M-regularity of the Fano surface
M-REGULARITY OF THE FANO SURFACE
ANDREAS HÖRING
Abstract. In this note we show that the Fano surface in the intermediate
Jacobian of a smooth cubic threefold is M -regular in the sense of Pareschi and
Popa.
1. Introduction
Let X3 ⊂ P4 be a smooth cubic threefold, then its intermediate Jacobian
J(X) := H2,1(X,C)∗/H3(X,Z)
is a principally polarised abelian variety (J(X),Θ) of dimension five that is not a Ja-
cobian of a curve [4, Thm.0.12]. The Fano scheme F parametrising lines contained
in X is a smooth surface, and the Abel-Jacobi map F → J(X) is an embedding
that induces an isomorphism Alb(F )) ≃ J(X) [4, Thm.0.6,0.9]. Furthermore the
cohomology class of F ⊂ J(X) is minimal, that is
[F ] =
There is only one other known family of examples of principally polarised abelian
varieties (A,Θ) of dimension n such that for 1 ≤ d ≤ n− 2, a minimal cohomology
class Θ
(n−d)!
can be represented by an effective cycle of dimension d: the Jacobians
of curves J(C) where the suvarietiesWd(C) ⊂ J(C) have minmal cohomology class.
O. Debarre has shown that on a Jacobian these are the only subvarieties having
minimal class [5, Thm.5.1], furthermore by a theorem of Z. Ran [11, Thm.5], the
only principally polarised abelian fourfolds with a subvariety of minimal class are
(products of) Jacobians of curves. In higher dimension few things are known about
subvarieties having minimal class.
In [9], G. Pareschi and M. Popa introduce a new approach to the characterisation
of these subvarieties: they consider the (probably more tractable) cohomological
properties of the twisted structure sheaf of the subvariety. More precisely we have
the following conjecture.
1.1. Conjecture. [5],[9] Let (A,Θ) be an irreducible principally polarised abelian
varieties of dimension n, and let Y be a nondegenerate subvariety (cf. [11, p.464])
of A of dimension d ≤ n− 2. The following statements are equivalent.
1.) The variety Y has minimal cohomology class, i.e. [Y ] = Θ
(n−d)!
2.) The twisted structure sheaf OY (Θ) is M -regular (cf. definition 1.4 below),
and h0(Y,OY (Θ)⊗ Pξ) = 1 for Pξ ∈ Pic
0(A) general.
Date: 4th April, 2007.
http://arxiv.org/abs/0704.0558v2
3.) Either (A,Θ) is the Jacobian of a curve of genus n and Y is a translate
of Wd(C) or −Wd(C), or n = 5, d = 2 and (A,Θ) is the intermediate
Jacobian of a smooth cubic threefold and Y is a translate of F or −F .
The implication 2) ⇒ 1) is the object of [9, Thm.B]. The implication 3) ⇒ 2)
has been shown for Jacobians of curves in [8, Prop.4.4]. We complete the proof of
this implication by treating the case of the intermediate Jacobian.
1.2. Theorem. Let X3 ⊂ P4 be a smooth cubic threefold, and let (J(X),Θ) be
its intermediate Jacobian. Let F ⊂ J(X) be an Abel-Jacobi embedded copy of the
Fano variety of lines in X. Then OF (Θ) is M -regular and h
0(F,OF (Θ)⊗ Pξ) = 1
for Pξ ∈ Pic
0 J(X) general.
Since the properties considered are invariant under isomorphisms, the theorem
implies the same statement for −F .
The study of the remaining open implications of conjecture 1.1 is a much harder
task than the proof of theorem 1.2. In an upcoming paper we will start to investi-
gate this problem under the additional hypothesis that (A,Θ) is the intermediate
Jacobian of a generic smooth cubic threefold. In this case we can show the following
statement.
1.3. Theorem. [6] Let X3 ⊂ P4 be a general smooth cubic threefold. Let (J(X),Θ)
be its intermediate Jacobian, and let F ⊂ J(X) be an Abel-Jacobi embedded copy
of the Fano variety of lines in X. Let S ⊂ J(X) be a surface that has minimal
cohomology class, i.e. [S] = Θ
. Then S is a translate of F or −F .
Notation and basic facts.
We work over an algebraically closed field of characteristic different from 2. We
will denote by D ≡ D′ the linear equivalence of divisors, and by D ≡num D
′ the
numerical equivalence.
For (A,Θ) a principally polarised abelian variety (ppav), we identify A with
Â = Pic0(A) via the morphism induced by Θ. If ξ ∈ A is a point, we denote by Pξ
the corresponding point in Â = Pic0(A) which we consider as a numerically trivial
line bundle on A.
1.4. Definition. [10] Let (A,Θ) be a ppav of dimension n, and let F be a coherent
sheaf on A. For all n ≥ i > 0, we denote by
V iF := {ξ ∈ A | h
i(A,F ⊗ Pξ) > 0}
the i-th cohomological support locus of F . We say that F is M -regular if
codimV i
for all i ∈ {1, . . . , n}.
If l ⊂ X is a line, we will denote by [l] the corresponding point of the Fano
surface F and by Dl ⊂ F the incidence curve of l, that is, Dl parametrises lines in
X that meet l. Furthermore we have by [4, §10], [12, §6] and Riemann-Roch that
OF (Θ) ≡num 2Dl,(1.5)
KF ≡num 3Dl,(1.6)
Dl ·Dl = 5,(1.7)
χ(F,OF (Θ)) = 1.(1.8)
2. Prym construction of the Fano surface
We recall the construction of the Fano surface as a special subvariety of a Prym
variety [3, 2]: let C̃ := Dl0 ⊂ F be the incidence curve of a general line l0 ⊂ X . Let
X ′ be the blow-up of X in l0. Then the projection from l0 induces a conic bundle
structure X ′ → P2 with branch locus C ⊂ P2 a smooth quintic. This conic bundle
induces a natural connected étale covering of degree two π : C̃ → C (cf. [1, Ch.I]
for details), and we denote by σ : C̃ → C̃ the involution induced by π.
The kernel of the normmorphism Nm : JC̃ → JC has two connected components
which we will denote by P and P1. The zero component P is called the Prym variety
associated to π, and it is isomorphic as a ppav to J(X) [1, Thm.2.1].
Let H ⊂ C be an effective divisor given by a hyperplane section in P2. Then H
has degree five and h0(C,OC(H)) = 3, so the complete linear system g
5 corresponds
to a P2 ⊂ C(5). We choose a divisor H̃ ∈ C̃(5) such that π(5)([H̃ ]) = [H ], where
π(5) : C̃(5) → C(5) is the morphism induced by π on the symmetric products. Let
φH : C
(5) → JC and φ
: C̃(5) → JC̃ be the Abel-Jacobi maps given by H and H̃.
We have a commutative diagram
C̃(5)
The fibre of φ
(C̃(5)) → φH(C
(5)) over the point 0 (and thus the intersection of
(C̃(5)) with kerNm) has two connected components F0 ⊂ P and F1 ⊂ P1. If
we identify P and P1 via H̃ − σ(H̃), we obtain an identification F1 = −F0 [3,
p.360]. The (non-canonical) isomorphism of ppavs P ≃ J(X) transforms F0 into a
translate of the Fano surface F [3, Thm.4].
From now on we will identify P (resp. F0) and J(X) (resp. some Abel-Jacobi
emdedded copy of the Fano surface F ).
We will now prove two technical lemmata on certain linear systems on C̃. The
first is merely a reformulation of [2, §2,ii)].
2.9. Lemma. The line bundle O
(C̃) is a base-point free pencil of degree five such
that any divisor D ∈ |O
(C̃)| satisfies π∗D ≡ H.
Proof. We define a morphism µ : C̃ = Dl0 → l0 ≃ P
1 by sending [l] ∈ C̃ to
l∩l0. Since l0 is general and through a general point of l0 there are five lines distinct
from l0, the morphism µ has degree 5. If [l] ∈ F , then Dl · Dl0 = 5 by formula
(1.7), so for [l] 6= [l0] the divisor Dl0 ∩ Dl ∈ |OC̃(Dl)| is effective. Furthermore
π∗Dl ≡ H , since π∗Dl is the intersection of C ⊂ P
2 with the image of l under
the projection X ′ → P2. By specialisation the linear system |O
(C̃)| is not empty
and a general divisor D in it corresponds to the five lines distinct from l0 passing
through a general point of l0. Hence OC̃(C̃) ≃ µ
∗OP1(1) and π∗D ≡ H . �
2.10. Lemma. The sets
V ′0 := {ξ ∈ P | h
0(C̃,O
(C̃)⊗ Pξ) > 0}
V ′1 := {ξ ∈ P | h
0(C̃,O
(2C̃)⊗ Pξ) > 1}
are contained in translates of F ∪ (−F ).
Proof.
1) Let D ∈ |O
(C̃) ⊗ Pξ| be an effective divisor. Then π∗C̃ ≡ π∗D ≡ H . It
follows that D ∈ (φ
(C̃(5)) ∩ kerNm), so D is in F or −F .
2) We follow the argument in [2, §3]. By [2, §2,iv)] we have h0(C̃,O
(C̃ +
σ(C̃))) = 4, so h0(C̃,O
(2C̃)) is odd. It follows from the deformation invariance
of the parity [7, p.186f] that
V ′1 = {ξ ∈ P | h
0(C̃,O
(2C̃)⊗ Pξ) ≥ 3}.
Fix ξ ∈ P such that h0(C̃,O
(2C̃)⊗ Pξ) ≥ 3 and D ∈ |OC̃(2C̃)⊗ Pξ|. Let s and t
be two sections of O
(C̃) such that the associated divisors have disjoint supports,
then we have an exact sequence
0 → O
(D − C̃)
(t,−s)
(D)⊕2
(s,t)
(D + C̃) → 0.
This implies
h0(C̃,O
(D − C̃)) + h0(C̃,O
(D + C̃)) ≥ 2h0(C̃,O
(D)) = 6,
furthermore by Riemann-Roch h0(C̃,O
(D + C̃)) = h0(C̃,O
−D − C̃)) + 5.
Now K
−D ≡ σ(D) and h0(C̃,O
(σ(D) − C̃)) = h0(C̃,O
(D − σ(C̃))) imply
h0(C̃,O
(D − C̃)) + h0(C̃,O
(D − σ(C̃))) ≥ 1.
Hence D ≡ C̃ + D′ or D ≡ σ(C̃) + D′ where D′ is an effective divisor such that
′ ≡ H . We see as in the first part of the proof that the effective divisors D′
such that π∗D
′ ≡ H are parametrised by a set that is contained in a translate of
F ∪ (−F ). �
3. Proof of theorem 1.2.
Since OF (Θ) ≡num OF (2C̃) by formula (1.5), it is equivalent to verify the stated
properties for the sheaf OF (2C̃).
Step 1. The second cohomological support locus is contained in a translate of
F ∪ (−F ). By formula (1.6), we have KF ≡ OF (3C̃)⊗Pξ0 for some ξ0 ∈ P . Hence
by Serre duality h2(F,OF (2C̃)⊗Pξ) = h
0(F,OF (C̃)⊗P
ξ ⊗Pξ0), so it is equivalent
to consider the non-vanishing locus
V0 := {ξ ∈ P | h
0(F,OF (C̃)⊗ Pξ) > 0}.
If l ∈ F is a line on X , the corresponding incidence curve Dl ⊂ F is an effective
divisor numerically equivalent to C̃, so it is clear that ±F is (up to translation) a
subset of V0. In order to show that we have an equality, consider the exact sequence
0 → OF ⊗ Pξ → OF (C̃)⊗ Pξ → OC̃(C̃)⊗ Pξ → 0.
Clearly h0(F,OF ⊗ Pξ) = 0 for ξ 6= 0, so h
0(F,OF (C̃)⊗ Pξ) ≤ h
0(C̃,O
(C̃)⊗ Pξ)
for ξ 6= 0. Since a divisor D ∈ |O
(C̃)| satisfies π∗D ≡ H , we conclude with
Lemma 2.10.
Step 2. The first cohomological support locus is is contained in a union of trans-
late of F ∪ (−F ). Since χ(F,OF (2C̃)) = χ(F,OF (Θ)) = 1 (formula (1.8)), we
h1(F,OF (2C̃)⊗ Pξ) = h
0(F,OF (2C̃)⊗ Pξ) + h
0(F,OF (C̃)⊗ P
ξ ⊗ Pξ0)− 1.
Since
h0(F,OF (2C̃)⊗ Pξ) = h
0(F,OF (Θ)⊗ Pξ) ≥ 1
for all ξ ∈ P , the first cohomological support locus is contained in the locus where
h0(F,OF (C̃)⊗ P
ξ ⊗Pξ0) > 0 or h
0(F,OF (2C̃)⊗ Pξ) > 1. By step 1 the statement
follows if we show the following claim: the set
V1 := {ξ ∈ P | h
0(F,OF (2C̃)⊗ Pξ) > 1}
is contained in a union of translates of F ∪ (−F ).
Step 3. Proof of the claim and conclusion. Consider the exact sequence
0 → OF (C̃)⊗ Pξ → OF (2C̃)⊗ Pξ → OC̃(2C̃)⊗ Pξ → 0.
By the first step we know that h0(F,OF (C̃)⊗ Pξ) = 0 for ξ in the complement of
a translate of F ∪ (−F ), so
h0(F,OF (2C̃)⊗ Pξ) ≤ h
0(C̃,O
(2C̃)⊗ Pξ)
for ξ in the complement of a translate of F ∪ (−F ). The claim is then immediate
from Lemma 2.10. By the same lemma h0(C̃,O
(2C̃)⊗Pξ) = 1 for ξ ∈ P general,
so h0(F,OF (2C̃)⊗ Pξ) = h
0(F,OF (Θ)⊗ Pξ) = 1 for ξ ∈ P general. �
Remark. It is possible to strengthen a posteriori the statements in the proof:
since Theorem 1.2 holds, we can use the Fourier-Mukai techniques from [9] to see
that the cohomological support loci are supported exactly on the theta-dual of F
(ibid, Definition 4.2), which in our case is just F .
Acknowledgements. I would like to thank Mihnea Popa for suggesting to me
to work on this question. Olivier Debarre has shown much patience at explaining
to me the geometry of abelian varieties. For this and many discussions on minimal
cohomology classes I would like to express my deep gratitude.
References
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[9] G. Pareschi and M. Popa. Generic vanishing and minimal cohomology classes on abelian
varieties. arXiv:math.AG/0610166, 2006.
[10] G. Pareschi and M. Popa. GV-sheaves, Fourier-Mukai transform, and Generic Vanishing.
arXiv:math.AG/0608127, 2006.
[11] Z. Ran. On subvarieties of abelian varieties. Inventiones Math., 62:459–479, 1981.
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Andreas Höring, IRMA, Université Louis Pasteur, 7 rue René Descartes, 67084 Stras-
bourg, France
E-mail address: andreas.hoering@ujf-grenoble.fr
http://arxiv.org/abs/math/0610166
http://arxiv.org/abs/math/0608127
	1. Introduction
	2. Prym construction of the Fano surface
	3. Proof of theorem ??.
	References
