On the existence of ramified abelian covers

0.14 MB·English

by  Valery Alexeev

ON THE EXISTENCE OF RAMIFIED ABELIAN COVERS
2 VALERYALEXEEVANDRITAPARDINI
1
0
Abstract. GivenanormalcompletevarietyY,distincteffectiveWeildivisors
2
D1,...Dn ofY andpositiveintegersd1,...dn,wespellouttheconditionsfor
t the existence of an abelian cover X → Y branched with order di on Di for
c
i=1,...n.
O
Asanapplication,weprovethataGaloiscoverofanormalcompletetoric
3 varietybranchedonthetorus-invariantdivisorsisitselfatoricvariety.
2 2010 Mathematics Subject Classification: 14E20,14L30.

]
G
Dedicated to Alberto Conte on his 70th birthday.
A
. Contents
h
t
a 1. Introduction 1
m 2. Abelian covers 2
[ 3. Toric covers 5

References 8
1
v
4
7
1. Introduction
1
6
.
0 GivenaprojectivevarietyY andeffectivedivisorsD1,...DnofY,decidingwhether
1 there exists a Galois cover branched on D ,...D with given multiplicities is a
1 n
2 very complicated question, which in the complex case is essentially equivalent to
1
describing the finite quotients of the fundamental group of Y \(D ∪···∪D ).
: 1 n
v In Section 2 of this paper we answer this question for a normal variety Y in the
i case that the Galois group of the cover is abelian (Theorem 2.1), using the theory
X
developedin[Par91]and[AP12]. Inparticular,weprovethatwhentheclassgroup
r
a Cl(Y) is torsion free, every abelian cover of Y branched on D1,...Dn with given
multiplicities is the quotient of a maximal such cover, unique up to isomorphism.
In Section 3 we analyze the same question using toric geometry in the case
when Y is a normal complete toric variety and D ,...D are invariant divisors
1 n
and obtain results that parallel those in Section 2 (Theorem 3.5). Combining the
two approacheswe are able to show that a Galois coverof a normalcomplete toric
variety branched on the invariant divisors is toric (Theorem 3.6).
Acknowledgments. WewishtothankAngeloVistoliforusefuldiscussionsonthe
topic of this paper (cf. Remark 3.7).
Notation. G always denotes a finite group, almost always abelian, and G∗ :=
Hom(G,K)∗ the group of characters; o(g) is the order of the element g ∈ G and
|H| is the cardinality of a subgroup H <G. We work over an algebraically closed
field K whose characteristic does not divide the order of the finite abelian groups
1
2 VALERYALEXEEVANDRITAPARDINI
we consider.
If A is an abelian group we write A[d] := {a ∈ A | da = 0} (d an integer),
A∨ :=Hom(A,Z) and we denote by Tors(A) the torsion subgroup of A.
The smooth part of a variety Y is denoted by Y . The symbol ≡ denotes linear
sm
equivalence of divisors. If Y is a normal variety we denote by Cl(Y) the group of
classes, namely the group of Weil divisors up to linear equivalence.
2. Abelian covers
2.1. The fundamental relations. We quickly recallthe theory of abelian covers
(cf. [Par91],[AP12],andalso[PT95])inthemostsuitableformfortheapplications
considered here.
There are slightly different definitions of abelian covers in the literature (see,
for instance, [AP12] that treats also the non-normal case). Here we restrict our
attention to the case of normal varieties, but we do not require that the covering
mapbeflat,hencewedefineanabeliancoverasafiniteGaloismorphismπ: X →Y
of normal varieties with abelian Galois group G (π is also called a “G-cover”).
Recall that, as already stated in the Notations, throughout all the paper we
assume that G has order not divisible by charK.
To every component D of the branch locus of π we associate the pair (H,ψ),
where H < G is the cyclic subgroup consisting of the elements of G that fix the
preimage of D pointwise (the inertia subgroup of D) and ψ is the element of the
character group H∗ given by the natural representation of H on the normal space
to the preimage of D at a generalpoint (these definitions are well posed since G is
abelian). It can be shown that ψ generates the group H∗.
If we fix a d-th root ζ of 1, where d is the exponent of the group G, then a pair
d
(H,ψ) as above is determined by the generator g ∈H such that ψ(g)=ζo(g). We
follow this convention and attach to every component D of the branch locus of π
i
a nonzero element g ∈G.
i
If π is flat, which is always the case when Y is smooth, the sheaf π O decom-
∗ X
poses under the G-action as ⊕χ∈G∗L−χ1, where the Lχ are line bundles (L1 =OY)
and G acts on L−1 via the character χ.
χ
Given χ∈G∗ and g ∈G, we denote by χ(g) the smallestnon-negativeinteger a
ad
such that χ(g)=ζo(g). The main result of [Par91] is that the Lχ, Di (the building
data of π) satisfy the following fundamental relations:
(2.1) Lχ+Lχ′ ≡Lχ+χ′ +XDi ∀χ,χ′ ∈G∗
i
where εi = ⌊χ(gi)+χ′(gi)⌋. (Notice that the coefficients εi are equal either to
χ,χ′ o(gi) χ,χ′
0 or to 1). Conversely, distinct irreducible divisors D and line bundles L satisfy-
i χ
ing (2.1) are the building data of a flat (normal) G-cover X → Y; in addition, if
h0(O )=1 then X →Y is uniquely determined up to isomorphism of G-covers.
Y
Ifwefixcharactersχ ,...χ ∈G∗suchthatG∗isthedirectsumofthesubgroups
1 r
generated by the χ , and we set L :=L , m :=o(χ ), then the solutions of the
j j χj j j
fundamentalrelations(2.1)areinone-onecorrespondencewiththe solutionsofthe
ON THE EXISTENCE OF RAMIFIED ABELIAN COVERS 3
following reduced fundamental relations:
m χ (g )
j j i
(2.2) mjLj ≡X Di, i=1,...r
d
i
i
Notice that if Pic(Y)[d] = 0, then for fixed (D ,g ), i = 1,...n, the solution of
i i
(2.2) is unique, hence the branch data (D ,g ) determine the cover.
i i
In order to deal with the case when Y is normal but not smooth, we observe
thatthecomplementY \Y ofthe smoothparthascodimension>1;soanycover
sm
X → Y can be recovered from its restriction X′ → Y to the smooth locus, by
sm
taking the integral closure of Y in the extension in K(X′) ⊃K(Y). In addition, if
h0(O )=1,thenwehavealsoh0(O )=1andthecoverX′ →Y isdetermined
Y Ysm sm
by the building data L ,D . Using the identification Pic(Y )=Cl(Y )=Cl(Y),
χ i sm sm
we can regardthe L as elements of Cl(Y) and, taking the closure, the D as Weil
χ i
divisorsonY,andwecaninterpretthefundamentalrelationsasequalitiesinCl(Y).
In this sense, if Y is normal variety with h0(O ) = 1, then the G-covers X → Y
Y
are determined by the building data up to isomorphism.
2.2. The maximal cover. Let Y be a complete normal variety, let D ,...D be
1 n
irreducible effective divisors of Y and let d ,...d be positive integers (it is conve-
1 n
nient to allow the possibility that d =1 for some i). We set d:=lcm(d ,...d ).
i 1 n
We say that an abelian cover π: X → Y is branched on D ,...D with orders
1 n
d ,...d if:
1 n
• π istotallyramified,i.e. theinertiasubgroupsofthedivisorialcomponents
of the branch locus of π generate G (equivalently, π does not factorize
through a cover X′ →Y that is ´etale over Y );
sm
• the divisorial part of the branch locus of π is contained in D ;
Pi i
• the ramification order of π over D is equal to d .
i i
Letη: Y →Y bearesolutionofthesingularitiesandsetN(Y):=Cl(Y)/η Pic0(Y).
∗
e e
Sincethe mapη : Pic(Y)=Cl(Y)→Cl(Y)issurjective,N(Y)isaquotientofthe
∗
N´eron-SeverigroupNS(eY), hencee it is finitely generated. It followsthat η Pic0(Y)
∗
e e
is the largest divisible subgroup of Cl(Y) and therefore N(Y) does not depend on
the choice of the resolution of Y (this is easily checked also by a geometrical ar-
gument). The group Cl(Y)∨ coincides with N(Y)∨, hence it is a finitely generated
free abelian group of rank equal to the rank of N(Y).
Consider the map Zn → Cl(Y) that maps the i-th canonical generator to the
class of D , let φ: Cl(Y)∨ →⊕n Z be the map obtained by composing the dual
i i=1 di
mapCl(Y)∨ →⊕(Zn)∨ with (Zn)∨ =Zn →⊕n Z and let K be the image of
i=1 di min
φ. Let G be the abelian group defined by the exact sequence:
max
(2.3) 0→K →⊕n Z →G →0.
min i=1 di max
Then we have the following:
Theorem 2.1. Let Y be a normal variety with h0(O ) = 1, let D ,...D be
Y 1 n
distinct irreducible effective divisors, let d ,...d be positive integers and set d :=
1 n
lcm(d ,...d ). Then:
1 n
(1) If X →Y is a G-cover branched on D ,...D with orders d ,...d , then:
1 n 1 n
(a) themap ⊕n Z →Gthat maps 1∈Z tog descends toasurjection
i=1 di di i
G ։G;
max
4 VALERYALEXEEVANDRITAPARDINI
(b) the map Z →G is injective for every i=1,...n.
di max
(2) If the map Z → G is injective for i = 1,...n and N(Y)[d] = 0, then
di max
there exists a maximal abelian cover X → Y with Galois group G
max max
branched on D ,...D with orders d ,...d .
1 n 1 n
(3) If the map Z → G is injective for i = 1,...n and Cl(Y)[d] = 0,
di max
then the cover X →Y is unique up to isomorphism of G -covers and
max max
every abelian cover X →Y branched on D ,...D with orders d ,...d is
1 n 1 n
a quotient of X by a subgroup of G .
max max
Proof. Let H ,...H ∈ N(Y) be elements whose classes are free generators of the
1 t
abelian group N(Y)/Tors(N(Y)), and write:
t
(2.4) Di =XaijHj mod Tors(N(Y)), j =1,...t
j=1
Hence,thesubgroupK of⊕n Z isgeneratedbytheelementsz :=(a ,...a ),
min i=1 di j 1j nj
for j =1,...t.
Let X → Y be a G-cover branched on D ,...D with orders d ,...d and let
1 n 1 n
(D ,g ) be its branch data. Consider the map ⊕n Z → G that maps 1 ∈ Z
i i i=1 di di
to g : this map is surjective, by the assumption that X → Y is totally ramified,
i
and its restriction to Z is injective for i = 1,...n, since the cover is branched
di
on D with order d . If we denote by K the kernel of ⊕n Z → G, to prove
i i i=1 di
(1) it suffices to show that K ⊇ K . Dually, this is equivalent to showing that
min
G∗ ⊆ K⊥ ⊂ ⊕n (Z )∗. Let ψ ∈ (Z )∗ be the generator that maps 1 ∈ Z
min i=1 di i di di
to ζddi and write χ ∈ G∗ as (ψ1b1,...ψnbn), with 0 ≤ bi < di; if o(χ) = m then
(2.2) gives mLχ ≡ Pni=1 mdbiiDi. Plugging (2.4) in this equation we obtain that
n biaij is an integer for j =1,...t, namely χ∈K⊥ .
Pi=1 di min
(2) Let χ ,...χ be a basis of G∗ and, as above, for s = 1,...r write χ =
1 r s
(ψ1bs1,...ψnbsn), with 0 ≤ bsi < di. Since by assumption N(Y)[d] = 0, by the proof
of (1) the elements t ( n bsiaij)H , s = 1,...r, can be lifted to solutions
Pj=1 Pi=1 di j
L ∈N(Y)ofthereducedfundamentalrelations(2.2)foraG -coverwithbranch
s max
data (D ,g ), where g ∈ G is the image of 1 ∈ Z . Since the group Pic0(Y) is
i i i di
divisible, it is possible to lift the L to solutions L ∈Cl(Y). We let X →Y be
s s max
the G -cover determined by these solutions.
max
(3) Since Cl(Y)[d] = 0, any G-cover such that the exponent of G is a divisor
of d is determined uniquely by the branch data; in particular, this holds for the
cover X → Y in (2) and for every intermediate cover X /H → Y, where
max max
H <G . The claim now follows by (1). (cid:3)
max
Example 2.1. Take Y = Pn−1 and let D ,...D be the coordinate hyperplanes.
1 n
InthiscasethegroupK isgeneratedby(1,...1)∈⊕n Z ,hencebyTheorem2.1
i=1 di
thereexistsanabeliancoverofPn−1 branchedoverD ,...D withordersd ,...d
1 n 1 n
iff d divides lcm(d ,...,d ,...d ) for every i = 1,...n. For d = ··· = d = d,
i 1 i n 1 n
thenG =Zn/<(1,...1b)>andX →Pn−1 is the coverPn−1 →Pn−1 defined
max d max
by [x ,...x ]7→[xd,...xd].
1 n 1 n
In general, X is a weighted projective space P( d ,... d ) and the cover is
max d1 dn
given by [x1,...xn]7→[xd11,...xdnn].
ON THE EXISTENCE OF RAMIFIED ABELIAN COVERS 5
3. Toric covers
Notations 3.1. Here, we fix the notations which are standard in toric geometry.
A (complete normal) toric variety Y corresponds to a fan Σ living in the vector
space N ⊗R, where N ∼= Zr. The dual lattice is M = N∨. The torus is T =
N ⊗C∗ =Hom(M,C∗).
The integral vectors r ∈ N will denote the integral generators of the rays σ ∈
i i
Σ(1) of the fan Σ. They are in a bijection with the T-invariant Weil divisors D
i
(i=1,...,n) on Y.
Definition 3.2. A toric cover f: X → Y is a finite morphism of toric varieties
corresponding to the map of fans F: (N′,Σ′)→(N,Σ) such that:
(1) N′ ⊆N is a sublattice of finite index, so that N′⊗R=N ⊗R.
(2) Σ′ =Σ.
The proof of the following lemma is immediate.
Lemma 3.3. The morphism f has the following properties:
(1) It is equivariant with respect to the homomorphism of tori T′ →T.
(2) It is an abelian cover with Galois group G=ker(T′ →T)=N/N′.
(3) It is ramified only along theboundary divisors D , with multiplicities d ≥1
i i
defined by the condition that the integral generator of N′∩R r is d r .
≥0 i i i
Proposition 3.4. Let Y be a complete toric variety such that Cl(Y) is torsion
free, and X → Y be a toric cover. Then, with notations as above, there exists the
following commutative diagram with exact rows and columns.
0 0
(cid:15)(cid:15) (cid:15)(cid:15)
Cl(Y)∨ // K
(cid:15)(cid:15) (cid:15)(cid:15)
0 //⊕n Zd D∗ // ⊕n ZD∗ //⊕n Z //0
i=1 i i i=1 i i=1 di
p′ p
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //N′ // N //G //0
(cid:15)(cid:15) (cid:15)(cid:15)
0 0
(Here D∗ are formal symbols denoting a basis of Zn, resp. ⊕n Z ). Moreover,
i i=1 di
each of the homomorphisms Z →G is an embedding.
di
Proof. The third row appeared in Lemma 3.3, and the second row is the obvious
one.
It is well known that the boundary divisors on a complete normal toric variety
span the group Cl(Y), and that there exists the following short exact sequence of
lattices:
0→M −→⊕n ZD −→Cl(Y)→0.
i=1 i
6 VALERYALEXEEVANDRITAPARDINI
Since Cl(Y) is torsion free by assumption, this sequence is split and dualizing it
one obtains the central column. Since ⊕n ZD∗ → N is surjective, then so is
i=1 i
⊕n Z →G. The group K is defined as the kernel of this map.
i=1 di
Finally, the condition that Z → G is injective is equivalent to the condition
di
that the integral generator of N′∩R r is d r , which holds by Lemma 3.3. (cid:3)
≥0 i i i
Theorem 3.5. Let Y be a complete toric variety such that Cl(Y) is torsion free,
let d ,...d be positive integers and let K and G be defined as in sequence
1 n min max
(2.3). Then:
(1) There exists a toric cover branched on D of order d , i = 1,...n, iff the
i i
map Z →G is injective for i=1,...n.
di max
(2) If condition (1) is satisfied, then among all the toric covers of Y rami-
fied over the divisors D with multiplicities d there exists a maximal one
i i
X → Y, with Galois group G , such that any other toric cover
Tmax max
X → Y with the same branching orders is a quotient X = X /H by a
Tmax
subgroup H <G .
max
Proof. Let X → Y be a toric cover branched on D ,...D with orders d ,...d ,
1 n 1 n
let N′ be the corresponding sublattice of N and G=N/N′ the Galois group. Let
N′ be the subgroup of N generated by d r , i = 1,...n. Then one must have
min i i
N′ ⊆ N′, hence the map Z → N/N′ is injective since Z → G = N/N′ is
min di min di
injective by Proposition3.4. We set X →Y to be the coverfor N′ . Clearly,
Tmax min
the cover for the lattice N′ is a quotient of the cover for the lattice N′ by the
min
group H =N′/N′ .
min
Consider the secondand third rows of the diagramof Proposition3.4 as a short
exact sequence of 2-step complexes 0→A• →B• →C• →0. The associated long
exact sequence of cohomologies gives
Cl(Y)∨ // K //coker(p′) // 0
For N′ =N′ , the map p′ is surjective, hence Cl(Y)∨ →K is surjective too, and
min
K =K , N/N′ =G .
min min max
Vice versa, suppose that in the following commutative diagram with exact row
and columns each of the maps Z →G is injective.
di
0 0
(cid:15)(cid:15) (cid:15)(cid:15)
Cl(Y)∨ q //K //0
min
(cid:15)(cid:15) (cid:15)(cid:15)
0 //⊕n Zd D∗ // ⊕n ZD∗ //⊕n Z //0
i=1 i i i=1 i i=1 di
p
(cid:15)(cid:15) (cid:15)(cid:15)
N G
max
(cid:15)(cid:15) (cid:15)(cid:15)
0 0
We complete the firstrowon the left by adding ker(q). We haveaninduced homo-
morphism ker(q)→⊕Zd D∗, and we define N′ to be its cokernel.
i i
ON THE EXISTENCE OF RAMIFIED ABELIAN COVERS 7
Now consider the completed first and second rows as a short exact sequence of
2-step complexes 0→A• →B• →C• →0. The associated long exact sequence of
cohomologies says that ker(q)→⊕n Zd D∗ is injective, and the sequence
i=1 i i
0 //N′ // N //G // 0
max
is exact. It follows that N′ = N′ and the toric morphism (N′ ,Σ)→(N,Σ) is
min min
then the searched-for maximal abelian toric cover. (cid:3)
Wenowcombinetheresultsofthissectionwiththoseof§2toobtainastructure
result for Galois covers of toric varieties.
Theorem 3.6. Let Y be a normal complete toric variety and let X → Y be a
Galois cover with group G branched on the invariant divisors D ,...D . Then G
1 n
is abelian and X →Y is a toric cover.
Proof. Let U ⊂ Y be the open orbit and let X′ → U be the cover obtained by
restriction. By [Mi, Prop. 1], X′ →Y is, up to isomorphism, a homomorphism of
tori (recall that |G| is not divisible by charK by assumption). It follows that the
Galois group G is abelian.
Let d ,...d be the orders of ramification of X →Y on D ,...D .
1 n 1 n
Assume first that Cl(Y) has no torsion. (Notice that in this case any abelian
coverof Y is totally ramified). Then by Theorem2.1 every abeliancoverbranched
on D ,...D with orders d ,...d is a quotient of the maximal abelian cover
1 n 1 n
X → Y by a subgroup H < G . In particular, this is true for the cover
max max
X →Y ofTheorem3.5. SinceX andX havethesameGaloisgroupit
Tmax max Tmax
follows that X = X . Hence X → Y, being a quotient of X , is a toric
max Tmax Tmax
cover.
Consider now the general case. The group TorsCl(Y) is finite, and isomorphic
to N/hr i. The cover Y′ → Y corresponding to TorsCl(Y) is toric, and one has
i
TorsCl(Y′)=0.
Indeed, on a toric variety the group Cl(Y) is generated by the T-invariant Weil
divisors D . Thus, Cl(Y) is the quotient of the free abelian group ⊕ZD of all T-
i i
invariant divisors modulo the subgroup M of principal T-invariantdivisors. Thus,
TorsCl(Y)≃M′/M,whereM ⊂M′ ⊂⊕QD isthesubgroupofQ-linearfunctions
i
onN takingintegralvaluesonthevectorsr definedin3.1. ThenN′ :=M′∨ isthe
i
subgroupofN generatedbyther ,andthecoverY′ →Y isthecovercorresponding
i
to the mapoffans (N′,Σ)→(N,Σ). OnY′ one has N′ =hr i, so TorsCl(Y′)=0.
i
LetX′ →Y′ beaconnectedcomponentthepullbackofX →Y: itisanabelian
cover branched on the invariant divisors of Y′, hence by the first part of the proof
it is toric. The map X′ →Y is toric, since it is a composition of toric morphisms,
hence the intermediate cover X →Y is also toric. (cid:3)
Remark 3.7. The argument that shows that the Galois group is abelian in the
proofofTheorem3.6wassuggestedtousbyAngeloVistoli. Healsoremarkedthat
it is possible to prove Theorem 3.6 in a more conceptual way by showing that the
torus action on the cover X′ →U of the open orbit of Y extends to X, in view of
the properties of the integral closure. However our approach has the advantage of
describing explicitly the fan/building data associated with the cover.
8 VALERYALEXEEVANDRITAPARDINI
References
[AP12] V. Alexeev, R. Pardini, Non-normal abelian covers, Compositio Math. 148 (2012),
no.04,1051–1084.
[Mi] M.Miyanishi,Onthealgebraicfundamentalgroupofanalgebraicgroup,J.Math.Kyoto
Univ.12(1972), 351–367.
[Par91] R. Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991),
191–213.
[PT95] R. Pardini, F. Tovena, On the fundamental group of an abelian cover, International J.
Math.6no.5(1995), 767–789.