SYZYGIES AND EQUATIONS OF KUMMER VARIETIES

arXiv:1205.1310v2 [math.AG] 30 Sep 2012

SOFIA TIRABASSI

Abstract. We study property Np for Kummer varieties embedded by a power of an
ample line bundle.

1. Introduction
Let X be an abelian variety. Its associated Kummer variety, KX , is the quotient of X
by the natural (Z/2Z)-action induced by the morphism iX : X → X defined by x 7→ −x.
Given a Kummer variety, KX , and an ample line bundle A on KX , a result of Sasaki ([25])
states that A⊗m is very ample and the embedding it defines is projectively normal as soon
as m ≥ 2. Later Khaled ([15]) and Kempf ([14]) proved that, under the same conditions,
the homogeneous ideal of KX is generated by elements of degree 2 and 3, while, if m ≥ 3
it is generated in degree 2. Concerning the case m = 1, if, furthermore, we assume that A
is a normally generated, very ample, line bundle on X, then the homogeneus ideal of KX
is generated in degree less than or equal to 4. In this paper we prove that these statements
are particular cases of more general results on the syzygies of the variety KX .
More precisely, let Z an algebraic variety over an algebraically closed field k and let A
an ample invertible sheaf on Z, generated by its global sections. With RA we will indicate
the section ring associated to the sheaf A :
M
H 0 (Z, A ⊗n ),
RA :=
n∈Z

while SA will be the symmetric algebra of H 0 (Z, A ). The ring RA is a finitely generated
graded SA -algebra and as such it admits a minimal free resolution
(1.1)

fp+1

fp

f2

f1

f0

E• = 0 → · · · −−−→ Ep −→ · · · −→ E1 −→ E0 −→ RA → 0

L
with Ei ≃ j SA (−aij ), aij ∈ Z>0 . In order to extend classical results of Castelnuovo,
Mattuck, Fujita and Saint-Donat on the projective embeddings of curves, Green ([8])
introduced the following:
Definition (Property Np ). Let p be a given integer. The line bundle A satisfies property
Np if, in the notations above,
E0 = SA
and
Ei = ⊕SA (−i − 1)

1 ≤ i ≤ p.

2010 Mathematics Subject Classification. 14M99, 14K99 Primary, 14K05, 14F05 Secondary.
Key words and phrases. Syzygies, Koszul cohomology, Kummer varieties.
1

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SOFIA TIRABASSI

Pareschi ([22]) extended the above condition as follows : we say that, given a non-negative
integer r, property N0r holds for A if, in the notation above, a0j ≤ 1 + r for every j.
r
holds for A and apj ≤ p + 1 + r
Inductively we say that A satisfies property Npr if Np−1
for every j.
Green in [8] proved that, if Z is a smooth curve of genus g and A a very ample line
bundle on Z then A satisfies Np if deg A ≥ 2g + 1 + p. He also conjectured the behaviour
of property Np for the canonical bundle of a non-hyperelliptic curve. Green’s conjecture
was recently proved for the general curve by Voisin ([27, 28]).
For what concerns higher dimensional varieties, the syzygies of the projective space
were studied by Green in [9], [21], and in the recent preprint [5]. For arbitrary smooth
varieties there is a general conjecture of Mukai and in [4] Ein–Lazarsfeld proved that, if
Z is of dimension n, denoting by ωZ its canonical line bundle on Z, then for any L very
ample on Z the sheaf
A := ωZ ⊗ L ⊗(n+1+d)
satisfies Np for every d ≥ p ≥ 1.
Abelian varieties distinguish themselves among other smooth varieties since, at least in
what concerns their syzygies, they tend to behave in any dimension like elliptic curves.
More precisely, Koizumi ([17]) proved that, given A ample on an abelian variety X, then
for m ≥ 3, A ⊗m embeds X in the projective space as a projectively normal variety.
Furthermore, a classical theorem of Mumford ([20]), improved by Kempf ([11]), states
that the homogeneous ideal of X is generated in degree 2 as long as m ≥ 4. These
results inspired Lazarsfeld to conjecture that A ⊗m satisfies Np for every m ≥ p + 3. A
generalized version of Lazarsfeld’s conjecture, involving property Npr rather than simply
Np , was proved in [22]; later in [24] Pareschi–Popa were able to recover and improve
Pareschi’s statements as a consequence of the powerful, Fourier-Mukai based, theory of
M -regularity that they developed in [23].
Given the results on projective normality and degree of defining equations of Sasaki and
Khaled, it was natural to conjecture that a bound m0 (p, r), independent of the dimension
of KX , could be found such that A ⊗m satisfies Npr for every m ≥ m0 (p, r). In this paper
we present some results in this direction. The main idea behind the proofs is that ample
line bundles on a Kummer variety KX have a nice description in terms of ample line
bundles on X. More precisely, denoting by πX : X → KX the quotient map, then for
∗ A ≃ A ⊗2 . Hence we can
every A ample on KX there exists A ample on X such that πX
⊗2m
use Pareschi–Popa machinery to find some results on A
and then study how the Z/2Z
action fits in the framework. Below we list the main results we obtained.
Theorem A. Fix two non-negative integers p and r such that char(k) does not divide
p + 1, p + 2. Let A be an ample line bundle on a Kummer variety KX . Then:
(a) A⊗n satisfies property Np for every n ∈ Z such that n ≥ p + 2;
(b) more generally A⊗n satisfies property Npr for every n such that (r + 1)n ≥ p + 2.
Since it consists in an improvement of existing results on the degrees of defining equations of Kummer varieties it is worth to emphasize individually the case p = 1 of the above
statement. Thanks to the geometric meaning of property Npr (Section 2.1) one can deduce
the following:
Particular Case B. Let be A a very ample line bundle on a Kummer variety KX . Then
the ideal of the image ϕA (KX ) in P(H 0 (X, A)) is generated by forms of degree at most 4.

SYZYGIES AND EQUATIONS OF KUMMER VARIETIES

3

The improvement with respect to what was classically known, thanks to the work of
Wirtinger, Andreotti–Mayer [1] and Khaled, is that this holds for all Kummer varieties,
and not only for the general ones. It might be interesting to see whether this fact has
some applications to moduli of abelian varieties.
Adding one hypothesis about the line bundle A we can get a somewhat better result
improving the work of Kempf and Khaled; namely:
Theorem C. Let p and r be two integers such that p ≥ 1, r ≥ 0 and char(k) does not
divide p + 1, p + 2. Let A be an ample line bundle on a Kummer variety KX , such that its
∗ A ≃ A ⊗2 with A an ample symmetric invertible sheaf on X which does not
pullback πX
have a base divisor. Then the followings hold:
(a) A⊗n satisfies property Np for every n ∈ Z such that n ≥ p + 1;
(b) more generally A⊗n satisfies property Npr for every n such that (r + 1)n ≥ p + 1.
Again, it is worth of single out the case p = 1 of the above Theorem, concerning the
equations of the Kummer variety KX .
Particular Case D. Suppose that char(k) does not divide 2 or 3 and let A be an ample
∗ A ≃ A ⊗2 with A without a base divisor. Then:
invertible sheaf on KX such that πX
(a) if n ≥ 2, then the ideal IKX ,A⊗n of the embedding ϕA⊗n is generated by quadrics;
(b) IKX ,A is generated by quadrics and cubics.
The key point of the proofs of Theorems 2.A and 2.C will be to reduce the problem on
the Kummer variety KX to a different problem on the abelian variety X. Namely we will
show that property Npr on the Kummer is implied by the surjectivity of a map of the type:
M
m
(*)
H 0 (X, F ⊗ α) ⊗ H 0 (X, H ⊗ α) →α H 0 (X, F ⊗ H ⊗ α)
b
[α]∈U

b is a non empty open subset of X,
b the abelian
where F and H are sheaves on X and U
variety dual to X, and mα is just the multiplication of global sections. Criteria for the
surjectivity of such maps are implicit in Kempf’s work ([12, 13]), for the case F a vector
bundle and H a line bundle (for an explicit argument due to Lazarsfeld see [22]). These
results were improved and extended to general coherent sheaves by Pareschi–Popa in [24].
This paper is organized in the following manner: in the next Section we explain some
background material such as the relationship between proprerty Npr and the cohomology
of the Koszul complex and a useful criterion for the surjectivity of a map of type (*). In
Section 3 we present some slightly modified version of results of Sasaki and Khaled. The
last section is entirely devoted to the proof of the main theorems.
Notations. Throughout the paper we will work over an algebraically closed field k of
characteristic different from 2; further restrictions on the field will be stated when needed.
The word ”variety” will mean a projective variety over k.
⋄ Given F a sheaf on a variety Z, then we denote by Bs(F ) the locus in Z where F
is not generated by its global sections, i. e. the locus of z ∈ Z where the evaluation
map H 0 (Z, F ) ⊗ k(z) → F ⊗ k(z) fails to be surjective.
⋄ If L is a line bundle on a variety Y , ϕL will be the rational map associated to L.
The symbol IY,L is the ideal of ϕL (Y ) in P(H 0 (Y, L)).

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SOFIA TIRABASSI

⋄ If we have a product of varieties, X1 ×· · ·×Xn then pi is the i-th projection. Given
Fi a sheaf on Xi , then F1 ⊠ · · · ⊠ Fn will stand for the sheaf p∗1 F1 ⊗ · · · ⊗ p∗n Fn
on X1 × · · · × Xn .
Given an abelian variety X of dimension g:
⋄ iX will be the inversion morphism; tx will be the translation by the point x;
⋄ The group law on the abelian variety is denoted by p1 + p2 , while given an integer
n the map nX : X −→ X will be the multiplication by n.
2. Background Material
2.1. Property Nrp and Koszul Cohomology. We begin by reviewing some well known
relations between property Np , or more generally property Npr , and the surjectivity of
certain multiplication maps of sections of vector bundles. Let Z be a projective variety
and L be an ample invertible sheaf on Z. We know from [8, Thm. 1.2] (see also [10] Thm.
1.2 or [18, p. 511]) that condition Np is equivalent to the exactness in the middle of the
complex
(2.1)

p+1
^

H 0 (L) ⊗ H 0 (L⊗h ) →

p
^

H 0 (L) ⊗ H 0 (L⊗h+1 ) →

p−1
^

H 0 (L) ⊗ H 0 (L⊗h+2 )

for any h ≥ 1. More generally, condition Npr is equivalent to exactness in the middle of
(2.1) for every h ≥ r + 1. Suppose that L generated by its global sections and consider
the following exact sequence:
0 → ML −→ H 0 (L) ⊗ OZ −→ L → 0.

(2.2)

It is well known (see, for example [18] or [5])that property Npr is implied by the surjectivity of
!
p
p+1
^
^
ML ⊗ L⊗h+1 .
H 0 (L) ⊗ H 0 (L⊗h ) −→ H 0
(2.3)
It follows that if

H 1 (Z,

(2.4)

p+1
^

ML ⊗ L⊗h ) = 0

for any h ≥ r + 1, then condition Npr is satisfied. If char(k) does not divide p,
direct summands of E ⊗p for any vector bundle E . This yields
Lemma 2.1. Assume that char(k) does not divide p + 1.

Vp

E is a

(a) If H 1 (Z, ML⊗p+1 ⊗ L⊗h ) = 0 for any h ≥ r + 1, then L satisfies Npr .
(b) Let W ⊆ H 0 (Z, L) be a free sublinear system and denote by MW the kernel of
⊗p
the evaluation map W ⊗ OZ → L. Assume that H 1 (Z, MW
⊗ L⊗h ) = 0. Then
⊗p+1
H 1 (Z, MW
⊗ L⊗h ) = 0 if and only if the multiplication map
⊗p
⊗p
W ⊗ H 0 (MW
⊗ L⊗h ) −→ H 0 (MW
⊗ L⊗h+1 )

is surjective.
Proof. The proof of (a) is straightforward, while (b) follows from the following exact
sequence:
⊗p+1
⊗p
⊗p
0 → MW
⊗ L⊗h −→ W ⊗ MW
⊗ L⊗h −→ MW
⊗ L⊗h+1 → 0.



SYZYGIES AND EQUATIONS OF KUMMER VARIETIES

5

2.1.1. Property Nrp for Small p’s’. By definition , if a variety Z is embedded in a projective
space by a very ample line bundle L satisfying property N0r , then the variety Z is h-normal
for every h ≥ r. In particular property N0 is equivalent to projective normality.
Although this is a standard fact, it is perhaps worth spelling out the meaning of property
N1r , thus providing a direct proof of Particular Case B and D.
Proposition 2.2. If L is a very ample line bundle on an algebraic variety Z satisfying
N1r , then the homogeneous ideal of Z is generated by homogeneous elements of degree at
most r + 2.
Proof. Denote by V the vector space H 0 (Z, L) and let Sk V be the component of degree
k of the symmetric algebra of V , S. Consider furthermore the two S-modules
M
I=
H 0 (P(V ), IZ,L(k))
and RL . Look at the following commutative diagram where the middle column is given
by the Koszul complex.
Sk−1 V ⊗

0

(3)

V


// Sk V ⊗ V

// Ik ⊗ V
(1)

0

V2

// H 0 (Z, ML ⊗ L⊗k )

// {RL }k ⊗ V

// 0


// {RL }k+1

// 0

(2)


// Sk+1 V



// Ik+1

Our aim is to see that the map (1) is surjective for every k ≥ r + 2. Suppose that L
satisfies property N1r , then in particular property N0r holds for L and the second and third
row are exact for every k ≥ r + 1. Since (2) is surjective, by the Snake Lemma, for every
k ≥ r + 1 the surjectivity of (1) is implied by the surjectivity of (3) for every k ≥ r + 2.
Now we can factor (3) in the following way:
Sk−1 V ⊗

V2

(3)

V
❙❙❙
❙❙❙
❙❙❙
❙
g ❙❙❙❙
))

H 0 (Z, L⊗k−1 ) ⊗

// H 0 (Z, ML ⊗ L⊗k )
❥❥44
❥❥❥❥
❥
❥
❥
❥❥
❥❥❥❥ f

V2

V

where g is the canonical mapping H 0 (P(V ), OP (k − 1)) −→ H 0 (Z, L⊗k−1 ) and f is the
map in (2.3). For every k ≥ r + 2, g is surjective because L satisfies N0r , while the

surjectivity of f is equivalent to property N1r ; hence (3) is surjective.
2.2. M-regular Sheaves and Multiplication Maps.
2.3. A review on M -regularity. In what follows we briefly recall some notions about
M -regular sheaves. We refer to [23, 24] for further details.
Given an abelian variety X and a coherent sheaf F on X, we introduce the cohomological
support loci of F :
b
V i (F ) := {α | hi (F ⊗ α) 6= 0} ⊆ X
It is said that a sheaf F satisfies index theorem with index j (or equivalently it is said to
be I.T.(j) if for every i 6= j the loci V i (F ) are empty.

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SOFIA TIRABASSI

Definition 1. A sheaf F over an abelian variety X is said to be Mukai-regular (or simply
M -regular) if, for every i > 0, codim V i (F ) ≥ i + 1.
2.3.1. Application to multiplication maps. M -regular sheaves and M -regularity theory,
introduced by Pareschi-Popa in a long series of paper ranging from 2002 to 2009, are
crucial to our purpose thank to their application in determining whether a map of the
form
M
mα
(2.5)
H 0 (X, F ⊗ α) ⊗ H 0 (X, H ⊗ α∨ ) −→
H 0 (X, F ⊗ H ),
[α]∈U

b an open set, is surjective. We
with F and H sheaves on an abelian variety X and U ⊆ X
will list below all the results of such kind that we will be using troughout the paper. The
first one is an extension of a theorem that had already appeared in the work of Kempf,
Mumford and Lazarsfeld.
Theorem 2.3 ([23], Theorem 2.5). Let F and H be sheaves on X such that F is M regular and H is locally free satisfying I.T. with index 0. Then (2.5) is surjective for any
b
non empty Zariski open set U ⊆ X.
In particular, if F and H are as above, then there exists N a positive integer such that
b the map
for the general [α1 ], . . . , [αN ] ∈ X
N
M

H 0 (X, F ⊗ αk ) ⊗ H 0 (X, H ⊗ α∨
k)

mαk

// H 0 (X, F ⊗ H )

k=1

is surjective.
We conclude this paragraph by presenting two results on multiplication maps of sections
we will be needing afterward.
Proposition 2.4. Let A be an ample line bundle on an abelian variety X. The map
(2.6)

m : H 0 (X, A ⊗2 ) ⊗ H 0 (X, A ⊗2 ⊗ α) −→ H 0 (X, A ⊗4 ⊗ α).

b If furthermore A does not have a base divisor, then
is surjective for the general [α] ∈ X.
0
the locus Z ⊆ Pic (X) in which it fails to be surjective has codimension at least 2.

Proof. The first part of the statement is classical, for a reference see, for example [2,
Proposition 7.2.2]. Pareschi–Popa provided another proof of this, together with a proof
of the second statement of the Proposition, using their Fourier–Mukai based methods (see
[24, Proposition 5.2, Proposition 5.6, and Theorem 5.8]).

3. Multiplication Maps on Abelian Varieties
Let A be an ample symmetric invertible sheaf on the abelian variety X and denote by
ψA : A → i∗X A its normalized isomorphism (see [19, p. 304]). The Z/2Z action on X
induced by the involution iX : X −→ X induces trough ψA a lifting of the action on A .
The composition
H 0 (X, A )

i∗X

// H 0 (X, i∗ A )
X

i∗X (ψA )

// H 0 (X, A )

is denoted by [−1]A . We let
H 0 (X, A )± = {s ∈ H 0 (X, A ) such that [−1]A s = ±s}

SYZYGIES AND EQUATIONS OF KUMMER VARIETIES

7

If A is totally symmetric (see [19, p. 304], e.g. if A is an even power of a symmetric
line bundle), then there exists1 a line bundle A on the Kummer variety KX such that
∗ A ≃ A and one can identify H 0 (K , A) with H 0 (X, A )+ . Conversely if A is an
πX
X
ample line bundle on KX , then there exists A , ample and symmetric on X, such that
∗ A ≃ A ⊗2 .
πX
A well known result by Khaled states that
Proposition 3.1 ([15]). Suppose that k = 2n is an even positive integer. Thus A ⊗k is
totally symmetric. Then
0
⊗k +
m+
) ⊗ H 0 (X, A ⊗h ⊗ α) −→ H 0 (X, A ⊗k+h ⊗ α)
α : H (X, A

is surjective for n ≥ 1, every h ≥ 3, and for every α ∈ Pic0 (X)
The main goal of this section is to prove that the same is true for every h = 2 and for
b If furthermore we assume that A does not have a base divisor, then we
general α ∈ X.
b where
will show that the locus of [α] ∈ X
(3.1)

0
⊗2n +
m+
) ⊗ H 0 (X, A ⊗2 ⊗ α) −→ H 0 (X, A ⊗2(n+1) ⊗ α)
α : H (X, A

fails to be surjective has codimension at least 2. We will do this by the methods of Khaled
([15]).
To this end, consider the isogeny
ξ : X × X −→ X × X
given by ξ = (p1 + p2 , p1 − p2 )
b we have an isomorphism
Lemma 3.2. For any [α] ∈ X
ξ ∗ (p∗1 (A ⊗ β) ⊗ p∗2 (A ⊗ α))

// p∗ (A ⊗2 ⊗ β ⊗ α) ⊗ p∗ (A ⊗2 ⊗ β ⊗ α∨ )
1
2

Proof. Observe that
ξ ∗ (p∗1 (A ⊗ β) ⊗ p∗2 (A ⊗ α))|X×{y} ≃ t∗y A ⊗ t∗−y A ⊗ t∗y α∨ ⊗ t∗−y α ≃
≃ A ⊗2 ⊗ β ⊗ α.
Restricting to {0} × X we get
ξ ∗ (p∗1 (A ⊗ β) ⊗ p∗2 (A ⊗ α))|{0}×X ≃ A ⊗ β ⊗ i∗ A ⊗ α) ≃ A ⊗2 ⊗ β ⊗ α∨ ;
the statement follows from the See-Saw principle.



Composing with the Künneth isomorphism we have a map
∗

ξ : H 0 (X, A ⊗ β) ⊗ H 0 (X, A ⊗ α) −→ H 0 (X, A ⊗2 ⊗ β ⊗ α) ⊗ H 0 (X, A ⊗2 ⊗ β ⊗ α∨ ).
Taking [α] = [β ∨ ] we want to characterize the image of H 0 (X, A ⊗ β)⊗H 0 (X, A ⊗ β ∨ )
in H 0 (X, A ⊗2 )+ ⊗ H 0 (X, A ⊗2 ⊗ β ⊗2 ) through ξ ∗ .
In order to achieve this goal, we consider TbA ⊗β , the involution of H 0 (X, A ⊗ β) ⊗
0
H (X, A ⊗ β ∨ ) defined by TbA ⊗β (s ⊗ t) = i∗ (ψβ ∨ )i∗ t ⊗ i∗ (ψβ )i∗ s.
Let us denote by
[H 0 (X, A ⊗ β) ⊗ H 0 (X, A ⊗ β ∨ )]±
the eigenspaces.
1This is a standard fact. A proof could be found in [26, Prop. 2.1.7]

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SOFIA TIRABASSI

b we have
Proposition 3.3. For every [α] ∈ X

ξ ∗ [H 0 (X, A ⊗ β) ⊗ H 0 (X, A ⊗ β ∨ )]± ⊆ H 0 (X, A ⊗2 )± ⊗ H 0 (X, A ⊗2 ⊗ β ⊗2 ).

The proof of this statement is just a slight modification of Khaled’s proof of [16, Proposition 2.2] and therefore we omit it. The complete proof can also be found in [26, Prop.
2.2.3]
b Then the
Theorem 3.4. Let A be an ample symmetric line bundle on X. Take [α] ∈ X.
multiplication map
m : H 0 (X, A ⊗2 ) ⊗ H 0 (X, A ⊗2 ⊗ α) −→ H 0 (X, A ⊗4 ⊗ α)

is surjective if and only if the following multiplication map is surjective
m+ : H 0 (X, A ⊗2 )+ ⊗ H 0 (X, A ⊗2 ⊗ α) −→ H 0 (X, A ⊗4 ⊗ α)
Proof. Obviously, if the map m+ is surjective, then also m is. Therefore we are left to
prove that the surjectivity of m implies the surjectivity of m+ . As observed by Khaled,
m is the composition of
ξ ∗ : H 0 (X, A ⊗2 ) ⊗ H 0 (X, A ⊗2 ⊗ α) → H 0 (X, A ⊗4 ⊗ α) ⊗ H 0 (X, A ⊗4 ⊗ α∨ )
with id⊗eA ⊗4 ⊗α∨ , where eA ⊗4 ⊗α∨ denotes the evaluation in 0 of the sections of A ⊗4 ⊗α∨ .
In fact, if we denote by ∆ : X → X × X the diagonal immersion, then m is just ∆∗
composed with the Künneth isomorphism. Now we can write ∆ = ξ ◦ f where f : X →
X × X is the morphism defined by x 7→ (x, 0X ). Now observe that, modulo Künneth
isomorhism,
f ∗ : H 0 (X, A ⊗4 ) ⊗ H 0 (X, A ⊗4 ⊗ α∨ ) → H 0 (X, A ⊗4 )
is exactly id ⊗ eA ⊗4 ⊗α∨ .
Observe that ξ ◦ ξ is the map (x, y) 7→ (2X (x), 2X (y)). Hence, taking β ∈ Pic0 (X) such
that β ⊗2 ≃ α, we can write
m ◦ ξ ∗ : H 0 (X, A ⊗ β) ⊗ H 0 (X, A ⊗ β ∨ ) → H 0 (X; A ⊗4 ⊗ α)
as id ⊗ eA ⊗4 ⊗α∨ ◦ (2∗X (−) ⊗ 2∗X (−)) = 2∗X ⊗ eA ⊗2 ⊗β ∨ . Thus we can consider the following
commutative diagram
M
ξ∗
// H 0 (A ⊗2 ) ⊗ H 0 (A ⊗2 ⊗ α)
H 0 (A ⊗ β) ⊗ H 0 (A ⊗ β ∨ )
[β ⊗2 ]=[α]

❚❚❚❚
❚
2∗X ⊗eA ⊗β ∨ ❚❚❚❚❚❚
❚))

❧❧❧
❧❧❧
❧
❧
❧❧❧
❧❧❧ m
❧
❧
uu ❧
❧

H 0 (A ⊗4 ⊗ α)

where the upper arrow is an isomorphism by projection formula. It follows that the
surjectivity of m is equivalent to the surjectivity of
M
M
2∗X ⊗ eA ⊗β ∨ :
H 0 (A ⊗ β) ⊗ H 0 (A ⊗ β ∨ ) → H 0 (A ⊗4 ⊗ α),
[β ⊗2 ]=[α]

[β ⊗2 ]=[α]

which, in turn, is equivalent the following condition:
(†)

b such that β ⊗2 ≃ α.
0∈
/ Bs(A ⊗ β ∨ ) for every β ∈ X

We claim that (†) implies the surectivity of m+ , the statement of the theorem will follow
directly.

SYZYGIES AND EQUATIONS OF KUMMER VARIETIES

9

Thanks to Proposition 3.3 and the isomorphism
L

[β ⊗2 ]≃[α] H

0 (A

⊗ β)

2∗X

// H 0 (A ⊗4 ⊗ α),

yielded by 2∗X and projection formula, it is enough to check that for every [β] satisfying
[β ⊗2 ] = [α]
2∗X ⊗ eA ⊗β ∨ [H 0 (A ⊗ β) ⊗ H 0 (A ⊗ β ∨ )]+ = 2∗X (H 0 (A ⊗ β)).
Hence for each s ∈ H 0 (A ⊗ β), we need a section σ ∈ (H 0 (A ⊗ β) ⊗ H 0 (A ⊗ β ∨ ))+ such
that 2∗X ⊗ eA ⊗β ∨ (σ) = 2∗X (s). Therefore take s ∈ H 0 (X, A ⊗ β) and denote by λ the
constant eA ⊗β ∨ ([i∗ ψβ ◦ i∗ ](s)). If λ 6= 0, take
σ :=

1
· (s ⊗ [i∗ ψβ ◦ i∗ ](s)).
λ

If, otherwise, λ = 0, it follows from (†) that there exists t ∈ H 0 (X, A ⊗ β) such that
eA ⊗β ∨ ([i∗ ψβ ◦ i∗ ](t)) = 1. Then, take σ to be the section
(s + t) ⊗ [i∗ ψβ ◦ i∗ ] (s + t) − t ⊗ [i∗ ψβ ◦ i∗ ](t)

∈

(H 0 (A ⊗ β) ⊗ H 0 (A ⊗ β ∨ ))+ .

Applying 2∗X ⊗ eA ⊗β ∨ to σ we get
2∗X ⊗ eA ⊗β ∨ (σ) = 2∗X ⊗ eA ⊗β ∨ ((s + t) ⊗ [i∗ ψβ ◦ i∗ ] (s + t) − t ⊗ [i∗ ψβ ◦ i∗ ](t)) =
= 2∗X (s + t) · 1 − 2∗X (t) · 1 = 2∗X (s).

Corollary 3.5.
(1) For every A ample symmetric invertible sheaf on X the multiplication map
(3.2)

m+ : H 0 (X, A ⊗2 )+ ⊗ H 0 (X, A ⊗2 ⊗ α) −→ H 0 (X, A ⊗4 ⊗ α)

is surjective for the generic [α] ∈ Pic0 (X).
(2) If furthermore A does not have a base divisor, then the locus where (3.2) is not
surjective has codimension at least 2.
Proof. The statement follows directly from the Theorem 3.4 and the corresponding statements (see Proposition 2.4) about the map m.

Now we are ready to prove the result that is the main point of this section.
Theorem 3.6. Let A be an ample symmetric invertible sheaf on X, then
(1) there exist a non-empty open subset U ⊆ Pic0 (X) such that for every n ∈ Z with
n ≥ 1 and and every α ∈ U the following map is surjective
(3.3)

0
⊗2n +
m+
) ⊗ H 0 (X, A ⊗2 ⊗ α) −→ H 0 (X, A ⊗2n+2 ⊗ α);
α : H (X, A

(2) if, furthermore, A does not have a base divisor, then the locus Z in Pic0 (X) where
(3.3) fails to be surjective has codimension at least 2.
Proof. We will proceede by induction on n, with base case given by Corollary 3.5.

10

SOFIA TIRABASSI

Case n > 1. Observe the following commutative diagram:
H 0 (X, A ⊗2 )+⊗ H 0 (X, A ⊗2(n−1) )+⊗ H 0 (X, A ⊗2 ⊗ α)

ϕα

// H 0 (X, A ⊗2 )+⊗ H 0 (X, A ⊗2(n−1)+2 ⊗ α)
ψα



H 0 (X, A ⊗2n )+⊗ H 0 (X, A ⊗2 ⊗ α)



mα

// H 0 (X, A ⊗2n+2 ⊗ α)

Let [α] ∈ Pic0 (X) be is such that mα is not surjective. Then, necessarily the map ψα ◦ ϕα
is not surjective. Since 2 + 2(n − 1) ≥ 3, by Proposition 3.1, ψα is surjective for every
b Per force, then, the map ϕα could not be surjective. It follows that the locus of
[α] ∈ X.
point [α] such that mα is not surjective is contained in the following set:
W := {[α] ∈ Pic0 (X) | ϕα is not surjective}.

By inductive hypothesis W has codimension ≥ 1, in the general case, or ≥ 2, when A has
not a base divisor. Therefore the Theorem is proved.

4. Equations and Syzygies of Kummer Varieties
Putting together the results of the previous paragraphs, in this last Section we will
prove Theorem A and Theorem C. First of all, observe that the case p = 0 of the Theorem
A follows directly as a corollary of Khaled’s work (see Proposition 3.1). Thus we may
suppose p ≥ 1.
Our strategy will be to use the part (b) of Lemma 2.1 and to reduce the problem to
checking the surjectivity of
(4.1)

⊗nh
⊗n(h+1)
H 0 (KX , A⊗n ) ⊗ H 0 (KX , MA⊗p
) −→ H 0 (KX , MA⊗p
)
⊗n ⊗ A
⊗n ⊗ A

for every h ≥ r+1. Let A be an ample symmetric line bundle on X such that A ⊗2 ≃ π ∗ A,
we split the proof in two steps.
In first place, the surjectivity of (4.1) is implied by the surjectivity of
(4.2) H 0 (X, A ⊗2n )+ ⊗ H 0 (X, π ∗ (MA⊗n )⊗p ⊗ A ⊗2nh ) −→ H 0 (X, π ∗ (MA⊗n )⊗p ⊗ A ⊗2n(h+1) ).

Before going any further we need to introduce some notation.
Notation 4.1. Suppose that A is an ample line bundle on KX and let A be an invertible
∗ A = A ⊗2 . Take n an integer such that A⊗n is globally generated
sheaf on X such that πX
and consider the following exact sequence of vector bundles:
0 → MA⊗n −→ H 0 (KX , A⊗n ) ⊗ OKX −→ A⊗n → 0.
By pulling back via the canonical surjection πX we get:
∗
0 → πX
(MA⊗n ) −→ H 0 (X, A ⊗2n )+ ⊗ OX −→ A ⊗2n → 0.

Hence, after defining,

Wn := H 0 (X, A 2n )+

∗ M ⊗n ≃ M
we have that πX
Wn .
A

From now on given a sheaf F on X, we will often write H 0 (F ) instead of H 0 (X, F ).
We will split the proof in two steps.
Step 1: Reduction to an M -regularity problem
Consider the vanishing locus V 1 (MWn ⊗ A ⊗2 ). We claim that it coincides with the
b such that the multiplication map
locus of [α] ∈ X
0
⊗2n +
m+
) ⊗ H 0 (A ⊗2 ⊗ α) −→ H 0 (A ⊗2(n+1) ⊗ α)
α : H (A

SYZYGIES AND EQUATIONS OF KUMMER VARIETIES

11

is not surjective. In fact, from the short exact sequence
0 → MWn ⊗ A ⊗2 ⊗ α −→ Wn ⊗ A ⊗2 ⊗ α −→ A ⊗2(n+1) ⊗ α → 0.
taking cohomology we deduce that the surjectivity fo m∗α is equivalent to the vanishing of
H 1 (MWn ⊗ A ⊗2 ⊗ α). Thanks to this characterization of the locus on Pic0 (X) where m+
α
fails to be surjective we were able to prove the following
Lemma 4.1. Let A and E be an ample symmetric sheaf on an abelian variety X and a
coherent sheaf on X, respectively. If E ⊗ A ⊗−2 is M -regular, then the multiplication map
H 0 (X, A ⊗2n )+ ⊗ H 0 (X, E ) −→ H 0 (X, A ⊗2n ⊗ E )

(4.3)

is surjective for every n ≥ 1.
Before proceeding with the proof we will state an immediate corollary of this Lemma,
that reduces our problem to a M -regularity problem.
⊗p
Corollary 4.2. If MW
⊗ A ⊗2(nh−1) is M -regular, then (4.2) is surjective.
n

Proof of the Lemma. By Proposition 3.6(1) we know that V 1 (MWn ⊗A ⊗2 ) a proper closed
b Therefore there exists an open set U
b0 ⊆ Pic0 (X) such that U
b0 ∩ V 1 (MWn ⊗
subset of X.
⊗2
A ) = ∅. Now observe the following commutative diagram.
M
H 0 (A ⊗2n )+ ⊗ H 0 (A ⊗2 ⊗ α) ⊗ H 0 (E ⊗ A ⊗−2 ⊗ α∨ )
b0
α∈U

❲❲❲❲❲
❲❲❲❲❲
❲❲❲❲❲
++

H 0 (A ⊗2n )+ ⊗ H 0 (E )

f

M

b0
α∈U


0

H (A

⊗2n+2

⊗ α) ⊗ H 0 (E ⊗ A ⊗−2 ⊗ α∨ )
❲❲h❲❲❲
❲❲❲❲❲
❲❲❲❲❲
++

g



H 0 (E ⊗ A ⊗2n )

b
The map f = ⊕m+
α is surjective by our choice of the set U0 , the map h is surjective by
M -regularity hypothesis together with Theorem 2.3. Thus g is necessarily surjective.

⊗p
Step 3: Conclusions. By Corollary 4.2 we are reduced to prove that MW
⊗ A ⊗2(nh−1)
n
is M -regular for every h ≥ r. This is proved in the next two statements.
⊗p
Proposition 4.3. Let p be a positive integer. Then MW
⊗ A ⊗m satisfies I.T. with index
n
0 (and hence it is M -regular) for every m ≥ 2p + 1

Note that Theorem A follows at once from this Proposition taking m = 2nr − 2.
Proof. We will proceede by induction on p.
Case p = 1. Let us consider the following exact sequence:
(4.4)

0 → MWn ⊗ A ⊗m ⊗ α −→ Wn ⊗ A ⊗m ⊗ α −→ A ⊗2n+m ⊗ α → 0.

12

SOFIA TIRABASSI

One can easily see that the vanishing of the higher cohomology of MWn ⊗ A ⊗m ⊗ α is
implied by:
(i) the vanishing of the higher cohomology of A ⊗m ⊗ α and
(ii) the surjectivity of the following multiplication map:
H 0 (X, A ⊗2n )+ ⊗ H 0 (X, A ⊗m ⊗ α) −→ H 0 (X, A ⊗2n+m ⊗ α).
Condition (i) holds for every α as long as m ≥ 1, while, thanks to Khaled result (see
Proposition 3.1), we know condition (ii) is satisfied for every α as long as m ≥ 3.
b By twisting (4.4) by M ⊗p−1
Case p > 1. Suppose now that p > 1 and take any α ∈ X.
Wn
⊗p
⊗m ⊗ α is implied by
we can observe that the vanishing of higher cohomology of MW
⊗
A
n
⊗p−1
(i) the vanishing of the higher cohomology of MW
⊗ A ⊗m ⊗ α and
n
(ii) the surjectivity of the following multiplication map:

⊗p−1
⊗p−1
(4.5) H 0 (X, A ⊗2n )+ ⊗ H 0 (X, MW
⊗ A ⊗m ⊗ α) −→ H 0 (X, MW
⊗ A ⊗2n+m ⊗ α).
n
n

By induction (i) holds as long as m ≥ 2p − 1. Thanks to Lemma 4.2 and Lemma 2.3 we
⊗p−1
know that if MW
⊗ A ⊗m−2 ⊗ α satisfies I.T. with index 0, then (4.5) is surjective. But
n
⊗p−1
⊗ A ⊗m−2 ⊗ α is I.T. with index 0 whenever
we use induction again and we get that MW
n
m − 2 ≥ 2p − 1, that is whenever m ≥ 2p + 1 and hence the statement is proved.

Proposition 4.4. In the notations above, take p ≥ 1 an integer. If A does not have a
⊗p
base divisor, then MW
⊗ A ⊗m is M -regular for every m ≥ 2p.
n
Again Theorem C, follows at once after taking m = 2(nr − 1).
Proof. For m ≥ 2p + 1 the statement is a direct consequence of the Proposition above,
hence we can limit ourselves to the case m = 2p. We will proceede by induction on p.
Case p = 1: We want to prove that codim V i (MWn ⊗ A ⊗2 ) > i for every i ≥ 1. From
the vanishing of the higher cohomology of A ⊗2 ⊗ α for every α ∈ Pic0 (X) we know that
the loci
V i (MWn ⊗ A ⊗2 ) = ∅ for every i ≥ 2.
b such that the multiRecall that the locus V 1 (MWn ⊗ A ⊗2 ) is the locus of points α ∈ X
plication
Wn ⊗ H 0 (A ⊗2 ⊗ α) −→ H 0 (A⊗2n+2 ⊗ α)
is not surjective. We know from Proposition 3.6 that, if A has not a base divisor, then
this locus has at least codimension 2 and hence the statement is proved.
b Consider the following exact sequence
Case p > 1 Take α ∈ X.

(4.6)

⊗p
⊗p−1
⊗p−1
0 → MW
⊗ A ⊗2p ⊗ α −→ Wn ⊗ MW
⊗ A ⊗2p ⊗ α −→ MW
⊗ A ⊗2p+n ⊗ α → 0
n
n
n

⊗p−1
From Proposition 4.3(a) we know that for every i ≥ 1 both H i (MW
⊗ A ⊗2p ⊗ α) and
n
⊗p−1
⊗p
H i (MW
⊗ A ⊗2p+n ⊗ α) vanish. Thus the loci V i (MW
⊗ A ⊗2p ) are empty for every
n
n
i ≥ 2. It remains to show that that
⊗p
codim V 1 (MW
⊗ A ⊗2p ) ≥ 2.
n

SYZYGIES AND EQUATIONS OF KUMMER VARIETIES

13

b where the following
As before one may observe that this locus is exactly the locus in X
multiplication map fails to be surjective:
⊗p−1
⊗p−1
Wn ⊗ H 0 (MW
⊗ A ⊗2p ⊗ α) −→ H 0 (MW
⊗ A ⊗2p+2n ⊗ α).
n
n

b h1 (M ⊗p−1 ⊗
In fact, taking cohomology in (4.6) and observing that for every [α] ∈ X,
Wn
A ⊗2p ⊗α) = 0, due to Proposition 4.3, we have the conclusion following the same argument
as in the case p = 0.
⊗p−1
⊗p
⊗A ⊗2(p−1)
⊗A ⊗2p ). By inductive hypothesis the sheaf MW
Now take [α] ∈ V 1 (MW
n
n
is M -regular. Corollary 2.3 implies that there exist a positive integer N and [β1 ], . . . , [βN ] ∈
b such that the following is surjective.
X
N
M

⊗p−1
⊗ A ⊗2(p−1) ⊗ βk∨ )
H 0 (A ⊗2n+2 ⊗ βk ⊗ α) ⊗ H 0 (MW
n

mβ k

// H 0 (M ⊗p−1 ⊗ A ⊗2p+2n ⊗ α)
Wn

k=1

Consider the commutative square
N
M
⊗p−1
⊗ A ⊗2p−2 ⊗ βk∨ )
H 0 (A ⊗2n )+ ⊗ H 0 (A ⊗2 ⊗ α ⊗ βk ) ⊗ H 0 (MW
n
k=1

❲❲❲❲❲
❲❲❲❲❲
❲❲❲❲❲
❲❲❲❲❲
❲++

⊗p−1
⊗ A ⊗2p ⊗ α)
H 0 (A ⊗2n )+ ⊗ H 0 (MW
n



N
M
⊗p−1
⊗ A ⊗2(p−1) ⊗ βk∨ )
H 0 (A ⊗2n+2 ⊗ α ⊗ βk ) ⊗ H 0 (MW
n
k=1

❲❲❲❲❲
❲❲❲❲❲
❲❲❲❲❲
❲❲❲❲❲
❲++



⊗p−1
⊗ A ⊗2n+2p )
H 0 (MW
n

The right arrow is not surjective by our choice of α. The bottom arrow is surjective,
hence the left arrow could not be surjective. Therefore
α∈

N
[

Zk ,

k=1

b such that the multiplication map
where Zk stands for the locus of [β] ∈ X
(4.7)

H 0 (A ⊗2n )+ ⊗ H 0 (A ⊗2 ⊗ β ⊗ βk ) → H 0 (A ⊗2n+2 ⊗ β ⊗ βk )

fails to be surjective. Thus one has that

⊗p
V 1 (MW
⊗ A ⊗2p ) ⊆
n

N
[

Zk .

k=1

By Theorem 3.6(2) the loci Zk have codimension at least 2, therefore
⊗p
codim V 1 (MW
⊗ A ⊗2p ) ≥ codim
n

N
[

Zk ≥ 2

k=1

and the statement is proved.



14

SOFIA TIRABASSI

Acknowledgments. This is part of my Ph. D thesis defended at the Mathmatics department of Università degli studi Roma TRE. I would like to thank my advisor G. Pareschi
for introducing me to the subject of syzygies and for many suggestions. I am very indebted
to C. Ciliberto and F. Viviani for many engaging mathematical conversations.
Part of this work was carried out during the tenure of an ERCIM ”Alain Bensoussan”
Fellowship Programme. This Programme is supported by the Marie Curie Cofunding of
Regional, National and International Programmes (COFUND) of the European Commission.
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Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa (Poland)
E-mail address: tirabassi@mimuw.edu.pl

