Schottky via the punctual Hilbert scheme

TLDR: In this article, a smooth projective curve of genus G$ can be reconstructed from its polarized Jacobian as a certain locus in the Hilbert scheme, defined by geometric conditions in terms of the polarization of the Jacobian.

Abstract: We show that a smooth projective curve of genus $g$ can be reconstructed from its polarized Jacobian $(X, \Theta)$ as a certain locus in the Hilbert scheme $\mathrm{Hilb}^d(X)$, for $d=3$ and for $d=g+2$, defined by geometric conditions in terms of the polarization $\Theta$. The result is an application of the Gunning-Welters trisecant criterion and the Castelnuovo-Schottky theorem by Pareschi-Popa and Grushevsky, and its scheme theoretic extension by the authors.

Full text

SCHOTTKY VIA THE PUNCTUAL HILBERT SCHEME
MARTIN G. GULBRANDSEN AND MARTÍ LAHOZ
Abstract.
We show that a smooth projective curve of genus
g
can be reconstructed from its
polarized Jacobian
(X, Θ)
as a certain locus in the Hilbert scheme
Hilb
d
(X)
, for
d = 3
and for
d = g + 2
, dened by geometric conditions in terms of the polarization
Θ
. The result is an
application of the GunningWelters trisecant criterion and the CastelnuovoSchottky theorem
by PareschiPopa and Grushevsky, and its scheme theoretic extension by the authors.
1.
Introduction
Let
(X, Θ)
be an indecomposable principally polarized abelian variety (
ppav
) of dimension
g
over an algebraically closed eld
k
of characteristic dierent from
2
. The polarization
Θ
is
considered as a divisor class under algebraic equivalence, but for notational convenience, we shall
x a representative
Θ ⊂ X
.
(X, Θ)
being indecomposable means that
Θ
is irreducible.
The geometric Schottky problem asks for geometric conditions on
(X, Θ)
which determine
whether it is isomorphic, as a
ppav
, to the Jacobian of a nonsingular genus
g
curve
C
. The
Torelli theorem then guarantees the uniqueness of the curve
C
up to isomorphism. One may
ask for a constructive version: can you write down the curve
C
, starting from
(X, Θ)
? Even
though one may embed
C
in its Jacobian
X
, there is no canonical choice of such an embedding,
so one cannot reconstruct
C
as a curve in
X
without making some choices along the way. We
refer to Mumford's classic [11] for various approaches and answers to the Schottky and Torelli
problems, and also to Arbarello [1], Beauville [2] and Debarre [3] for more recent results.
In this note, we show that any curve
C
sits naturally inside the punctual Hilbert scheme
of its Jacobian
X
. We give two versions: rstly, using the GunningWelters criterion [7, 14],
characterizing Jacobians by having many trisecants, we reconstruct
C
as a locus in
Hilb
3
(X)
.
Secondly, using the CastelnuovoSchottky theorem, quoted below, we reconstruct
C
as a locus
in
Hilb
g+2
(X)
. In fact, for any indecomposable
ppav
(X, Θ)
, we dene a certain locus in the
Hilbert scheme
Hilb
d
(X)
for
d ≥ 3
, and show that this locus is either empty, or one or two copies
of a curve
C
, according to whether
(X, Θ)
is not a Jacobian, or the Jacobian of the hyperelliptic
or nonhyperelliptic curve
C
. Then we characterize the locus in question for
d = 3
in terms of
trisecants, and for
d = g + 2
in terms of being in special position with respect to
2Θ
-translates.
2010
Mathematics Subject Classication.
Primary 14H42; Secondary 14C05.
Both authors are partially supported through the Research Council of Norway grant
Sheaves on abelian
varieties
, grant number 230986. M.L. is partially supported by MTM2012-38122-C03-02.
1

2 M. G. GULBRANDSEN AND M. LAHOZ
To state the results precisely, we introduce some notation. For any subscheme
V ⊂ X
, we
shall write
V
x
⊂ X
for the translate
V − x
by
x ∈ X
. Let
ψ : X → P
2
g
−1
be the (Kummer) map
given by the linear system
|2Θ|
.
Theorem A.
Let
Y ⊂ Hilb
3
(X)
be the subset consisting of all subschemes
Γ ⊂ X
with support
{0}
, with the property that
{x ∈ X Γ
x
⊂ ψ
−1
()
for some line
 ⊂ P
2
g
−1
}
has positive dimension. Then
Y
is closed and
(1)
if
X
is not a Jacobian, then
Y = ∅
;
(2)
if
X
∼
=
Jac(C)
for a hyperelliptic curve
C
, then
Y
is isomorphic to the curve
C
;
(3)
if
X
∼
=
Jac(C)
for a non-hyperelliptic curve
C
, then
Y
is isomorphic to a disjoint union
of two copies of
C
.
The proof is by reduction to the GunningWelters criterion; more precisely to the character-
ization of Jacobians by inectional trisecants. Note that the criterion dening
Y
only depends
on the algebraic equivalence class of
Θ
, and not the chosen divisor.
For the second version, we need some further terminology from [12] and [6].
Denition 1.1.
A nite subscheme
Γ ⊂ X
of degree at least
g + 1
is
theta-general
if, for all
subschemes
Γ
d
⊂ Γ
d+1
in
Γ
of degree
d
and
d + 1
respectively, with
d ≤ g
, there exists
x ∈ X
such that the translate
Θ
x
contains
Γ
d
, but not
Γ
d+1
.
Denition 1.2.
A nite subscheme
Γ ⊂ X
is
in special position
with respect to
2Θ
-translates
if the codimension of
H
0
(X, I
Γ
(2Θ
x
))
in
H
0
(O
X
(2Θ
x
))
is smaller than
deg Γ
for all
x ∈ X
.
Again note that these conditions depend only on the algebraic equivalence class of
Θ
. The term
special position makes most sense for
Γ
of small degree, at least not exceeding
dim H
0
(O
X
(2Θ
x
)) =
2
g
.
Our second version reads:
Theorem B.
Let
Y ⊂ Hilb
g+2
(X)
be the subset consisting of all subschemes
Γ ⊂ X
with
support
{0}
, which are theta-general and in special position with respect to
2Θ
-translates. Then
Y
is locally closed, and
(1)
if
X
is not a Jacobian, then
Y = ∅
;
(2)
if
X
∼
=
Jac(C)
for a hyperelliptic curve
C
, then
Y
is isomorphic to the curve
C
minus
its Weierstraÿ points;
(3)
if
X
∼
=
Jac(C)
for a non-hyperelliptic curve
C
, then
Y
is isomorphic to a disjoint union
of two copies of
C
minus its Weierstraÿ points.
The proof of Theorem B is by reduction to the CastelnuovoSchottky theorem, which is the
following:

SCHOTTKY VIA THE PUNCTUAL HILBERT SCHEME 3
Theorem 1.3.
Let
Γ ⊂ X
be a nite subscheme of degree
g + 2
, in special position with respect
to
2Θ
-translates, but theta-general. Then there exist a nonsingular curve
C
and an isomor-
phism
Jac(C)
∼
=
X
of
ppav
s, such that
Γ
is contained in the image of
C
under an AbelJacobi
embedding.
Here, an
AbelJacobi embedding
means a map
C → Jac(C)
of the form
p 7→ p − p
0
for some
chosen base point
p
0
∈ C
. This theorem, for reduced
Γ
, is due to PareschiPopa [12] and, under
a dierent genericity hypothesis, Grushevsky [4, 5]. The scheme theoretic extension stated above
is by the authors [6]. The scheme theoretic generality is clearly essential for the application in
Theorem B.
We point out that the GunningWelters criterion is again the fundamental result that under-
pins Theorem 1.3, and thus Theorem B. More recently, Krichever [9] showed that Jacobians are
in fact characterized by the presence of a single trisecant (as opposed to a positive dimensional
family of translations), but we are not making use of this result.
2.
Subschemes of AbelJacobi curves
For each integer
d ≥ 1
, let
Y
d
⊂ Hilb
d
(X)
be the closed subset consisting of all degree
d
subschemes
Γ ⊂ X
such that
(i) the support of
Γ
is the origin
0 ∈ X
,
(ii) there exists a smooth curve
C ⊂ X
containing
Γ
, such that the induced map
Jac(C) → X
is an isomorphism of
ppav
's.
We give
Y
d
the induced reduced scheme structure.
We shall now prove analogues of (1), (2) and (3) in Theorems A and B for
Y
d
with
d ≥ 3
:
Proposition 2.1.
With
Y
d
⊂ Hilb
d
(X)
as dened above, we have:
(1)
If
X
is not a Jacobian, then
Y
d
= ∅
.
(2)
If
X
∼
=
Jac(C)
for a hyperelliptic curve
C
, then
Y
d
is isomorphic to the curve
C
.
(3)
If
X
∼
=
Jac(C)
for a non-hyperelliptic curve
C
, then
Y
d
is isomorphic to a disjoint union
of two copies of
C
.
As preparation for the proof, consider a Jacobian
X = Jac(C)
for some smooth curve
C
of
genus
g
. It is convenient to x an AbelJacobi embedding
C → X
; any other curve
C
0
⊂ X
for
which
Jac(C
0
) → X
is an isomorphism is of the form
±C
x
for some
x ∈ X
. Such a curve
±C
x
contains the origin
0 ∈ X
if and only if
x ∈ C
. Hence
Y
d
is the image of the map
φ = φ
+
`
φ
−
: C
`
C → Hilb
d
(X)
that sends
x ∈ C
to the unique degree
d
subscheme
Γ ⊂ ±C
x
supported at
0
, with the positive
sign on the rst copy of
C
and the negative sign on the second copy.

4 M. G. GULBRANDSEN AND M. LAHOZ
More precisely,
φ
is dened as a morphism of schemes as follows. Let
m: X × X → X
denote
the group law, and consider
m
−1
(C) ∩ (C × X)
as a family over
C
via rst projection. The bre over
p ∈ C
is
C
p
. Let
N
d
= V (m
d
0
)
be the
d − 1
'st order innitesimal neighbourhood of the origin in
X
. Then
Z = m
−1
(C) ∩ (C × N
d
) ⊂ C × X
is a
C
-at family of degree
d
subschemes in
X
; its bre over
p ∈ C
is
C
p
∩N
d
. This family denes
φ
+
: C → Hilb
d
(X)
, and we let
φ
−
= −φ
+
(where the minus sign denotes the automorphism of
Hilb
d
(X)
induced by the group inverse in
X
).
Lemma 2.2.
The map
φ
+
: C → Hilb
d
(X)
is a closed embedding for
d > 2
.
In the proof of the Lemma, we shall make use of the dierence map
δ : C × C → X
, sending
a pair
(p, q)
to the degree zero divisor
p − q
. We let
C − C ⊂ X
denote its image. If
C
is
hyperelliptic, we may and will choose the AbelJacobi embedding
C ⊂ X
such that the involution
−1
on
X
restricts to the hyperelliptic involution
ι
on
C
. Thus, when
C
is hyperelliptic,
C − C
coincides with the distinguished surface
W
2
, and the dierence map
δ
can be factored via the
symmetric product
C
(2)
:
C × C
1×ι
∼
=
-
C × C
C
(2)
?
-
X
δ
?
We note that the double cover
C × C → C
(2)
, that sends an ordered pair to the corresponding
unordered pair, is branched along the diagonal, so that via
1 × ι
, the branching divisor becomes
the antidiagonal
(1, ι): C → C × C
.
As is well known, the surface
C − C
is singular at
0
, and nonsingular everywhere else. The
blowup of
C − C
at
0
coincides with
δ : C × C → C − C
when
C
is nonhyperelliptic, and with
the addition map
C
(2)
→ W
2
when
C
is hyperelliptic.
Proof of Lemma 2.2.
To prove that
φ
+
is a closed embedding, we need to show that its restriction
to any nite subscheme
T ⊂ C
of degree
2
is nonconstant, i.e. that the family
Z|
T
is not a product
T × Γ
. For this it suces to prove that if
Γ
is a nite scheme such that
(1)
m
−1
(C) ⊃ T × Γ,
then the degree of
Γ
is at most
2
.

SCHOTTKY VIA THE PUNCTUAL HILBERT SCHEME 5
Consider the following commutative diagram:
(2)
X × X
(m,pr
2
)
∼
=
-
X × X
m
−1
(C) ∩ (X × C)
∪
6
∼
=
-
C × C
∪
6
X
δ
?
pr
1
-
First suppose
T = {p, q}
with
p 6= q
. The claim is then simply that
C
p
∩ C
q
, or equivalently
its translate
C ∩ C
q−p
, is at most a nite scheme of degree
2
. Diagram (2) identies the bre
δ
−1
(q − p)
on the right with precisely
C ∩ C
q−p
on the left. But
δ
−1
(q − p)
is a point when
C
is
nonhyperelliptic, and two points if
C
is hyperelliptic.
Next suppose
T ⊂ C
is a nonreduced degree
2
subscheme supported in
p
. Assuming
Γ
satises
(1), we have
Γ ⊂ C
p
, so
m
−1
(C) ∩ (X × C
p
) ⊃ T × Γ
or equivalently
m
−1
(C) ∩ (X × C) ⊃ T
p
× Γ
−p
.
We have
T
p
⊂ C − C
, and Diagram (2) identies
δ
−1
(T
p
)
on the right with
m
−1
(C) ∩ (T
p
× C)
on the left.
Suppose
C
is nonhyperelliptic. Then
δ
is the blowup of
0 ∈ C −C
, and
δ
−1
(T
p
)
is the diagonal
∆
C
⊂ C × C
together with an embedded point of multiplicity
1
(corresponding to the tangent
direction of
T
p
⊂ C −C
). Diagram (2) identies the diagonal in
C ×C
on the right with
{0}× C
on the left. Thus
m
−1
(C) ∩(T
p
× C)
is
{0} × C ⊂ X × C
with an embedded point. Equivalently,
m
−1
(C) ∩ (T × C
p
)
is
{p} × C
p
with an embedded point, say at
(p, q)
. This contains no constant
family
T × Γ
except for
Γ = {q}
, so
Γ
has at most degree
1
.
Next suppose
C
is hyperelliptic. We claim that
δ
−1
(T
p
)
is the diagonal
∆
C
⊂ C × C
with
either two embedded points of multiplicity
1
, or one embedded point of multiplicity
2
. As in
the previous case, this implies that
m
−1
(C) ∩ (T × C
p
)
is
{p} × C
p
with two embedded points of
multiplicity
1
or one embedded point of multiplicity
2
, and the maximal constant family
T × Γ
it contains has
Γ
of degree
2
. It remains to prove that
δ
−1
(T
p
)
is as claimed.
We have
W
2
= C − C
, and the blowup at
0
is
C
(2)
→ W
2
= C − C
. The preimage of
T
p
is
the curve
(1 + ι): C → C
(2)
, together with an embedded point of multiplicity
1
, say supported
at
q + ι(q)
. Now the two to one cover
C × C → C
(2)
is branched along the diagonal
2C ⊂ C
(2)
,
If
q 6= ι(q)
, then the preimage in
C × C
is just
(1, ι): C → C × C
, together with two embedded
points of multiplicity
1
, supported at
(q, ι(q))
and
(ι(q), q)
. If
q = ι(q)
, i.e.
q
is Weierstraÿ,
then we claim the preimage in
C × C
is
(1, ι): C → C × C
together with an embedded point of
multiplicity
2
. This follows once we know that the curves
2C
and
(1 + ι)(C)
in
C
(2)
intersect

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Schottky via the punctual hilbert scheme" ?

The authors show that a smooth projective curve of genus g can be reconstructed from its polarized Jacobian ( X, Θ ) as a certain locus in the Hilbert scheme Hilb ( X ), for d = 3 and for d = g + 2, de ned by geometric conditions in terms of the polarization Θ. The result is an application of the Gunning Welters trisecant criterion and the Castelnuovo Schottky theorem by Pareschi Popa and Grushevsky, and its scheme theoretic extension by the authors. 

Q2. What is the simplest way to reconstruct a curve?

The geometric Schottky problem asks for geometric conditions on (X,Θ) which determinewhether it is isomorphic, as a ppav, to the Jacobian of a nonsingular genus g curve C. TheTorelli theorem then guarantees the uniqueness of the curve C up to isomorphism. 

Q3. What is the proof of the Lemma?

In the proof of the Lemma, the authors shall make use of the di erence map δ : C × C → X, sending a pair (p, q) to the degree zero divisor p − q. 

Q4. What is the criterion for a nite subscheme?

Γd ⊂ Γd+1 in Γ of degree d and d + 1 respectively, with d ≤ g, there exists x ∈ X such that the translate Θx contains Γd, but not Γd+1.De nition 1.2. 

Q5. What is the simplest way to prove that + is a closed embedding?

To prove that φ+ is a closed embedding, the authors need to show that its restriction to any nite subscheme T ⊂ C of degree 2 is nonconstant, i.e. that the family Z|T is not a product T × Γ. 

Q6. What is the argument for a hyperelliptic curve?

If C is hyperelliptic with hyperelliptic involution ι, however, the authors nd that φ+ factors through C/ι ∼= P1, and Y2 ∼= P1, and the authors cannot reconstruct C from Y2 alone. 

Q7. What is the morphism of C C?

The blowup of C − C at 0 coincides with δ : C × C → C − C when C is nonhyperelliptic, and with the addition map C(2) →W2 when C is hyperelliptic. 

Q8. What is the bre over p C?

The bre over p ∈ C is Cp. Let Nd = V (md0) be the d− 1'st order in nitesimal neighbourhood of the origin in X. ThenZ = m−1(C) ∩ (C ×Nd) ⊂ C ×Xis a C- at family of degree d subschemes in X; its bre over p ∈ C is Cp∩Nd. 

Q9. What is the preimage of C C?

If q = ι(q), i.e. q is Weierstraÿ, then the authors claim the preimage in C ×C is (1, ι) : C → C ×C together with an embedded point of multiplicity 2. 

Q10. What is the criterion for Y in Proposition 2.1?

The authors write F̂ for the Fourier Mukai transform [10, 8] of a WIT-sheaf F on X [10, Def. 2.3]:F̂ is a sheaf on the dual abelian variety, which the authors will identify with X using the principalpolarization. 

Q11. What is the simplest way to prove that + is a nonreduced?

Next suppose T ⊂ C is a nonreduced degree 2 subscheme supported in p. Assuming Γ satis es (1), the authors have Γ ⊂ Cp, som−1(C) ∩ (X × Cp) ⊃ T × Γor equivalentlym−1(C) ∩ (X × C) ⊃ Tp × Γ−p. 

Q12. What is the criterion for de ning y?

A nite subscheme Γ ⊂ X is in special position with respect to 2Θ-translates if the codimension of H0(X,IΓ(2Θx)) in H 0(OX(2Θx)) is smaller than deg Γ for all x ∈ X. 

Q13. What was the author's contribution to the project?

The authors were partially funded by the NILS project UCM-EEA-ABEL-03-2010 to visit respectively the University of Barcelona and the Stord/Haugesund Uni-versity College.