∈M ; z^iu=0} . If M_n=⊂M_n-1⊂⋯⊂M_1⊂M_0=M  is
the first canonical filtration of $M$, we have  M_i⊂M^(n-i) for 0≤i≤n.

One defines in the same way the second canonical filtration of $\ke$:

\[\ke^{(0)}=\nsp\subset \ke^{(1)}\subset\cdots\subset
\ke^{(n-1)}\subset \ke^{(n)}=\ke  .\]

Let $P$ be a closed point of $X$.
Let $M$ be a _X_n,P-module of finite type. Then $M$ is called quasi free if there exist non negative integers m_1,…,m_n and an
isomorphism  M≃⊕_i=1^nm_i_X_i,P . The integers
m_1,…,m_n are uniquely determined: it is easy to recover them from
the first canonical filtration of $M$. We say that (m_1,…,m_n) is the
type of $M$.

Let $\ke$ be a coherent sheaf on X_n. We say that $\ke$ is quasi free
at $P$ if _P is quasi free, and that $\ke$ is quasi locally free
on a nonempty open subset U⊂X if it is quasi free at every point of
$U$. If U=X we say that $\ke$ is quasi locally free. In this case the types
of the modules _x, x∈X, are the same, and for every x∈X
there exists a neighbourhood V⊂X_n of $x$ and an isomorphism

\begin{equation}\label{equ20}\ke_{|V} \ \simeq \ \sigg_{n=1}^n m_i\ko_{X_i\cap
V} \ .
\end{equation}

The proof of the following theorem is the same as in the case (X)=1
(th. 5.1.3 of [9]).

Theorem:  The following two assertions are

(i) The $\ko_{X_n,P}$-module $M$ is quasi free.

(ii) All the $M_i/M_{i+1}$ are free $\ko_{X,P}$-modules.

It follows that

Theorem:  The following two assertions are

(i) $\ke$ is quasi locally free.

(ii) All the _i/_i+1 are locally free on $X$.

For every coherent sheaf $\ke$ on X_n there exists a nonempty open subset
U⊂X such that  is quasi locally free on $U$.

Proposition:  Let $\ke$ be a quasi locally
free sheaf on X_n. Then

(i)  $\ke$ is reflexive,

(ii) for every positive integer
$i$ we have  ^i__X_n(,_X_n)=0.

(iii) for every vector bundle $\F$ on X_n and every integer $i$ we
have ^i__X_n(,)≃H^i(X_n,^∨) .

The assertion (iii) follows easily from (i), with the Ext spectral sequence.
Assertions (i) and (ii) are local, so we can replace X_n with  U×Z_n, where $U$ is an open subset of X_n. Let U_i=U×Z_i, for 1≤i≤n. The results follow from $(\ref{equ20})$ and the
free resolutions
....[r]     _U_n[r]^-×t^i[r]
_U_n[r]^-×t^n-i[r]     _U_n[r]^-×t^i
[r]     _U_i

Serre duality for quasi locally free sheaves –
Suppose that $X$ is projective of dimension $d$. It follows from
proposition <ref> that

Proposition:  For every i≥0 and every
quasi locally free sheaf $\ke$ on X_n, there is a functorial isomorphism

\[H^i(X_n,\ke) \ \simeq \ H^{d-i}(X_n,\ke^\vee\ot\omega_{X_n})^* \ . \]

Corollary:  We have
ω_X_n|X≃ω_XL^1-n .

The proof is similar to that of proposition 7.2, III, of [24].

Simple vector bundlesvb

We use the notations of <ref>.
Let $\E$ be a vector bundle on X_n and  E=_|X.
We say that $\E$ is simple if the only endomorphisms of $\E$ are the
homotheties, i.e. if the canonical map  H^0(_X_n)→()  is

Lemma: Suppose that the restriction map
H^0(_X_n)→H^0(_X_n-1)  is surjective, that
_|X_n-1 is simple on X_n-1 and  (E,EL^n-1)=H^0(X,L^n-1). Then $\E$ is simple.

Let f∈(). Then the induced morphism
_|X_n-1→_|X_n-1  is the multiplication by some
λ∈H^0(_X_n-1). Let  μ∈H^0(_X_n)  be such
that  μ_|X_n-1=λ. Then the restriction of  f-μI_X_n
to X_n-1 vanishes. It follows that  f-μI_X_n  can be
factorized in the following way
[r]     _|X=E[r]     EL^n-1=^(1)[r]
Since  (E,EL^n-1)=H^0(X,L^n-1), f-μI_X_n is the
multiplication by some η∈H^0(X,L^n-1)⊂H^0(_X_n),
and f=(μ+η)I_X_n.

It follows easily by induction on $n$ that

Corollary: Suppose that $L$ is a non trivial
ideal sheaf, or (L)<0 if $X$ is a curve, and that E=_|X is
simple. Then $\E$ is simple.

§ CANONICAL CLASS ASSOCIATED TO A VECTOR BUNDLE

Let $Z$ be a scheme over $\C$

and $E$ a vector bundle on $Z$
of rank $r$. To $E$ one associates an element ∇_0(E) of H^1(Z,EE^*Ω_Z), called the canonical class of $E$ (cf. [17],
3-). If $L$ is a line bundle on $Z$, then  ∇_0(L)∈H^1(Z,Ω_Z).
If $Z$ is smooth and projective, L=_Z(Y), where Y⊂Z is a
smooth hypersurface, then ∇_0(L) is the cohomology class of $Y$.

Let (Z_i)_i∈I be an open cover of $Z$ such that $E$ is defined by a
cocycle (θ_ij), θ_ij∈(r,_Z_ij). Then
((dθ_ij)θ_ij^-1) is a cocycle (in the sense of
<ref>) which represents ∇_0(E).

We have  EE^*≃(E)⊕_Z, so  H^1(EE^*Ω_Z)≃H^1(Z,(E)Ω_Z)⊕H^1(Z,Ω_Z). The
component of ∇_0(E) in H^1(Z,Ω_Z) is

If L_1, L_2 are line bundles on $Z$, then

\begin{equation}\label{equ21}\nabla_0(L_1\ot L_2) \ = \
\nabla_0(L_1)+\nabla_0(L_2) \ .
\end{equation}

Let $S$ be a scheme over $\C$ and s∈S a closed point. Let $Z$ be a
scheme and p_Z:Z×S→Z, p_S:Z×S→S the projections. Let
$\E$ be a vector bundle over Z×S. We have

\[\nabla_0(\E) \ \in \ H^1(Z\times S,\E\ot\E^*\ot\Omega_{Z\times S})=
H^1(Z\times S,\E\ot\E^*\ot p_Z^*(\Omega_Z))\oplus H^1(Z\times S,\E\ot\E^*\ot
p_S^*(\Omega_S)) \ . \]

By restricting to Z×{s} we get a canonical map

\[r_s:H^1(Z\times S,\E\ot\E^*\ot\Omega_{Z\times S}) \lra
H^1(Z,\E_s\ot\E^*_s\ot\Omega_Z) \ . \]

From lemma 3.2.1 of [17] we have

\begin{equation}\label{equ2}
r_s(\nabla_0(\E)) \ = \ \nabla_0(\E_s) \ .
\end{equation}

The case of smooth projective varietiescan_smo

Let $Z$ be a scheme over $\C$, $T$ a projective integral variety and
$\kl$ a line bundle on Z×T, viewed as a family of line bundles on $Z$
parametrized by $T$. For any closed point t∈T, let

Proposition:  The map

\[\xymatrix@R=5pt{\nabla_\kl:T\ar[r] & H^1(Z,\Omega_Z)\\ t\fmaps[r] &
\nabla_0(\kl_t)}\]

is constant.

Let t∈T and S⊂T an affine open neighbourhood of $t$. Then we
have H^1(Z×S,Ω_Z×S)≃_S(S)H^1(Z,ω_Z). Let

\[\nabla_0(\kl_{|Z\times S}) \ = \ \sigg_{k=1}^p f_k\ot\nabla_k \ , \]

with  f_1,…,f_p∈_S(S), ∇_1,…,∇_p∈H^1(Z,_Z). The by <ref> we have  ∇_(s)=_k=1^p
f_k(s)∇_k  for every s∈S. Hence ∇_ is a regular map
to a finite dimensional vector space. Since $T$ is projective, ∇_
must be constant.

Corollary:  If $Z$ is smooth and projective,
and  L∈(Z), then ∇_0(L) is invariant by deformation of $L$.

This follows from the fact that the connected component of
(Z) containing $L$ parametrizes a projective family of line bundles
containing $L$.

The differential morphism – Let  η∈H^1(Z,_Z), represented by a cocycle (η_ij) (with respect to an
open cover (Z_i)_i∈I of $Z$), η_ij∈_Z(Z_ij). Then
(dη_ij) is a cocycle which represents  d_Z(η)∈H^1(,Ω_Z) (which depends only on $\eta$). This defines a linear map

\[d_Z:H^1(Z,\ko_Z)\lra H^1(Z,\Omega_Z) \ . \]

The following result and its proof are classical:

Proposition: If $Z$ is smooth and projective,
then  d_Z=0 .

Let $\kl$ be a Poincaré bundle on  ^0(Z)×Z. We can compute the
canonical classes in analytic cohomology. In particular, there exist an open
cover (Z_i) of $Z$ and a neighbourhood $S$ of  _Z∈^0(Z) (for
the usual topology) such that _|Z×S is represented by a cocycle
(θ_ij), where θ_ij is an invertible analytic function on
Z_ij×S. Let  u∈H^1(Z,_Z)=T__Z(^0(Z)).
There exists an analytic map  ι:U→S (where $U$ is a neighbourhood
of 0 in $\C$), which is an embedding such that  ϕ'(0):→T__Z(^0(Z))  sends 1 to $\BS{u}$. Let

\[\alpha_{ij}^U \ = \ \alpha_{ij|Z_{ij}\times U} \ , \qquad
\alpha_{ij}^0 \ = \ \alpha_{ij|Z_{ij}\times\{0\}} \ . \]

we can write

\[\alpha_{ij}^U(x,t) \ = \ \alpha_{ij}^0(x)+t\alpha_{ij}^0(x)\beta_{ij}(x,t) \ ,
\]

where β_ij is an analytic map. Let  β_ij^0  =
β_ij|Z_ij×{0}. We have  β_ij^0 + β_jk^0
=β_ik^0  on Z_ijk, and the family (β_ij^0) is a cocycle
which represents $\BS{u}$.

A simple computation shows that we can write

\[\frac{d\alpha_{ij}^U}{\alpha_{ij}^U} \ = \
\frac{d\alpha_{ij}^0}{\alpha_{ij}^0}+td\beta_{ij}+\rho_{ij}dt+t^2\mu_{ij} \ , \]

where ρ_ij is an analytic function on  Z_ij×U  and
μ_ij∈H^0(Z_ij×U,Ω_Z_ij×U). It follows that

\[d(\BS{u}) \ = \ (\nabla_\kl\circ\iota)'(1) \ . \]

Since ∇_∘ι is constant by proposition <ref>, we have

Similarly, in the general case, let $Y$ be a scheme and $\kl$ a line bundle on
Z×Y. Let y∈Y be a closed point and  ω_y:T_yY→H^1(Z,_Z)  the Kodaïra-Spencer map. Then we have

Proposition: We have

The dualizing sheaf of a primitive double schemedua_X2

Let $X$ be a smooth, projective and irreducible variety. Let X_2 be a
primitive double scheme, with underlying smooth variety $X$, and associated
line bundle $L$ on $X$. We have a canonical exact sequence
0[r]     L[r]     _X_2[r]^-ζ    _X[r]     0
associated with  η∈H^1(X,T_XL) (cf. <ref>).
We will prove

Theorem:  The map
H^1(ζ):H^1(X_2,_X_2)→H^1(X,_X)  is surjective.

Corollary: We have
ω_X_2≃_X_2  if and only if  ω_X≃L .

If  ω_X_2≃_X_2  then  ω_X≃L  by
corollary <ref>. Conversely, suppose that  ω_X≃L. Then
ω_X_2|X≃_X. We have an exact sequence
0[r]     ω_X=Lω_X_2[r]^-ρ    _X[r]     0  .
and the associated long exact sequence
0[r]     H^0(X,ω_X)[r]
H^0(X,_X)[r]^-δ    H^1(X,L)  .
On the other hand, from the exact sequence

\[0\lra L\lra\ko_{X_2}\lra\ko_X\lra 0\]

we deduce the following
H^1(X_2,_X_2)[rr]^-H^1(ξ)         H^1(X,_X)[r]^-δ'
H^2(X,L)  .
From Serre duality on X_2 we get the commutative square
H^0(X,_X)[rr]^-δ[d]^≃        H^1(X,L)[d]^≃

H^2(X,L)^*[rr]^-^tδ'         H^1(X,_X)^*
From theorem <ref> we have δ'=0. Hence δ=0 and
H^0(ρ) is surjective. If  σ∈H^0(X_2,ω_X_2)  is such
that  H^0(ρ)(σ)=1, the corresponding morphism
_X_2→ω_X_2  is an isomorphism, so

We now prove theorem <ref>, i.e. that the connecting morphism
δ:H^1(X,_X)→H^2(X,L)  is null. We use the notations of
<ref>. Let  β∈H^1(X,_X), defined by a cocycle
(β_ij), β_ij∈_X(U_ij). We can view β_ij as
an element of H^0(_U_ij×Z_2). Then  δ(β) is
represented by the cocycle (σ_ijk), with

\[\sigma_{ijk} \ = \ \beta_{ij}+\delta_{ij}^*(\beta_{jk})-\beta_{ik}\]

(cf. <ref>). We have then

\begin{equation}\label{equ22}\sigma_{ijk} \ = \ D_{ij}(\beta_{jk}) \ .
\end{equation}

\[\Theta:H^1(X,T_X\ot L)\ot H^1(X,\Omega_X)\lra H^2(X,L)\]

be the canonical map. Then it follows from $(\ref{equ22})$ that
δ(β)=Θ(ηd_X(β)). Since  d_X(β)=0  by
proposition <ref>, we have  δ(β)=0.

§ EXTENSIONS OF FAMILIES OF VECTOR BUNDLES TO HIGHER

Let Y=X_n a primitive multiple scheme of multiplicity $n$, with
underlying smooth projective irreducible variety $X$, and associated line
bundle $L$ on $X$.

If X_n can be extended to a primitive multiple scheme X_n+1
of multiplicity n+1, we have a canonical exact sequence of sheaves on

\begin{equation}\label{equ1}\Sigma_{X_{n+1}}:\qquad
0\lra L^n\lra\Omega_{X_{n+1}|X_n}\lra\Omega_{X_n}\lra 0 \ , \end{equation}

corresponding to

Let $S$ be a smooth algebraic variety over $\C$, and
p_X_n:X_n×S→X_n, p_S:X_n×S→S the projections. We
can see X_n×S as a primitive multiple scheme of multiplicity $n$,
with induced smooth variety X×S and associated line bundle
p_X_n^*(L)_|X×S. If X_n is extended to X_n+1,
X_n+1×S as a primitive multiple scheme of multiplicity n+1
extending X_n×S. We have Ω_X_n×S=p_X_n^*(Ω_X_n)⊕p_S^*(Ω_S) and

\[\sigma_{X_{n+1}\times S} \ \in \ \Ext^1_{\ko_{X_n\times
S}}(p_{S}^*(\Omega_S),p_{X_n}^*(L^n)) \ . \]

Then Σ_X_n+1×S is p_X_n^*(Σ_X_n)⊕p_S^*(Σ), where $\Sigma$ is the exact sequence
0[r]     0[r]     Ω_S[r]     Ω_S[r]     0  .
It follows that the component of σ_X_n+1×S in
^1__X_n×S(p_S^*(Ω_S),p_X_n^*(L^n)) vanishes.

The family of extensions to X_n+1 of a vector bundle on

Let $\E$ be a vector bundle on X_n, and  E=_|X. We keep the
notations of <ref>.

Let V=^1__X_n+1(,EL^n), and =(V). Let
p:X_n+1×→X_n+1, q:X_n+1×→ be the
projections. If

\[0\lra p^*(E\ot L^n)\ot q^*(\ko_\P(1))\lra\ke\lra p^*(\E)\lra 0\]

is an exact sequence of sheaves on X_n+1×, the induced sequence on
0[r]     EL^nV^*[d][r]     p_*()[r]
[d][r]     0

p_*(p^*(EL^n)q^*(_(1)))         p_*(p^*())
is also exact. This defines a linear map

\[{\bf F}:\Ext^1_{\ko_{X_{n+1}\times\P}}(p^*(\E),p^*(E\ot L^n)\ot
q^*(\ko_\P(1)))\lra\Ext^1_{\ko_{X_{n+1}}}(\E,E\ot L^n)\ot V^*=V\ot V^* \ . \]

Proposition:  ${\bf F}$ is an isomorphism.

We prove first the surjectivity of $\bf F$. Let ϕ∈V^*,
inducing ϕ∈H^0(q^*(_(1))), σ∈V, corresponding
to the exact sequence

\[0\lra E\ot L^n\lra\ke\lra\E\lra 0 \ . \]

We have a commutative diagram
0[r]     EL^n[r][d]^Iϕ     [r][d]
[d][r]     0

0[r]     EŁ^nV^*[r]     [r]     [r]     0  ,
where the exact sequence in the bottom corresponds to σϕ∈VV^* (cf. [7], prop. 4.3.2). This diagram is the image (by p_*) of
the following commutative diagram on X_n+1×
0[r]     p^*(EL^n)[r][d]^Iϕ
p^*()[r][d]     p^*()[d][r]     0

0[r]     p^*(EL^n)q^*(_(1))[r]     p^*()[r]
p^*()[r]     0  ,
where the exact sequence in the bottom is associated to σ'∈^1__X_n+1×(p^*(),p^*(EL^n)q^*(_(1)))  such that F(σ')=σϕ. It follows
that  σϕ∈(F), and that $\bf F$ is surjective.

Now we prove that $\bf F$ is injective. For any coherent sheaves $\ke$, $\kf$
on a scheme $Y$, let Γ_, be the natural exact sequence

\[0\lra H^1(Y,\HHom(\ke,\kf))\lra\Ext^1_{\ko_Y}(\ke,\kf)\lra

which is a part of Γ_, (cf. <ref>). For
Γ_,EL^nV^* on X_n+1 we have

\[H^1(\HHom(\E,E\ot L^n\ot V^*)) \ = \ \Ext^1_{\ko_X}(E,E\ot L^n)\ot V^* \ , \]

\[H^0(\EExt^1_{\ko_{X_{n+1}}}(\E,E\ot L^n\ot V^*)) \ = \ \End(E)\ot V^* \ . \]

For Γ_p^*(),p^*(EL^n)q^*(_(1)) on X_n+1×
we have

\[\HHom(p^*(\E),p^*(E\ot L^n)\ot q^*(\ko_\P(1))) \ = \ p^*(\HHom(E,E\ot
L^n))\ot q^*(\ko_\P(1)) \ , \]

\[H^1(\HHom(p^*(\E),p^*(E\ot L^n)\ot q^*(\ko_\P(1)))) \ = \
\Ext^1_{\ko_X}(E,E\ot L^n)\ot V^* \ . \]

We have

\[\EExt^1_{\ko_{X_{n+1}\times\P}}(p^*(\E),p^*(E\ot L^n)\ot q^*(\ko_\P(1))) \ =
\ \EEnd(p^*(E))\ot q^*(\ko_\P(1)) \ , \]

\[H^0(\EExt^1_{\ko_{X_{n+1}\times\P}}(p^*(\E),p^*(E\ot L^n)\ot q^*(\ko_\P(1))))
\ = \ \End(E)\ot V^* \ . \]

Let  W=^1__X_n+1×(p^*(),p^*(EL^n)q^*(_(1))). We have then a commutative diagram (a morphism from
Γ_p^*(),p^*(EL^n)q^*(_(1)) to Γ_,EL^nV^*)
0[r]     ^1__X(E,EL^n)V^*[r][d]
W[r][d]^F     (E)V^*[d]

0[r]     ^1__X(E,EL^n)V^*[r]
^1__X_n+1(,EL^n)V^*[r]     (E)V^*

which implies easily the injectivity of $\bf F$.

The universal family of extensions of $\E$ – Let

\[\bm{\sigma}\in\Ext^1_{\ko_{X_{n+1}\times\P}}(p^*(\E),p^*(E\ot L^n)\ot
q^*(\ko_\P(1))) \ \simeq \ V\ot V^* \ , \]

corresponding to I_V, and

\[0\lra p^*(E\ot L^n)\ot q^*(\ko_\P(1))\lra\ke\lra p^*(\E)\lra 0\]

the corresponding extension on  X_n+1×. The sheaf $\ke$ is flat
on $\P$, and for every y∈, the restriction of this exact sequence to

\[0\lra E\ot L^n\ot y^*\lra\ke_y\lra \E\lra 0\]

is exact (by lemma <ref>), associated to the inclusion
i:y↪V, i∈y^*V=^1__X_n+1(,EL^ny^*).

Let  δ:^1__X_n+1(,EL^n)→(E)  be the
canonical map, and  V_=δ^-1(I_E). Then
_=_|X_n+1×V_  is a family of locally free sheaves on
X_n+1 parametrized by V_, flat on V_. Hence it is a vector
bundle on  X_n+1×V_.

Let p_X:X_n+1×S→X_n+1, p_S:X_n+1×S→S be the
projections. Let

\begin{eqnarray*}A & = & H^1(X_{n+1}\times S,\HHom(p_X^*(\E),p_X^*(E\ot
L^n))), \\
B & = & H^0(X_{n+1}\times S,\EExt^1(p_X^*(\E),p_X^*(E\ot L^n))),\\
A' & = & H^0(\ko_S)\ot\Ext^1_{\ko_X}(E,E\ot L^n),\\
B' & = & H^0(\ko_S)\ot\End(E).
\end{eqnarray*}

We have canonical isomorphisms α:A→A', β:B→B'.

Lemma:  If $S$ is affine, then we have a
commutative diagram
0[r]     A[r][d]^α    ^1__X_n+1×S(p_X^*(), p_X^*(EL^n))[r][d]^γ_S     B[d]^β

0[r]     A'[r]      H^0(_S)^1__X_n+1(, EL^n)[r]
where the bottom exact sequence is H^0(_S)Γ_,EL^n, and
γ_S is an isomorphism.

Analogous to the proof of proposition <ref>.

Proposition:  Let $\ke$ be a vector bundle on
X_n+1×S such that for every closed
point s∈S, _s|X_n≃. Let s_0∈S be a closed point.
Then there exists a neighbourhood S_0⊂S of s_0 and a morphism
f:S_0→V_  such that  f^♯(_)≃_|X_n+1×S_0.

Let  =(p_X^*(),_|X_n×S). The sheaf
p_S*() is a vector bundle on $S$. For every s∈S we have
p_S*()_s=(,_s|X_n). There exists an affine  neighbourhood
S_0 of s_0 and  ψ∈H^0(S_0,p_S*())  such that for every
s∈S_0, ψ(s) is an isomorphism. This section induces an
isomorphism ψ:p_X^*()_|X_n×S_0→_|X_n×S_0.

Let  W=(p_X^*(E),_|X×S); p_S*(W) is a vector bundle
on $S$, and for every s∈S, p_S*(W)_s=(E,_s|X). Let
ψ∈H^0(S_0,p_S*(W))  induced by $\psi$. This section induces
an isomorphism

From the canonical exact sequence
0[r]     _|X×Sp_X^*(L^n)[r]^-i     [r]^-p
_|X_n×S[r]     0
we get
0[r]     p_X^*(EL^n)_|X×S_0[rr]^-i∘ψ
p_X^*()_|X_n+1×S_0[r]     0  ,
associated to  σ∈^1__X_n+1×S_0(p_X^*()_|X_n+1×S_0, p_X^*(EL^n)_|X×S_0).
Then  f=γ_S_0(σ)  is a morphism S_0→V_  such that

Families and their extension to higher multiplicityFAM

Let $\E$ be a vector bundle on X_n×S, and  E=_|X×S.
Let p_X:X_n+1×S→X_n+1, p_S:X_n+1×S→S be the

Lemma:  Let s∈S be a closed point. Then
we have  ^1__X_n+1×S(,_X_n+1×{s})=0.

Let x∈X and W⊂X_n+1×S a neighbourhood
of (x,s) such that $\E$ is free on $W$. It suffices to prove that

\[\Tor^1_{\ko_W}(\ko_{W\cap(X_n\times S)},\ko_{W\cap(X_{n+1}\times\{s\})})
\ = \ 0 . \]

For this it suffices to show that

\[\Tor^1_{\ko_{X_{n+1}\times S}}(\ko_{X_n\times S},\ko_{X_{n+1}\times\{s\}})
\ = \ 0 \ . \]

Let U_n+1 a neighbourhood of $x$ in X_n+1, U_n the
corresponding neighbourhood in X_n, such that _X,X_n+1|U_n+1
is globally generated by t∈_X_n+1,x(U_n+1). The result follows
from the locally free resolution of _U_n×S:
_U_n+1×S[rr]^-×t         _U_n×S[r]     0   .

It follows that an extension  0→Ep_X^*(L^n)→→→0 restricts on  X_n+1×{s}  to an extension  0→E_sL^n→_s→_s→0.

Let $\E$ be a vector bundle on X_n×S, and  E=_|X×S. We

\[\nabla_0(\E) \ \in \ \Ext^1_{\ko_{X_n\times
S}}(\E,\E\ot p_{X_n}^*(\Omega_{X_n}))\oplus\Ext^1_{\ko_{X_n\times
S}}(\E,\E\ot p_S^*(\Omega_S)) \ . \]

We have  Δ()=σ_X_n+1×S∇_0(). The
component of σ_X_n+1×S in ^1_Ω_X_n×S(p_S^*(Ω_S),p_X_n^*(L^n))
vanishes. Let ∇_1 be the component of ∇_0() in ^1__X_n×S(,p_X_n^*(Ω_X_n)). We have then

\begin{equation}\label{equ4}
\Delta(\E) \ = \ \ov{\sigma}_{X_{n+1}\times S}\nabla_1 \ .
\end{equation}

Obstruction to the extension of the family – Let p_S,n, p_S,1 denote the projections  X_n×S→S,
X×S→S  respectively.
We suppose that for s∈S,

(i) $\dim(\Hom(\E_s,\E_s\ot\Omega_{X_n}))$  is independent of $s$.

(ii) $\dim(\End(E_s))$ is independent of $s$,

(iii) $\dim(\Ext^1_{\ko_X}(E_s,E_s\ot L^n))$ and
$\dim(\Ext^1_{\ko_{X_n}}(\E_s,\E_s\ot \Omega_{X_n}))$ are independent of $s$,

(iv) $\dim(\Ext^2_{\ko_X}(E_s,E_s\ot L^n))$ is independent of $s$.

It follows that  =R^1p_S,n*(^*p_X_n^*(Ω_X_n))  and
=R^2p_S,1*(E^*EL^n)  are vector bundles on $S$, with

\[\B_s=\Ext^1_{\ko_{X_n}}(\E_s,\E_s\ot\Omega_{X_n}) \ , \
\D_s=\Ext^2_{\ko_X}(E_s,E_s\ot L^n) \ . \]

Let s∈S be a closed point and

\[R_s:\Ext^2_{\ko_{X\times S}}(E,E\ot p_X^*(L^n))\lra
\Ext^2_{\ko_X}(E_s,E_s\ot L^n)\]

be the canonical map.

Proposition:  We have  R_s(Δ())=

Let  B=^1__X_n×S(,p_X_n^*(Ω_X_n)),
We have a canonical map  ϕ:B→D (multiplication by
σ_X_n+1×S), and ψ:→, where for every
s∈S, ψ_s is the canonical map

\[\Ext^1_{\ko_{X_n}}(\E_s,\E_s\ot\Omega_{X_n})\lra\Ext^2_{\ko_{X_n\times
S}}(\E_s,\E_s\ot L^n)\]

(multiplication by σ_X_n+1), and a canonical diagram
B[rr]^-Φ[d]         D[d]

H^0(S,)[rr]         H^0(S,)
where the vertical arrows come from Leray's spectral sequence. Evaluation at
$s$ gives the diagram
B[rr]^-Φ[d]         D[d]

^2__X(E_s,E_sL^n)  .
The result follows from $(\ref{equ4})$, $(\ref{equ2})$ and $(\ref{equ3})$.

Corollary:  Suppose that $S$ is affine, and
that for every closed point s∈S we have  Δ(_s)=0. Then we
have  Δ()=0.

We have, since $S$ is affine, by Leray's spectral sequence

\[\Ext^2_{\ko_{X\times S}}(E,E\ot p_X^*(L^n)) \ = \ H^0(S,\D) \ , \]

and the result follows from proposition <ref>.

In other words, if $S$ is affine, and if for every s∈S, _s can be
extended to a vector bundle on X_n+1, then $\E$ can be extended to a
vector bundle on X_n+1×S.

The universal family of extensions – We
suppose that the hypotheses of <ref> are satisfied, that $S$ is affine,
and that for every s∈S, $\Delta(\E_s)=0$. From the exact sequence of
<ref> it follows that
(^1__X_n+1(_s,E_sL^n)) is independent of $s$.

The relative Ext-sheaf (cf. [26]) =_p_S^1(,Ep_X^*(L^n))
is locally free, with, for every closed point s∈S,
_s=^1__X_n+1(_s,E_sL^n)). Let =(). Let
p:X_n+1×→X_n+1×S, q:X_n+1×→ be the
projections. If

\[0\lra p^*(E\ot p_X^*(L^n))\ot q^*(\ko_\kp(1))\lra\ke\lra p^*(\E)\lra 0\]

is an exact sequence of sheaves on X_n+1×, the induced sequence
on X_n+1×S
0[r]     EL^n^*[d][r]     p_*()[r]
[d][r]     0

p_*(p^*(Ep_X^*(L^n))q^*(_(1)))         p_*(p^*())
is also exact. This defines a linear map

\[\bm{\Phi}:\Ext^1_{\ko_{X_{n+1}\times\kp}}(p^*(\E),p^*(E\ot p_X^*(L^n))\ot
q^*(\ko_\kp(1)))\lra H^0(S,\V\ot\V^*) \ . \]

The proof of the following result is similar to that of proposition <ref>:

Proposition:  $\bm{\Phi}$ is an isomorphism.

\[\bm{\sigma}\in\Ext^1_{\ko_{X_{n+1}\times\kp}}(p^*(\E),p^*(E\ot p_X^*(L^n))\ot
q^*(\ko_\kp(1))) \ \simeq \ H^0(\V\ot\V^*) \ , \]

corresponding to I_, and

\[0\lra p^*(E\ot p_X^*(L^n))\ot q^*(\ko_\kp(1))\lra\bm{\ke}\lra p^*(\E)\lra 0\]

the corresponding extension on  X_n+1×. The sheaf $\bm{\ke}$ is
flat on $\kp$. For every s∈S, and for every y∈_s, the
restriction of this exact sequence to  X_n+1×{y}

\[0\lra E_s\ot L^n\ot y^*\lra\bm{\ke}_y\lra \E_s\lra 0\]

is exact (by lemma <ref>), associated to the inclusion

The sheaf $\V$ is associated to the presheaf $\kv$ defined by

\[\kv(U) \ = \ \Ext^1_{\ko_{X_{n+1}\times U}}(p^*(\E)_{|X_{n+1}\times U},
p^*(E\ot p_X^*(L^n))_{|X_{n+1}\times U})\]

for every open subset U⊂S. We have a canonical morphism

\[\Ext^1_{\ko_{X_{n+1}\times U}}(\E_{|X_{n+1}\times U},
(E\ot p_X^*(L^n))_{|X_{n+1}\times U})\lra\qquad\qquad\qquad\qquad \]

\[\qquad\qquad\qquad\to H^0(\EExt^1_{\ko_{X_{n+1}\times
U}}(\E_{|X_{n+1}\times U},(E\ot p_X^*(L^n))_{|X_{n+1}\times
U}))=\End(E_{|X_{n+1}\times U})\]

(coming from the Ext spectral sequence) inducing a morphism
δ:→p_*((E)). For every s∈S, δ_s is
the canonical map

\[\Ext^1_{\ko_{X_{n+1}}}(\E_s,E_s\ot L^n)\lra
H^0(\EExt^1_{\ko_{X_{n+1}}}(\E_s,E_s\ot L^n))=\End(E_s)\]

We have a canonical section $\mu$ of p_*((E)), such that
μ(s)=I_E_s for every s∈S. Let

\[\V_\E \ = \ \bm{\delta}^{-1}(\imm(\mu)) \ , \]

and _ the image of _ in $\kp$, which is a smooth subvariety of
$\kp$. The projection _→_ is an isomorphism.
The sheaf  _=_|X_n+1×_  is a vector
bundle on  X_n+1×_ (and X_n+1×_). We have a
canonical exact sequence on X_n+1×_ :

\[0\lra\pi^\sharp(E\ot p_X^*(L^n))\lra\ke_\E\lra\pi^\sharp(\E)\lra 0\]

(where $\pi$ is the projection _→S).

The case of non affine $S$ – We don't suppose as
in corollary <ref> that $S$ is affine. It is easy to see that the
schemes __|U, __|U on affine open subsets U⊂S, can be glued in an obvious way to define the schemes _, _
over $S$. Similarly the vector bundles __|__|U define the
vector bundle _ on X_n+1×_.

Proposition:  Let f:T→S be a morphism
of schemes and $\kf$ a vector bundle on X_n+1×T such that
_|X_n×T≃f^♯(). Then there exists a morphism
δ:T→_  over $S$ such that for every closed point t∈T
there exists an open neighbourhood T_0 ot $t$ such that δ^♯(_)_|X_n+1×T_0≃_|X_n+1×T_0.

Let _f=^1_f(f^♯(),f^♯(Ep_X^*(L^n)) be
the relative Ext-sheaf. By [1], theorem 1.9, for every affine open subset
U⊂S containing f(t), the canonical morphism

\[\EExt^1_{\ko_{X_{n+1}\times U}}(\E_{|X_{n+1}\times U},E_{|X_{n+1}\times U}\ot

\[\qquad\qquad\qquad\qquad\EExt^1_{\ko_{X_{n+1}\times
f^{-1}(U)}}(f^\sharp(\E_{|X_{n+1}\times U}),f^\sharp(E_{|X_{n+1}\times U}))
\]

is an isomorphism. It follows that we have a canonical isomorphism

\begin{equation}\label{equ6}\V_f \ \simeq \ f^\sharp(\V) \ .
\end{equation}

We define in an obvious
way the schemes _f^♯()≃_f^♯() analogous to
_≃_ respectively, which are closed subschemes of _f.
We have a canonical morphism of schemes  ϕ:_f→, coming from
$(\ref{equ6})$, and we have
ϕ(_f^♯())⊂V_. In the same way, we have a
canonical exact sequence

\begin{equation}\label{equ5}
p_X^*(L^n))\lra\ke_{f^\sharp(\E)}\lra{\pi'}^\sharp(\E)\lra 0
\end{equation}

on X_n+1×_f^♯() (where π' is the projection
_f^♯()→T), and
_f^♯()≃ϕ^♯(_). The canonical exact sequence
0[r]     f^♯(Ep_X^*(L^n))[r]     [r]
f^♯()[d][r]     0
_|X_n×T defines a section
σ:T→_f^♯()  of  _f^♯()→T. Let

\[\delta=\phi\circ\sigma:T\lra\kp_\E=\V_\E \ . \]

The inverse image of $(\ref{equ5})$ by $\phi$ is an exact sequence
0[r]     f^♯(Ep_X^*(L^n))[r]
σ^♯(_f^♯())[r][d]      f^♯()[r]     0  .

Over f^-1(U) this exact sequence corresponds to the same element of
f^♯(Ep_X^*(L^n))_|X_n+1×f^-1(U)), and this implies that
there exists a commutative diagram
0[r]     f^♯(Ep_X^*(L^n))_|X_n+1×f^-1(U)[d][r]     δ^♯(_))_|X_n+1×f^-1(U)[d]^ψ_U[r]     f^♯()_|X_n+1×f^-1(U)[d][r]     0

0[r]     f^♯(Ep_X^*(L^n))_|X_n+1×f^-1(U)[r]
_|X_n+1×f^-1(U)[r]     f^♯()_|X_n+1×f^-1(U)[r]     0
where ψ_U is an isomorphism. We take  T_0=f^-1(U).

The affine bundle structure of _ – Let  =R^1p_S*((,Ep_X^*(L^n))). It is a vector
bundle on $S$ (by <ref>, (ii)), and for every closed point s∈S we
have _s=^1__X(E_s,E_sL^n). For every open subset
U⊂S, we have a canonical injective map
(U)[r]     ^1__X_n+1×U(p^*()_|X_n+1×U,p^*(Ep_X^*(L^n))_|X_n+1×U)=(U)
which defines an injective morphism of vector bundles  →. It follows
that _ has a natural structure of affine bundle over $S$, with
associated vector bundle $\A$. By corollary <ref>, for every affine open
subset U⊂S, _|U has a section, hence is actually a vector
bundle isomorphic to _|U.

Since  Δ(_s)=0  for every s∈S, we have
Δ()∈(λ), where $\lambda$ is the canonical map

\[H^1(S,R^1p_{S*}(\HHom(\E,E\ot p_X^*(L^n))))\lra H^2(X_n\times S,\HHom(\E,E\ot

(cf. <ref>).
The following is a consequence of proposition <ref>:

Theorem:  We have

Families of simple vector bundlesFAM_s

We keep the notations of <ref> and <ref>. We don't suppose that $S$
is affine. To the hypotheses of <ref> we add the following: for every
s∈S, _s is simple, and for every m∈_, _,m is

We have obvious subbundles of the bundle $\A$ of <ref>:

\[\ko_S\ot\imm(\delta^0) \ \subset \ \ko_S\ot H^1(X,L^n) \ \subset \ \A
\]

(cf. <ref>).
Let  _=_/(_S(δ^0)) (cf. <ref>), and
ω:_→_ the projection, which is an affine bundle with
associated vector bundle  __(δ^0).

Invariance of _ – Let ' be a
vector bundle on  X_n×S, E'='_|X×S, and f:→'
an isomorphism, inducing obvious isomorphisms  p_f:_→_',
q_f:_→_'. Let ' be the vector bundle on $S$ analogous
to $\A$, corresponding to ', i.e. for every s∈S,
'_s=^1__X(E'_s,E'_sL^n). Then there are canonical obvious
isomorphisms  '≃, _'≃_, an they do not depend
on $f$.

Let s∈S. Then there exists an open neighbourhood S_0 of $s$ such

– $\A_{|S_0}$ is trivial and we have a subbundle
⊂_|S_0 such that  _|S_0=(_S_0(δ^0))

– there exists a section  ν:S_0→_|S_0  of

It follows that the affine bundle  ω:_|S_0→_|S_0
has a section $\tau$, so that we can view _|S_0 as a closed
subvariety of _|S_0. Let  η:_|S_0→S_0  be the

Proposition: Let m_0∈π^-1(s).
There exists an open affine neighbourhood M(m_0)⊂π^-1(S_0)
of $m$ such that

_|X_n+1×M(m_0)≃ω^*(_|X_n+1×_|S_0)_|X_n+1×M(m_0) .

\[\U \ = \ \ke_{\E|X_{n+1}\times\pi^{-1}(S_0)} \ , \quad
\V \ = \ \omega^\sharp(\ke_{\E|X_{n+1}\times\kq_{\E|S_0}}) \ . \]

For every m∈π^-1(S_0) we have _m≃_m (by proposition
<ref>), and (_m,_m)≃H^0(_X_n+1).

Let  p_0:X_n+1×π^-1(S_0)→π^-1(S_0)  be the projection.
Then  p_0*((,))  is a vector bundle. It follows that there
exists an open affine neighbourhood  M(m_0)⊂π^-1(S_0)  of $m$
such that there exists a section  σ∈H^0(M(m_0),p_0*((,)))
such that for every m∈M(m_0), σ(m)∈(_m,_m) is an
isomorphism. We obtain in this way an isomorphism  _|X_n+1×M(m_0)≃_|X_n+1×M(m_0).

Using proposition <ref> we can find finite open covers (S_i)_1≤i≤k of $S$, (P_i)_1≤i≤k of _, such that
P_i⊂π^-1(S_i), and vector bundles _i on
X_n+1×η^-1(S_i) such that  _|X_n+1×P_i≃ω^♯(_i)_|X_n+1×P_i.

Theorem:  Let f:T→S be a morphism
of schemes and $\kf$ a vector bundle on X_n+1×T such that
_|X_n×T≃f^♯(). Then there exists a unique morphism
γ_:T→_  such that for every closed point t∈T,
there exists $i$, 1≤i≤k, such that f(t)∈S_i, a neighbourhood
U⊂T of $t$ such that  γ_(U)⊂η^-1(S_i)  and _|X_n+1×U≃γ_^♯(_i)_|X_n+1×U.

This follows from propositions <ref> and <ref>. The unicity of
γ_ comes from the fact that if 1≤i,j≤k, and
q_i∈η^-1(S_i), q_j∈η^-1(S_j) are distinct, then by
proposition <ref>, the vector bundles _i,q_i and _j,q_j
on X_n+1 are not isomorphic.

Since points in __s correspond exactly to extensions of _s to
X_n+1, we have

Let f:T→S be a morphism of schemes and _1, _2 vector
bundles on X_n+1×T, and  θ_i:_i|X_n×T≃f^♯()  for i=1,2. Then the induced isomorphism
θ=θ_2^-1θ_1:_1|X_n×T→_2|X_n×T induces and isomorphism of affine bundles over $T$
__2≃__1  which is independent of θ_1,

Moduli spaces of vector bundlesmod

We use the notations of <ref> and <ref>.

Let χ_n be a set of isomorphism classes of vector bundles on X_n.
Suppose that there is a fine moduli space for χ_n, defined by a smooth
irreducible variety M_n, an open cover (M_i)_i∈I of $M$, and
for every i∈I,a vector bundle _i on X_n×M_i (cf.
We suppose that for every ∈χ_n, if E=_|X,

(i) $\ke$ and $E$ are simple,

(ii) $\dim(\Ext^1_{\ko_X}(E,E\ot L^n))$ and
$\dim(\Ext^1_{\ko_{X_n}}(\ke,\ke\ot\Omega_{X_n}))$ are independent of $\ke$,

(iii) $\dim(\Ext^2_{\ko_X}(E,E\ot L^n))$ is independent of $\ke$.

(iv) $\Delta(\ke)=0$ .

(v) every vector bundle $\kf$ on X_n+1 such that
_|X_n≃  is simple.

In particular the conditions of <ref> are verified (for the vector bundles
_i). Hence the smooth variety __i and the vector bundle
__i are defined. Let χ_n+1 the set of isomorphism
classes of vector bundles $\kf$ on X_n+1 such that

The condition (v) is satisfied if the hypotheses of lemma <ref> are
satisfied by every ∈χ_n. This is the case if $L$ is a non trivial
ideal sheaf. Condition (v) is also satisfied if χ_n consists of line

For every i∈I, we will apply theorem <ref> to _i:
we obtain an affine bundle η_i:__i→M_i, an open cover
(M_i,k)_k∈J_i of M_i, and for every k∈J_i a vector
bundle _i,k on  X_n+1×η_i^-1(M_i,k).

Finally we get a finer open cover (N_j)_j∈J of M_n, and for
every j∈J, a vector bundle _j on X_n×N_j satisfying the
conditions of <ref>). In particular, for every j,k∈J, we can find
an open cover of N_j∩N_k such that for every $U$ in this cover,
we have  _j|N_i∩N_k∩U≃_k|N_i∩N_k∩U.
Moreover, for every j∈J, we have a vector bundle _j on N_j, an
affine bundle  η_j:__j→N_j with associated vector bundle
_j, and a vector bundle _̋j on  X_n+1×__i.

From proposition <ref>, all the vector bundles _j, the affine
bundles __j can be glued to define respectively a vector bundle $\bf
A$ on M_n and an affine bundle  on M_n with
associated vector bundle $\bf A$.

Lemma: The scheme $\kq$ is separated.

According to [25], proposition 2.6.1, we need to prove the
following statement: let $Z$ be a smooth
variety, $U$, $V$ affine open subsets of $Z$ such that U∩V is affine,
$E$ (resp. $F$) a vector bundle on $U$ (resp. $V$), and  ϕ:E_|U∩V→F_|U∩V  an isomorphism. Let $Y$ be the scheme obtained by gluing
$E$ and $F$ using $\phi$. Then $Y$ is separated. But this is obvious.

Let χ_n+1 denote the set of isomorphism classes of vector bundles on
X_n+1 whose restriction to X_n belongs to χ_n. From theorem
<ref> follows easily

Theorem: The smooth variety $\kq$, the open
cover (__j) of $\kq$ and the vector bundles _̋j on
X_n+1×__j  define a fine moduli space for χ_n+1.

This proves theorem <ref> if N_n=M_n. The proof in general case is
similar, using the appropriate definition of N_n given in <ref>.

The subvariety of extensible vector bundlesextens

We use the notations of <ref>, except hypothesis (iv). We will indicate
briefly how to define the subvariety N⊂M of bundles that can be
extended to X_n+1. If this variety is smooth then one can easily modify
the preceding proofs to obtain a fine moduli space for the vector bundles on
X_n+1 whose restriction to X_n belongs to $\chi$, and this moduli
space is an affine bundle over $N$.

Let  p_X:X×M_i→X, p_M:p_X:X×M_i→M_i  be the
projections, and  E_i=_i|X×M_i. Then

\[\W_i \ = \ R^1p_{M*}(E_i^*\ot E_i\ot p_X^*(L^n))\]

is a vector bundle on M_i, and since all the vector bundles in $\chi$ are
simple, these vector bundles can be glued naturally to define a vector bundle
$\W$ on $M$. We have a canonical section of each _i

\[\xymatrix@R=5pt{\sigma_i:M_i\ar[r] & \W_i\\ m\fmaps[r] & \Delta(\E_{i,m}) .}\]

These section can be glued to define  σ∈H^0(M,). The variety $N$
is the zero subscheme scheme of $\sigma$.

§ PICARD GROUPS

We use the notations of <ref>. Let Y=X_n a primitive multiple scheme of
multiplicity n≥2, with underlying smooth projective irreducible variety
$X$, and associated line bundle $L$ on $X$.

Inductive definitionPic_univ

If  P=^d(X)  is an irreducible component of (X), let
_n,P be the set of line bundles $\kb$ on X_n such that
_|X∈P. Let P_0 be the component that contains
_X. Then  _n,P_0≠∅ (it contains
_X_n). Suppose that for every $\bf P$ such that _n,P is
not empty, there exists a fine moduli space ^P(X_n) for
_n,P, which is a smooth variety, defined by an open cover
(P_i)_i∈I of ^P(X_n), and for every i∈I, a line
bundle Ł_i on X_n×P_i.

The component ^P_0(X_n) is an algebraic group. The
multiplication with an  element of B∈_n,P defines an isomorphism

Suppose that X_n can be extended to a primitive multiple scheme X_n+1
of multiplicity n+1. From <ref>, we have a map
Δ:(X_n)→H^2(X,L^n)  such that  ∈(X_n)  can be
extended to a line bundle on X_n+1 if and only if  Δ()=0.
Suppose that some ∈_n,P can be extended to X_n+1. The
set of line bundles in _n,P that can be extended to X_n+1 is
a smooth closed subvariety

\[\Gamma^{\bf P}(X_{n+1}) \ \subset \ \Pic^{\bf P}(X_n) \ .\]

We have  Γ^P(X_n+1)=η_(Γ^P_0(X_n+1)), and Γ^P_0(X_n+1) is a subgroup of
^P_0(X_n): from <ref> and
<ref>, $(\ref{equ21})$, Γ^_X(X_n+1) is the kernel of the
morphism of groups

\[\Delta:\Pic^{{\bf P}_0}(X_n)\to H^2(X,L^n) \ , \]

(the obstruction morphism).

Let $Z$ be a scheme and Ł a line bundle on X_n+1×Z. Let
z∈Z be a closed point. Then there is an open neighbourhood $U$ of $z$,
i∈I, and a morphism  ϕ:U→P_i, such that  Ł_|X_n×U≃ϕ^♯(Ł_i).

Lemma: We have
(ϕ)⊂Γ^P(X_n+1) .

Let p_X:X×U→X, p_U:X×U→U be the
projections. As in [17], 7.1, we can associate to every line bundle
_0 on X_n×U an element Δ(_0) of H^2(X×U,p_X^*(L^n)) such that _0 can be extended to a line bundle on
X_n+1×U  if and only if  Δ(_0)=0. So we have
δ(Ł_|X_n+1×U)=0. Consider the
canonical morphism

\[\rho:H^2(X\times U,p_X^*(L^n))\lra H^0(U,R^2p_{U*}(p_X^*(L^n))) \ = \
H^0(\ko_U)\ot H^2(X,L^n) \ . \]

As in proposition <ref>, ρ(Δ(_0))=Δ∘ϕ. Hence
Δ∘ϕ=0, i.e. (ϕ)⊂Γ^P(X_n+1) .

From theorem <ref> we have

Theorem:  Suppose that _n+1,P
is nonempty. Then there exists a fine moduli space ^P(X_n+1)
for _n+1,P. It is a smooth irreducible variety, and an affine
bundle over Γ^P(X_n+1) with associated vector bundle
_Γ^P(X_n+1)(H^1(X,L^n)/(δ^0)) .

The kernel of the obstruction morphismKern_ob_m

As in <ref> we assume that we have an affine cover
(U_i)_i∈I of $X$ and trivializations δ_i:U_i^(n)→U_i×Z_n. We can write

\[(\delta_{ij}^*)_{|\ko_X(U_{ij})} \ = \
I_{|\ko_X(U_{ij})}+tD_{ij}+t^2\Phi_{ij} \ , \]

where D_ij is a derivation of _X(U_ij).

Suppose that X_n can be extended to a primitive multiple
scheme X_n+1 of multiplicity n+1. Let (δ'_ij) be a family
which defines X_n+1, δ'_ij being an automorphism of
U_ij×Z_n+1 inducing δ_ij. We have then

\[\delta'_{ij}(t) \ = \ \alpha_{ij}t \ , \]

for some α_ij∈[_X(U_ij)[t]/(t^n)]^*.

Let $\D$ be a line bundle on
X_n that can be extended to a line bundle on X_n+1. According to
<ref> this is equivalent to  Δ()=0 (cf. <ref>). Now
let ' be a line bundle on X_n such that
'_|X_n-1≃_|X_n-1. Then by formula $(\ref{equ3})$, '
can be extended to a line bundle on X_n+1 if and only if

\[\ov{\sigma}_{X_{n+1}}(\nabla_0(\D')-\nabla_0(\D)) \ = \ 0 \ . \]

Suppose that $\D$ is defined by a cocycle (θ_ij), with
θ_ij∈[_X(U_ij)(t)/(t^n)]^*, which is the set of elements
a∈_X(U_ij)(t)/(t^n) such that  a^(0)∈_X(U_ij)
is invertible. The cocycle relation is

\[\theta_{ik} \ = \ \theta_{ij}\delta_{ij}^*(\theta_{jk})\]

(cf. <ref>).
We can assume that ' is defined by a cocycle
(θ'_ij), with θ'_ij of the form

\[\theta'_{ij} \ = \ \theta_{ij}+\rho_{ij}t^{n-1} \ , \]

with  ρ_ij∈_X(U_ij). Let
β_ij=ρ_ij/θ^(0)_ij . The cocycle relation
for (θ'_ij) is equivalent to

\begin{equation}\label{equ17}\beta_{ik} \ = \
\beta_{ij}+(\alpha^{(0)}_{ij})^{n-1}\beta_{jk} \ , \end{equation}

i.e. (β_ij) represents an element $\beta$ of H^1(X,L^n-1).

The line bundles $\D$, ' are extensions to X_n of a line bundle
_n-1 on X_n-1. Let D=_n-1|X. We have canonical exact

\[0\lra D\ot L^{n-1}\lra\D\lra\D_{n-1}\lra 0 \ , \quad
0\lra D\ot L^{n-1}\lra\D'\lra\D_{n-1}\lra 0 \ , \]

corresponding to  σ,σ'∈^1__X_n(_n-1,DL^n-1)  respectively. From <ref> we have an exact sequence
0[r]     H^1(X,L^n-1)[r]^-Ψ    ^1__X_n(_n-1,DL^n-1)[r]      ,
and  σ'-σ=Ψ(β).

Now  ∇_0(')-∇_0()  is represented by the cocycle
) (cf. <ref>). We have

\[\frac{d\theta'_{ij}}{\theta'_{ij}}-\frac{d\theta_{ij}}{\theta_{ij}} \ = \
(n-1)\beta_{ij}t^{n-2}dt+t^{n-1}d\beta_{ij} \ . \]

Let  Ł=_X,X_n+1|X_2, which is a line bundle on X_2.
Recall that δ^1_Ł^n-1 denotes the connecting morphism
H^1(X,L^n-1)→H^2(X,L^n)  coming from the exact sequence  0→L^n→Ł^n-1→L^n-1→0  on X_2.

Theorem: We have σ_X_n+1(∇_0(')-∇_0())=nδ^1_Ł^n-1(β)

The line bundle Ł^n-1 is defined by the family
((α_ij^(0)+α_ij^(1)t)^n-1). We have
\[(\alpha_{ij}^{(0)}+\alpha_{ij}^{(1)}t)^{n-1} \ = \
\big(\alpha_{ij}^{(0)}\big)^{n-1}+(n-1)
\big(\alpha_{ij}^{(0)}\big)^{n-2}\alpha_{ij}^{(1)}t \ . \]

Hence from lemma <ref>, nδ^1_Ł^n-1(β) is represented by
(μ_ijk), μ_ijk∈_X(U_ijk)[t]/(t^n), with

\[\mu_{ijk} \ = \
\alpha^{(0)}_{ij}D_{ij}(\beta_{jk})\big) \ . \]

We consider the exact sequence

\[0\lra L^n\lra\Omega_{X_{n+1}|X_n}\lra\Omega_{X_n}\lra 0 \ , \]

and the associated map

\[\delta:H^1(X_n,\Omega_{X_n})\lra H^2(X,L^n) \ . \]

We construct Ω_X_n+1 as in <ref>. Let

\[A_{ij} \ = \ (n-1)\beta_{ij}t^{n-2}dt+t^{n-1}d\beta_{ij} \ . \]

Then we have

\[\ov{\sigma}_{X_{n+1}}(\nabla_0(\D')-\nabla_0(\D)) \ = \ \delta(\nabla_0(\D')
-\nabla_0(\D)) \ , \]

and it is represented by the family (ν_ijk|X_n), ν_ijk∈H^0(Ω_U_ijk×Z_n+1), where

\[\nu_{ijk} \ = \ A_{ij}-A_{ik}+{\bf d}_{ij}(A_{jk}) \ . \]

A long computation, using the equalities $(\ref{equ17})$,
D_ik=D_ij+α_ij^(0)D_jk  and

\[\alpha_{ik}^{(1)} \ = \ \alpha_{ij}^{(0)}D_{ij}(\alpha_{jk}^{(0)})+
\big(\alpha_{ij}^{(0)}\big)^2\alpha_{jk}^{(1)}+\alpha_{jk}^{(0)}\alpha_{ij}^{(1)}
\ , \]

shows that
β_jk+α^(0)_ijD_ij(β_jk))t^n-1dt, which, by lemma
<ref> and <ref>, implies theorem <ref>.

Let $\bf P$ be an irreducible component of (X). Recall that
Γ^P(X_n+1) is the smooth subvariety of ^P(X_n)
of line bundles which can be extended to X_n+1. The following result is
an easy consequence of theorem <ref>:

Proposition: Let $Z$ be the image of he
restriction morphism  Γ^P(X_n+1)→Γ^P(X_n). Then
$Z$ is smooth and Γ^P(X_n+1) is an affine bundle over $Z$ with
associated vector bundle  _Z(δ^1_Ł^n-1) .

§ PRODUCTS OF CURVES

Let $C$ and $D$ be smooth projective irreducible curves, X=C×D and
π_C:X→C, π_D:X→D  the projections. Let g_C (resp.
g_D) be the genus of $C$ (resp. $D$). Suppose that g_C≥2,

Basic propertiesbas_pro

– Let L_C (resp. L_D) be a line bundle on $C$ (resp. $D$) and
L=π_C^*(L_C)π_D^*(L_D). Then we have canonical isomorphisms

\begin{eqnarray*}
{\rm (i)} \ \ \ H^0(X,L) & \simeq & H^0(C,L_C)\ot H^0(D,L_D) ,\\
{\rm (ii)} \ \ H^1(X,L) & \simeq & \big(H^0(X,L_C)\ot H^1(D,L_D)\big)\oplus
\big(H^1(C,L_C)\ot H^0(D,L_D)\big) ,\\
{\rm (iii)} \ H^2(X,L) & \simeq & H^1(C,L_C)\ot H^1(D,L_D) .
\end{eqnarray*}

– Let M_C (resp. M_D) be a line bundle on $C$ (resp. $D$) and

Using <ref>, <ref>, and the direct sum decompositions of (ii) and
(iii), it is easy to see that the only components of the canonical map
H^1(X,L)H^1(X,M)→H^2(X,LM)  that could not vanish are the two
canonical maps

\[\big(H^1(L_C)\ot H^0(L_D)\big)\ot\big(H^0(M_C)\ot H^1(M_D)\big)\lra
H^1(L_C\ot M_C)\ot H^1(L_D\ot M_D) \ , \]

\[\big(H^0(L_C)\ot H^1(L_D)\big)\ot\big(H^1(M_C)\ot H^0(M_D)\big)\lra
H^1(L_C\ot M_C)\ot H^1(L_D\ot M_D) \ , \]

induced by

\[H^1(L_C)\ot H^0(M_C)\to H^1(L_C\ot M_C) \ , \quad
H^0(L_D)\ot H^1(M_D)\to H^1(L_D\ot M_D) \ , \]

\[H^0(L_C)\ot H^1(M_C)\to H^1(L_C\ot M_C) \ , \quad
H^1(L_D)\ot H^0(M_D)\to H^1(L_D\ot M_D)\]

– Similarly, the canonical map

\[H^1(X,\pi_C^*(L_C\ot M_C)\ot
\pi_D^*(L_D))\ot H^1(C,\pi_C^*(M_C^*))\to H^2(X,L)\]

comes from the two canonical maps

\[\big(H^0(L_C\ot M_C)\ot H^1(L_D)\big)\ot H^1(M_C^*)\lra H^1(L_C)\ot
H^1(L_D) \ , \]

\[\big(H^1(L_C\ot M_C)\ot H^0(L_D)\big)\ot\big(H^0(M_C^*)\ot
H^1(\ko_D)\big)\lra H^1(L_C)\ot H^1(L_D)\]

induced by  H^0(C,L_CM_C)H^1(C,M_C^*)→H^1(C,L_C)  and

\[H^1(L_C\ot M_C)\ot H^0(M_C^*)\lra H^1(L_C) \ , \quad
H^0(L_D)\ot H^1(\ko_D)\lra H^1(L_D)\]

– We have
Ω_X≃π_C^*(ω_C)⊕π_D^*(ω_D), hence ω_X≃π_C^*(ω_C)π_D^*(ω_D) and

\[H^1(X,\Omega_X) \ = \ \big(H^0(\omega_C)\ot H^1(\ko_D)\big)\oplus
H^1(\omega_C)\oplus\big(H^1(\ko_C)\ot H^0(\omega_D)\big)\oplus
H^1(\omega_D) \ . \]

With respect to this decomposition, the canonical class of $L$ is
∇_0(L)=∇_0(L_C)+∇_0(L_D) .

We have  T_XL=(π_C^*(T_C)L)⊕(π_D^*(T_D)L), and

\begin{equation}\label{equ19}H^1(X,T_X\ot L) \ = \ \big(H^0(T_C\ot L_C)\ot
\big(H^1(T_C\ot L_C)\ot H^0(L_D)\big)\oplus\end{equation}

\[ \qquad\qquad\qquad\qquad\quad
\big(H^0(L_C)\ot H^1(T_D\ot L_D)\big)\oplus
\big(H^1(L_C)\ot H^0(T_D\ot L_D)\big) \ . \]

Primitive double schemesprod_2

Line bundles on X_2 – We use the notations of <ref>. The non trivial primitive double schemes
with associated smooth variety $X$ and associated line bundle $L$ are
parametrized by (H^1(X,T_XL)). Let  η∈H^1(X,T_XL)
and X_2 the associated double scheme. Using the decomposition
$(\ref{equ19})$, we see that $\eta$ has four components:
η=(η_1,η_2,η_3,η_4) .

We denote by ϕ_L_C, ϕ_L_D the canonical maps  H^0(T_CL_C)H^1(ω_C)→H^1(L_C)  and H^0(T_DL_D)H^1(ω_D)→H^1(L_D)  respectively.

Let  Ψ:H^1(X,T_XL)H^1(X,Ω_X)→H^2(X,L)  be the
canonical morphism. Recall that $M$ can be extended to a line bundle on X_2
if and only if  Δ(M)=0, where
Δ(M)=Ψ(η∇_0(M)) (cf. [17], theorem 7.1.2). From
<ref>, we have

\begin{equation}\label{equ23}\Delta(M) \ = \ (\phi_{L_C}\ot
\phi_{L_D})(\eta_4\ot\nabla_0(M_D)) \ . \end{equation}

The canonical class of a line bundle on a smooth projective curve is in fact an
integer. For every line bundle $\kl$ on $C$ we have
∇_0()=()c, where  c=∇_0(_C(P)), for any P∈C.

K3 carpetsK3

Let X_2 be a primitive double scheme such that (X_2)_red=X, with
associated line bundle $L$ on $X$. We say that X_2 is a K3 carpet if
ω_X_2≃_X_2. According to corollary <ref>, X_2
is a K3 carpet if and only if  L≃ω_X.

Suppose that  L≃ω_X. We have an isomorphism
H^1(ω_C)≃ such that if $\kl$ is a line bundle of degree $d$ on
$C$, we have ∇_0()=d (and similarly on $D$). Let

\[\eta \ = \ (\eta_1,\eta_2,\eta_3,\eta_4) \ \in H^1(X,T_X\ot\omega_X)\]

(using the decomposition $(\ref{equ19})$). Here we have

\[\eta_1\in\C \ , \quad \eta_2\in H^1(C,\ko_C)\ot H^0(D,\omega_D) \ , \quad
\eta_3\in H^0(C,\omega_C)\ot H^1(D,\ko_D) \ , \quad \eta_4\in\C \ . \]

If  M=π_C^*(M_C)π_D^*(M_D),

\[\Delta(M) \ = \ \eta_1\nabla_0(M_C)+\eta_4\nabla_0(M_D) \ , \]

hence $M$ can be extended to a line bundle on X_2 if and only if

Corollary:  The scheme X_2 is projective
if and only if  η_1=η_4=0  or η_1η_4<0  and
η_1/η_4 is rational.

We have by <ref>

\[H^2(T_X\ot L^2) \ \simeq \ \big(H^1(C,\omega_C)\ot H^1(D,\omega_D^2)\big)
\oplus\big(H^1(C,\omega_C^2)\ot H^1(D,\omega_D)\big) \ = \ 0 \ . \]

Hence by <ref>, X_2 can be extended to a primitive multiple scheme of
multiplicity 3 if and only ω_X can be extended to a line bundle on
X_2. This is the case if and only  (g_C-1)η_1+(g_D-1)η_4=0, and
then X_2 is projective.

An example of extensions of double primitive schemesext_db

Let $\Theta$ be a theta characteristic on $C$ such that h^0(C,Θ)>0. We
take  L_C=Θ, L_D=ω_D. So we have

\[H^1(L) \ = \ \big(H^1(\Theta)\ot H^0(\omega_D)\big)\oplus\big(H^0(\Theta)\ot
H^1(\omega_D)\big) \ , \quad H^2(L)=H^1(\Theta)\ot H^1(\omega_D) \ . \]

In the decomposition $(\ref{equ19})$ the first summand is 0, and

\[H^1(T_X\ot L) \ = \ \big(H^1(\Theta^{-1})\ot
H^1(\ko_D)\big)\oplus H^1(\Theta) \ . \]

Let X_2 be a non trivial primitive double scheme, corresponding to η=(η_1,η_2η_3,η_4)∈(H^1(T_X×L)). We have
η_1=0, and we will also suppose that η_4=0. Such double schemes
are parametrized by

We have, from $(\ref{equ23})$, Δ(M)=0. Hence every line bundle on $X$
can be extended to a line bundle on X_2, and X_2 is projective. Such
extensions of $M$ are parametrized by an affine space, with underlying vector
space H^1(L).

Proposition:  Suppose that  η_4=0
and  η_3≠0. Then there exists an extension of X_2 to a
primitive multiple scheme X_3 of multiplicity 3, and X_3 is projective.

By theorem <ref>, to prove the existence of X_3 it
suffices to show that if

\[\chi:H^1(T_X\ot L)\ot H^1(L)\lra H^2(T_X\ot L^2)\]

is the canonical map, then  χ({η}H^1(L))=H^2(T_XL^2) . We

\begin{eqnarray*} H^2(T_X\ot L^2) & = & H^2(p_D^*(\omega_D^2))\oplus
p_D^*(\omega_D)) \ = \  H^2(p_C^*(\omega_C)\ot p_D^*(\omega_D))\\
& = & H^1(\omega_C)\ot H^1(\omega_D) \ \simeq \ \C \ ,
\end{eqnarray*}

and $\chi$ is equivalent to the canonical map

\[\chi':\big(H^0(\Theta)\ot H^1(\ko_D)\big)\ot\big(H^1(\Theta)\ot
H^0(\omega_D)\big)\lra H^1(\omega_C)\ot H^1(\omega_D) \ , \]

and we have to prove that  χ'({η_3}(H^1(Θ)H^0(ω_D))=H^1(ω_C)H^1(ω_D) .
Let (σ_i)_1≤i≤g_D be a basis of H^1(_D), and
(σ_i^*)_1≤i≤g_D the dual basis of H^0(ω_D). We can
write  η_3=_i=1^gμ_iσ_i, with  μ_i∈H^0(Θ). If 1≤j≤g_D and ρ∈H^1(Θ), we have
χ'(η_3(ρσ_j^*))=ρμ_j. Since η_3≠0, we
can choose $j$ such that μ_j≠0. We have an exact sequence on $C$
0[r]     Θ[rr]^-×μ_j         Θ^2=ω_C[r]
[r]     0  ,
where $\T$ is a torsion sheaf. It follows that
H^1(×μ_j):H^1(Θ)→H^1(ω_C)  is surjective. Since
H^2(L^2)=, every line bundle on X_2 can be extended to
X_3. This proves proposition <ref>.

Picard groupspic_b

We use the notations of <ref>.
Let M_C∈(C), M_D∈(D) be such that $(\ref{equ23})$ is
satisfied, so that π_C^*(M_C)π_D^*(M_D) can be extended to a line
bundle on X_2. Let $\bf P$ be the irreducible component of (C×D) that contains π_C^*(M_C)π_D^*(M_D). Let  ^P(X_2)⊂(X_2)  be the irreducible component of line bundles $\kl$
such that _|X∈P. From <ref>, it is an affine bundle
over $\bf P$, with associated vector bundle  _PH^1(X,L).

Let C'=C, X=C×D, and p_C:X×C'→C, p_C':X×C'→C', p_D:X×C'→D,p_X:X×C'→X, p_C×C:X×C'→C×C' the projections.

Let  n=(M_C)  and  Δ_C⊂C×C  the diagonal.
Then  Ł=p_C×C^*((nΔ_C))p_D^*(M_D)  is a family of
lines bundles on $X$ parametrized by C'. These line bundles
belong to $\bf P$, and $\L$ defines an immersion C'→^P(X).

We have

\[\Delta(\L) \ = \ \Delta(p_{C\times C}^*(\ko(n\Delta_C)))+
\Delta(p_D^*(M_D)) \ , \]

Let  q_C×C:p_C×C^*(Ω_C×C)→Ω_X×C', q_D:p_D^*(ω_D)→Ω_X×C'
be the injective canonical morphisms. We have

\[\nabla_0(p_{C\times C}^*(\ko(n\Delta_C))) \ = \ H^1(q_{C\times
\ , \quad \nabla_0(p_D^*(M_D)) \ = \ H^1(q_D)(\nabla_0(M_D)) \ . \]

We have  H^1(X×C',Ω_X×C')=A_C⊕A_C'⊕A_D, with

\begin{eqnarray*}A_{C} & = &
H^1(\ko_{C'}))\big) \ ,\\
A_{C'} & = &
H^1(\ko_{C}))\big) \ ,\\
A_{D} & = &
H^1(\ko_{C'}))\big) \ ,\\
\end{eqnarray*}

and  (H^1(q_D))  =  H^1(ω_D),

\[\imm(H^1(q_{C\times C})) \ =
H^1(\omega_{C})\oplus(H^0(\omega_{C})\ot H^1(\ko_{C'}))\oplus
H^1(\omega_{C'})\oplus(H^0(\omega_{C'})\ot H^1(\ko_{C})) \ . \]

We have  ∇_0(p_D^*(M_D))=∇_0(M_D), and by <ref>,

We see  X_2×C' as a primitive multiple scheme, with underlying
smooth scheme X×C' and associated line bundle p_X^*(L). We have

\[H^1(T_{X\times C'}\ot p_X^*(L)) \ \simeq \ H^1(T_X\ot L)\oplus
(H^0(T_X\ot L)\ot H^1(\ko_{C'}))\oplus B \ , \]

\[B \ = \ (H^1(L_{C})\ot H^0(L_D)\ot H^0(\omega_{C'}))\oplus\qquad
\qquad\qquad\qquad\quad \]

\[\qquad\qquad\qquad\qquad (H^0(L_{C})\ot H^1(L_D)\ot H^0(\omega_{C'}))\oplus
(H^0(L_{C})\ot H^0(L_D)\ot H^1(\omega_{C'})) ,\]

The element η' of H^1(T_X×C'p_X^*(L)) associated to
X_2×C' is $\eta$ (in the factor H^1(T_XL)).

We have

\[H^2(p_X^*(L)) \ = \ (H^1(L_C)\ot

\[\qquad\qquad\qquad(H^1(L_C)\ot H^0(L_D)\ot H^1(\ko_{C'}))\oplus
(H^0(L_C)\ot H^1(L_D)\ot H^1(\ko_{C'})) \ . \]

Let  β_L_C:H^0(ω_C)H^1(T_CL_C)→H^1(L_C)  be the
canonical map. As in <ref> we can compute Δ(Ł) and find that it
has the component  Δ(M)=0  in the factor  H^1(L_C)H^1(L_D),and the component  (β_L_CI_H^1(ω_C)H^0(L_D))(η_2I_H^0(ω_C))  in the factor H^1(L_C)H^0(L_D)H^1(_C'). It follows that

Theorem:  If  n(β_L_CI_H^1(ω_C)H^0(L_D))(η_2I_H^0(ω_C))≠0, then the
affine bundle ^P(X_2) is not a vector bundle over $\bf P$.

The case of K3-carpets – We have
η_1,η_4∈, η_1(M_C)+η_4(M_D)=0,η_2∈H^1(C,_C)H^0(D,ω_D). Theorem <ref> implies
that if  (M_C)≠0  and η_2≠0, then ^P(X_2) is not a vector bundle over $\bf P$.

§ MODULI SPACES OF VECTOR BUNDLES ON PRIMITIVE MULTIPLE

Let $C$ be a smooth irreducible projective curve of genus g≥2, and
Y=C_2  a ribbon over $C$, i.e. a primitive multiple curve of
multiplicity 2 such that Y_red=C. Let  L=_C,Y  be the
associated line bundle on $C$.

Construction of the moduli spacesrib

We suppose that  (L)<0.
Let $r$, $d$ be coprime integers such that r>0. Let M_C(r,d) be
the moduli space of stable sheaves on $C$ of rank $r$ and degree $d$. It is a
smooth projective variety of dimension  r^2(g-1)+1. On  M_C(r,d)×C  there is an universal bundle $E$. For every closed point m∈M_C(r,d), E_m is the stable vector bundle corresponding to $m$.

Let  p_M:M_C(r,d)×C→M_C(r,d), p_C:M_C(r,d)×C→C  be
the projections. The vector bundle of <ref>,

\[\A \ = \ R_1p_{M*}\big(\HHom(E,E\ot p_C^*(L))\big)\]

is of rank  r^2((L)+g-1). For every  m∈M_C(r,d),

The conditions of <ref> are satisfied. Hence we obtain a fine moduli space
M_Y(r,d) for the vector bundles $\E$ on $Y$ such that  _|C∈M_C(r,d). We have a canonical morphism

\[\tau_{r,d}:{\bf M}_Y(r,d)\lra M(r,d)\]

associating _|C to $\E$, and M_Y(r,d) is an affine bundle on
M(r,d) with associated vector bundle $\A$.

The same kind of construction can be applied in higher multiplicity to obtain
theorems <ref> and <ref>.

Picard groupspic_rib

We don't suppose here that (L)<0. We will use the results of <ref>.
For r=1, we have  M(r,d)=^d(C), the variety of
line bundles of degree $d$ on $C$. As on $C$ we have a decomposition of the
Picard group of $Y$:

\[\Pic(Y) \ = \ \bigcup_{d\in\Z}\Pic^{d}(Y) \ , \]

where  ^d(Y)=M_Y(1,d)  is the variety of line bundles on
$Y$ whose restriction to $C$ has degree $d$. With the notations of <ref>,
^d(Y)=^P(Y), where $\bf P$ is the set of line bundles of
degree $d$ on $C$. Consider the restriction morphism

\[\tau_{1,d}:\Pic^{d}(Y)\lra \Pic^{d}(C)\]

which is an affine bundle over ^d(C), with associated line bundle
_^d(C)H^1(C,L) .

Let  0→L→Ω_Y|C→ω_C→0  be the canonical exact
sequence, associated to σ_Y∈H^1(C,T_CL).
Let  p_Y:Y×C→Y  be the projection. Let

\[\Sigma_{Y\times C}:\qquad
0\lra p_Y^*(L)\lra\Omega_{Y\times C|C\times C}\lra\Omega_{C\times C}\lra 0 \ ,
\]

be the exact sequence $(\ref{equ1})$ corresponding to Y×C,
associated with

\[\sigma_{Y\times C} \ \in \ \Ext^1_{\ko_{C\times
C}}(p_2^*(\omega_C),p_1^*(L)) \ . \]

Then Σ_Y×C is p_1^*(Σ_Y)⊕p_2^*(Σ),
where $\Sigma$ is the exact sequence
0[r]     0[r]     ω_C[r]     ω_S[r]     0  .
It follows that the component of σ_Y×C in
^1__C×C(p_2^*(ω_C),p_1^*(L)) vanishes, and that

\[\sigma_{Y\times C} \ = \ \sigma_Y \in H^1(C,T_C\ot
L)\subset\Ext^1_{\ko_{C\times C}}(p_1^*(\omega_{C}),p_1^*(L))\]

(cf. $(\ref{equ10})$).

According to theorem <ref> it is equivalent to say that ^d(Y)
is not a vector bundle over ^d(C), and that a Poincaré
bundle $\D$ on  C×^d(C)  cannot be extended to a line
bundle of  Y×^d(C). To see this we can suppose that d=-1.
We have then an embedding

\[\xymatrix@R=5pt{C\flinc[r] & \Pic^{-1}(C)\\ P\fmaps[r] & \ko_C(-P) \ .} \]

We will show that with suitable hypotheses, _|C×C cannot be
extended to Y×C.
Let  p_1:C×C→C (resp. p_2:C×C→C) be the
first (resp. second) projection. It suffices to prove that for some
N∈(C), _|C×Cp_2^*(N) cannot be extended to
Y×C. So we can assume that  _|C×C=_C×C(-Γ), where  Γ⊂C×C  is the diagonal.

We see Y×C as a primitive multiple scheme, with associated line
bundle p_1^*(L) on C×C. We will show that
Δ(_C×C(-Γ))∈H^2(C×C,p_1^*(L))  is non zero.
This implies that _C×C(-Γ) cannot be extended to Y×C (cf. <ref>).

We have

\[\Delta(\ko_{C\times C}(-\Gamma)) \ \in \ H^2(C\times
C,p_1^*(L)) \ = \ L(H^0(C,\omega_C),H^1(C,L)) \ , \]

and  Δ(_C×C(-Γ))=σ_Y×C.∇_0(_C×C(-Γ)) (from [17], theorem 7.1.2).
It follows easily from <ref> that Δ(_C×C(-Γ))
is the image of σ_Y by the canonical map

\[\eta:H^1(C,T_C\ot L)\lra L(H^0(C,\omega_C),H^1(C,L)) \ . \]

It follows that if  η(σ_Y)≠0, then ^d(Y) is not a
vector bundle over ^d(C).

We will prove

Theorem :
^d(Y) is not a vector bundle over ^d(C) if either $C$
is not hyperelliptic and  (L)≤2-2g, or L=ω_C.

The duality morphismcoh_prod

We will use the results of <ref>.

Let $Z$ be smooth irreducible projective curve. Let  p_Z:Z×C→Z
and  p_C:Z×C→C  be the projections. Let $N$ be a vector bundle
on $C$. Then we have (from <ref>)

\begin{equation}\label{equ10}
H^1(Z\times C, p_C^*(N)) \ \simeq \ H^1(C,N)\oplus\big(H^0(C,N)\ot
H^1(Z,\ko_Z)\big) \ .
\end{equation}

We have also  H^2(Z×C,p_C^*(N))≃H^1(C,N)H^1(Z,_Z) .

Canonical bundles – We have
Ω_Z×C=p_Z^*(ω_Z)⊕p_C^*(ω_C), hence

\begin{equation}\label{equ11}H^1(Z\times C,\Omega_{Z\times C}) \ \simeq \

\[ \qquad H^1(C,\omega_C)\oplus\big(H^0(Z,\ko_Z)\ot

\[ \quad\quad\qquad\qquad\quad\qquad = \ \C\oplus
\oplus\C\oplus L(H^0(Z,\omega_Z),H^0(C,\omega_C)) \ . \]

We have  H^2(Z×C,Ω_Z×C)≃H^1(Z,ω_Z)H^1(C,ω_C)= .

Lemma: The duality morphism

\[H^1(Z\times C,\Omega_{Z\times C})\times H^1(Z\times C,\Omega_{Z\times C})\lra
H^2(Z\times C,\omega_{Z\times C})=\C\]

α_1a_2-α_2a_1+tr(f_2ϕ_1)-tr(f_1ϕ_2)  .

This follows from <ref>. For the minus signs, see <ref>.

Proof of theorem <ref>th4_pro

The canonical class of _C×C(-Γ) – We
have  ∇_0(_C×C(-Γ))=-cl_Γ, where cl_Γ is the cohomology class of $\Gamma$. It is the element of
H^1(C×C,Ω_C×C) corresponding to the canonical linear form

\[\tau:H^1(C\times C,\Omega_{C\times C})\lra

using the duality described in <ref>. For every

\[(a_1,f_1,a_2,f_2) \ \in \
\C\oplus\End(H^0(C,\omega_C))\oplus\C\oplus\End(H^0(C,\omega_C)) \ . \]

we have

\[\tau(a_1,f_1,a_2,f_2) \ = \ a_1+a_2+tr(f_1)+tr(f_2) \ . \]

It follows that

\[{\rm cl}_\Gamma \ = \ (1,I_{H^0(C,\omega_C)},-1,-I_{H^0(C,\omega_C)}) \ . \]

Proof of theorem <ref> – The case  L=ω_C
is obvious. So we suppose that $C$ is not hyperelliptic and  (L)≤2-2g. The transpose of the evaluation morphism _CH^0(C,ω_C)→ω_C  is injective and its cokernel $F$ is
stable (from [27], corollary 3.5). From the exact sequence

\[0\lra T_C\ot L\lra L\ot H^0(C,\omega_C)^*\lra F\ot L\lra 0 \ , \]

we have an exact sequence
0[r]     H^0(C,FL)[r]     H^1(C,T_CL)[r]^-η
L(H^0(C,ω_C),H^1(C,L))  .
Since  (FL)≤0  and FL is stable, we have
H^0(C,F^*L)=, and $\eta$ is injective. Since  σ_Y≠0
($Y$ is not trivial), we have  Δ(_C×C(-Γ))≠0.
This proves theorem <ref>.

Moduli spaces of vector bundles of rank r≥2 and degree

Theorem : Suppose that $C$ is not
hyperelliptic, $Y$ not trivial and  (L)≤2-2g.
Then M_Y(r,-1) is not a vector bundle over M(r,-1).

Let $E$ be a stable vector bundle on $C$ of rank $r$ and degree $0$ and x∈C. Then for every non zero linear form  φ:E_x→, the kernel of
the induced morphism φ̃:E→_x is a stable vector bundle:
let F⊂(φ̃) be a subbundle such that
0<(F)<r; then since F⊂E and $E$ is stable, we have
(F)<0. Hence  (F)/(F)<-1/r.

Let =(E^*) and π:→C, p_:×C→C,
p_C:×C→C the projections. Let Δ⊂C×C  be the diagonal. We have a canonical obvious surjective morphism on
sheaves on ×C

\[{\bf p}:p_C^*(E)\ot p_\P^*(\ko_\P(-1))\lra\ko_{(\pi\times p_C)^{-1}(\Delta)}
\ . \]

For every x∈C and non zero φ∈E_x^*, p_φ
is the morphism Eφ→_x induced by $\varphi$. The kernel of
$\bf p$ is a vector bundle $\E$ on ×C that is a family of stable
vector bundles on $C$, of rank $r$ and degree -1, parametrized by $\P$.

Let $\ke$ be a universal bundle on M(r,-1)×C.
Since r≥2, E^* has a rank 1 subbundle $D$, inducing a section $s$
of $\pi$. Then =s^*() is a family of stable vector bundles on $C$, of
rank $r$ and degree -1, parametrized by $C$. Hence there exists a morphism
τ:C→M(r,-1) and a line bundle $H$ on $C$ such that
≃(τ×I_C)^*()q^*(H) (where q:C×C→C is
the first projection).

We have a canonical immersion  i:C→^-1(C), sending $x$ to
_C(-x)(E). Let  det:M(r,-1)→^-1(C)  be
the determinant morphism. Then  det∘τ  is the identity
morphism (from $C$ to i(C)=C). It follows that $\tau$ is an immersion.

Now suppose that M_Y(r,-1) is a vector bundle over M(r,-1). By
theorem <ref>, $\ke$ can be extended to a vector bundle on  M_Y(r,-1)×Y, hence _|τ(C)×C can be extended to a
vector bundle on τ(C)×Y. Hence (_|τ(C)×C)
can be extended to a line bundle on i(C)×C. But this is impossible by
the proof of theorem <ref>.

Picard groups of primitive multiple schemes of multiplicity 3geq0

Let C_3 be an extension of C_2 of multiplicity 3. We use the notations
of <ref>. Let  P=^d(C). Every line bundle on C_2 can be
extended to C_3, since the obstructions lie in $H^2$ of vector bundles on
$C$. Hence we have  Γ^P(C_3)=^d(C_2).

Let ^d(C_3) denote the component of (C_3) of line bundles
whose restriction to $C$ is of degree $d$. From <ref> we see that
^d(C_3) is a smooth irreducible variety, and an affine
bundle over ^d(C_2) with associated vector bundle
_^d(C_2)(H^1(C,L^2)/(δ^0)) , where
δ^0:H^0(_C_2)→H^1(C,L^2)  is the map coming from the exact
sequence 0→L^2→_C_3→_C_2→0 .

We have an exact sequence

\[0\lra H^0(C,L)\lra H^0(\ko_{C_2})\lra H^0(\ko_C)=\C\lra 0 \ , \]

so that (δ^0) is the same as the image of the restriction of
δ^0 to H^0(C,L).

Now suppose that we have another extension C'_3 of C_2 to multiplicity
$3$. Then _C,C_3 and _C,C'_3 are two extensions of $L$ to a
line bundle on C_2, corresponding to
σ,σ'∈^1__C_2(_C,L)  respectively. We have also
two maps  δ^0,δ'^0:H^0(C,L)→H^1(C,L^2).

According to
<ref>, we have  σ'-σ=Ψ(μ), for some  μ∈H^1(C,L). Let χ:H^0(C,L)H^1(C,L)→H^1(C,L^2)  be the
canonical product.

From lemma <ref> follows easily

Proposition:  We have, for every  η∈H^0(C,L), (δ'^0-δ^0)(η)=χ(ημ).

This implies that it is possible that (δ^0)≠0. If C_2 is
the trivial ribbon (cf. <ref>), then δ^0=0, so we obtain a
formula for δ'^0.

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