Projective varieties covered by isotrivial families

0.11 MB·English

by  Anupam Bhatnagar

PROJECTIVE VARIETIES COVERED BY TRIVIAL
FAMILIES
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1 ANUPAMBHATNAGAR
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Abstract. Letkbeafieldofcharacteristiczero,Casmoothprojective
t
c curve defined over k. Let X,Y be projective schemes over C, with Y
O reduced and g : X → Y a morphism defined over C such that g# :
OY →g∗OX is an isomorphism. If X →C is a trivial family, then the
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generic fiberof Y →C is isotrivial.
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]
G
A
1. Introduction
.
h
Letk beafieldof characteristic zero, C asmoothprojective curvedefined
t
a over k with function field F.
m
[ Definition 1.1. Let S be a scheme over k and π : X → S a flat family of

schemes. Then
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v (a) π is called trivial if thereexists a schemeX definedover k such that
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1 X ∼= X × S.
6 0 k
(b) π is called isotrivial if there exists a finite unramified cover S′ → S
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5 such that πS′ :X ×S S′ → S′ is trivial.
.
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Let X,Y be projective schemes over C, with Y reduced and g : X → Y
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1 a morphism defined over F such that g# : OY → g∗OX is an isomorphism.
1 We show that if the family X → C is trivial then the generic fiber of the
v: family Y → C is isotrivial. (cf. Thm. 2.7)
i Sketch of proof : The deformations of Y are controlled by its differentials.
X
We study the deformations locally i.e. over a discrete valuation ring R on
r
a the curve C, and show that the fundamental exact sequence associated to
p
the diagram X → Y → Spec(R) → Spec(k)
(1) 0 → p∗Ω → Ω → Ω → 0
Spec(R)/Spec(k) Y/Spec(k) Y/Spec(R)
is split exact and consequently the deformation of Y is governed by that
of X. Next we consider an infinitesimal deformation of Y → Spec(R) over
the henselization of R (denoted by R˜) and show that the sequence above
remains split exact at every level of the deformation. Finally we use a result
of M. Greenberg to pass from R˜ to R.
Date: October 27, 2011. 2010 Mathematics Subject Classification: 14D15, 13D10,
37P55. Keywords and Phrases: Deformation Theory, Algebraic Dynamics, Isotrivial
families.
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2 ANUPAMBHATNAGAR
We finishthearticle by givingan examplewheretheresultcan beapplied
in the setting of algebraic dynamics. As an application we shall use this
result in a forthcoming paper with Lucien Szpiro, on parametrization of
points of canonical height zero of an algebraic dynamical system.
Notation. Throughout this paper k denotes a field of characteristic zero,
C a smooth projective curve over k with function field F. Given a scheme
X over C we denote its generic fiber by X .
F
Acknowledgements. I thank Lucien Szpiro for introducing me to the sub-
ject of algebraic dynamics and for sharing this question during my PhD.
Many thanks to Raymond Hoobler for several interesting conversations to-
wardsthispaper. ThankstoMadhavNoriforsuggestionstowardsimplifying
the proof of Proposition 2.1.
2. Main Result
Remark: In this section we assume that k is algebraically closed. The
proofs work without this hypothesis with some minor modifications. For
ease of notation we denote Spec(R) and Spec(k) by R and k respectively in
the sheaves of differentials.
Proposition 2.1. Let X,Y be projective schemes over C with Y reduced,
X → C a trivial family, and g : X → Y a C-morphism such that g# :O →
X
g∗OY is an isomorphism. Let U be a nonempty open subset of C such that
Y is flat over U and for Q ∈ U, let R be the local ring at Q. Let
g p
X → Y → Spec(R) → Spec(k)
be morphisms of schemes, then the associated sequence of differentials
(2) p∗Ω → Ω → Ω → 0
R/k Y/k Y/R
is split exact.
Proof. X → C is a trivial family therefore the sequence of differentials on
X
(3) 0 → g∗p∗Ω → Ω → Ω → 0
R/k X/k X/R
is split exact. Since pullbacks preserve right exactness, the sequence of
differentials on Y pulled back to X along g
(4) g∗p∗Ω → g∗Ω → g∗Ω → 0
R/k Y/k Y/R
is exact. g : X → Y induces morphisms between the above two exact
sequences, yielding the following commutative diagram:
(5) g∗p∗ΩR/k // g∗ΩY/k // g∗ΩY/R // 0
q
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // g∗p∗ΩR/k // ΩX/k // ΩX/R // 0
PROJECTIVE VARIETIES COVERED BY TRIVIAL FAMILIES 3
We now show that q is the identity map. Since differentials contain local
information it suffices to check the commutativity of the left square locally.
Let S ⊂ X,T ⊂ Y be open affine subsets and A = Γ(S,O ) and B =
X
Γ(T,O ). From the k-algebra homomorphisms k → R → A and k → R →
Y
B we get a A-module homomorphism α : Ω ⊗ A → Ω and a B-
R/k R A/k
module homomorphism β : Ω ⊗ B → Ω . The commutativity of
R/k R B/k
the left square is equivalent to commutativity of the following diagram (of
A-modules):
Ω ⊗ B⊗ A // Ω ⊗ A
R/k R B B/k B
(cid:15)(cid:15) (cid:15)(cid:15)
Ω ⊗ B⊗ A // Ω
R/k R B A/k
whichisequivalenttothecommutativityoffollowingtriangle(ofR-modules):
Ω // Ω
R/k B/k
FFFFFFFF## (cid:15)(cid:15)
Ω
A/k
and the commutativity of the triangle is clear. Thus q is the identity map.
Since (3) is split exact and q is the identity map, we conclude that (4) is
split exact. Since g∗ preserves direct sums, we have
g∗g∗ΩY/k ∼= g∗g∗ΩY/R⊕g∗g∗p∗ΩR/k
The natural map ΩY/k → g∗g∗ΩY/k induces the following commutative dia-
gram:
p∗ΩR/k // ΩX/k // ΩX/R // 0
g#
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // g∗g∗p∗ΩR/k // g∗g∗ΩY/k // g∗g∗ΩY/R // 0
Note that the bottom row is split exact, p∗ΩR/k ∼= OY and g∗g∗p∗ΩR/k ∼=
g∗OX. By assumption g# : OY → g∗OX is an isomorphism thus the se-
quence (2) is split exact. (cid:3)
We now consider an infinitesimal deformation of Y over the henselian
discretevaluationring,denotedR˜ andproceedtoshowthatthefamilyY →
F
Spec(F) is isotrivial. Before we proceed we need the following definitions:
Definition 2.2. Let X,S be schemes of finite type over k and f : X → S
a morphism of schemes. If f is smooth then the sequence
(6) 0→ f∗Ω → Ω → Ω → 0
S X X/S
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is exact. This extension is non-trivial in general and is given by a class
c∈ Ext1(Ω ,f∗Ω ). Since f∗Ω is locally free, one has
X/S S S
Ext1OX(ΩX/S,f∗ΩS) ∼=Ext1OX(OX,TX/S ⊗f∗ΩS)∼= H1(X,TX/S ⊗f∗ΩS)
The image of c by the canonical map
H1(X,TX/S⊗f∗ΩS) → H0(S,R1f∗(TX/S⊗f∗ΩS)) = H0(S,R1f∗TX/S⊗ΩS)
is called the Kodaira-Spencer class of X/S. One can view this class as a
morphism also i.e. the Kodaira-Spencer morphism
κX/S :TS → R1f∗TX/S
The fiber (κ ) = κ : T → H1(X ,T ) is the Kodaira-Spencer map
X/S s s S,s s Xs
at s ∈S.
The Kodaira-Spencer map at s measures how X deforms in the family
s
X/S in the neigbourhood of s.
Definition 2.3. ([1], pp. 255) A local ring A is henselian if every finite
A-algebra B is a product of local rings. We define the henselization of A
to be a pair (A˜,i), where A˜ is a local henselian ring and i : A → A˜ is
a local homomorphism such that: for any local henselian ring B and any
local homomorphism u : A → B there exists a unique local homomorphism
u˜ :A˜→ B such that u= u˜◦i.
Let R = O , the local ring at Q ∈ U,m its maximal ideal and let R˜
U,Q
denote the henselization of R. Define R = R˜/m˜n+1 for each n ≥ 0. There
n
arenaturalmapsSpec(R˜) → Spec(R)andSpec(Rn−1)→ Spec(Rn)induced
by the projections Rn → Rn−1 for n ≥ 1. Define Y˜ := Y ×R Spec(R˜),
Y := Y˜ × Spec(R ) for each n ≥ 0. We have the following commutative
n R n
diagram of schemes (we do not write Spec in the bottom row):
(7) YF // Y oo Y˜ oo ... oo Yn oo ... oo Y0
p pn p0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
F // R oo R˜ oo ... oo Rn oo ... oo k
Proposition 2.4. For each n ≥ 0 the sequence of differentials associated to
Y → Spec(R ) i.e.
n n
(8) 0 → p∗Ω → Ω → Ω → 0
n Rn/k Yn/k Yn/Rn
is split exact. Moreover, Y → Spec(R ) is trivial.
n n
Proof. Pulling back (2) along the natural map Spec(R )→ Spec(R) we get
n
the sequence (8). Since pullbacks preserve direct sums, the sequence (8)
is split exact. Hence the Kodaira-Spencer class of Y /Spec(R ) is trivial.
n n
In other words, Y → Spec(R ) is trivial. It follows that Y ∼= Y ×
n n n 0 k
Spec(R ) for each n ≥0. (cid:3)
n
PROJECTIVE VARIETIES COVERED BY TRIVIAL FAMILIES 5
Definition 2.5. If V,W andT areS-schemes, an S-isomorphism from V to
W parametrized by T will mean a T-isomorphism from V × T → W × T.
S S
The set of all such isomorphisms will be denoted by Isom (V,W)(T).
S
The association T 7→ Isom (V,W)(T) defines a contravariant functor
S
Isom (V,W) : (S −schemes)◦ → (Sets)
S
The functor Isom (V,W) is representable whenever V,W are flat and pro-
S
jective over S. For a proof of the representability of the Isom functor we
refer the reader to ([2] pp. 132-133). We denote the scheme representing
the functor Isom (V,W) by Isom (V,W).
S S
ToconcludethatthefamilyY → Spec(F)istrivialweneedthefollowing
F
result of Greenberg:
Theorem 2.6. Let R˜ be a henselian discrete valuation ring, with t the
generator of the maximal ideal. Let Z˜ be a scheme of finite type over R˜.
Then Z˜ has a point in R˜ if and only if Z˜ has a point in R˜/tn for every
n≥ 1.
Proof. [3]. (cid:3)
Theorem 2.7. Let X,Y be projective schemes over C, with Y reduced,
X → C a trivial family and g : X → Y a C-morphism such that g# :O →
Y
g∗OX is an isomorphism. Then YF → Spec(F), the generic fiber of Y → C
is isotrivial. In particular,
YF′ ∼= Y0×k Spec(F′)
′
where F is a finite extension of F.
Proof. Observe that Y˜ and Y × Spec(R˜) are flat, projective over Spec(R˜).
0 k
Let Isom (Y˜,Y × Spec(R˜))(T) be the set of isomorphisms from
R˜ 0 k
Y˜ × T → (Y × Spec(R˜))× T
R˜ 0 k R˜
Let Z′ = Isom (Y˜,Y × Spec(R˜)) be the scheme representing the functor
R˜ 0 k
Isom (Y˜,Y × Spec(R˜)). Note that Z′ is of finite type over R˜ and for
R˜ 0 k
Z = Isom (Y,Y ) we have Z′ = Z × Spec(R˜). By Proposition 2.4, Y ∼=
R 0 R n
Y × Spec(R ) for each n ≥ 0. Thus Z′ has a R -point for every n ≥ 0. By
0 k n n
the previous theorem Z′ has a R˜-point i.e. Y˜ ∼= Y × Spec(R˜). Note that
0 k
Spec(R˜) = limSpec(R ) and Z′(Spec(R˜)) = limZ(Spec(R )). Hence there
−→ n ←− n
exists an etale cover R′ of R such that YR′ ∼= Y0×k Spec(R′). Thus F′ (the
quotient field of R′) satisfies the requirements of the theorem. (cid:3)
In summary, we have shown that if X,Y are projective schemes over C
withY reduced,X → C atrivialfamilyandg :X → Y satisfiesOY ∼= g∗OX,
then Y → Spec(F) is isotrivial. Now we extend this result in the realm of
F
algebraic dynamics.
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3. Further Questions
′
Let k,C,F be as in the Introduction and let C be a finite unramified
′
extension of C defined over the field k .
Definition 3.1. Adynamical systemisapair(X,φ)whereX isaprojective
variety and φ :X → X is a non-constant morphism.
Definition 3.2. Let (X,φ) be a dynamical system defined over C i.e. X
and φ are defined over C. We say the pair (X,φ) is trivial if there exists
a dynamical system (X ,φ ) defined over k such that X ∼= X × C and
0 0 0 k
φ= φ × Id . (X,φ) is isotrivial if it is trivial after base change to C′.
0 k C
Question. Let(X,φ) and(Y,ψ) bedynamicalsystems definedover C and
g : X → Y a morphism defined over C such that g◦φ = ψ◦g. If (X,φ) is
trivial, does it imply that (Y,ψ) is isotrivial?
The following example shows that one cannot answer the above question
in the affirmative without additional assumptions.
Example. LetX beanabelianvarietydefinedoverk,φ = [2],thedoubling
morphism on X and Y = X. Let F be a field such that tr.deg (F) = 1,
k
P ∈ X(F) and τ be the translation by P on X. Let g = τ : X → X. We
P p
have the following diagram:
[2]
//
X X
τP τP
(cid:15)(cid:15) (cid:15)(cid:15)
ψ //
X X
Sinceτ isanautomorphism,wecandefineψ := τ ◦[2]◦τ−1. ForQ ∈ X(k),
P P P
ψ(Q) = 2Q−P. Thus ψ cannot be defined over any finite extension of k.
This example illustrates that even though Y is defined over k it is not
necessary that ψ will be defined over some finite extension of k.
References
[1] J.A.Dieudonn´e.Apanoramaofpuremathematics.PureandAppliedMathematics,
97. Academic Press, Inc.,New York,1982.
[2] B. Fantechi, L. Gottsche, L. Illusie, S. Kleiman, N. Nitsure. Fundamental Algebraic
Geometry, Mathematical Surveys and Monographs, 123, American Mathematical
Society,Providence, RI,2005.
[3] M. Greenberg. Rational points in henselian discrete valuation rings. Publ. Math.
I.H.E.S., 31, pp.59-64, 1966.
[4] L. Szpiro. Correspondence with Anupam Bhatnagar, 2010.
Courant Institute of Mathematical Sciences; New York University; 251
Mercer Street, New York, NY 10012, U.S.A.
E-mail address: [email protected]