on space filling curves and albanese varieties

GAFA, Geom. funct. anal.Vol. 11 (2001) 1192 – 12001016-443X/01/0601192-9 $ 1.50+0.20/0

c© Birkhauser Verlag, Basel 2001

GAFA Geometric And Functional Analysis

ON SPACE FILLING CURVES AND ALBANESEVARIETIES

O. Gabber

In section 1 we prove a theorem that can be used to construct smoothhypersurface sections of smooth projective varieties over finite fields passingthrough prescribed points using hypersurfaces of high degree divisible byp and answer questions of N.M. Katz [K]. Recently B. Poonen [P] provedstronger Bertini theorems. In section 2 we give remarks on Albanese va-rieties and explain how the results of section 1 imply that every abelianvariety can be embedded in a Jacobian.

1 A Theorem on Hypersurface Sections

Let p be a prime number and X → S be a morphism of Fp-schemes. If L isa line bundle on X we have a unique connection ∇ : L⊗p → Ω1

X/S ⊗ (L⊗p)which vanishes on local sections of L⊗p that are p-th powers of local sectionsof L.

If E is a vector bundle (i.e. a locally free sheaf of finite rank) on ascheme X and σ ∈ Γ(X,E) we have the zero locus V (σ) locally defined bythe components of σ relative to a trivialization Or E and for every pointx ∈ X we can consider the “value” σ(x) in the “fibre”

E|x = Ex ⊗OX,xκ(x) .

In the following theorem we consider a projective scheme X over a finitefield k of characteristic p, O(1) an ample line bundle on X whose tensorpowers are denoted by O(n), U ⊂ X an open subscheme, s ∈ N>0, E avector bundle on U everywhere of rank at least s, π : Ω1

U E an epimor-phism.

Then we have for m divisible by p an O(p)-linear map π∇ : OU (m) →E(m).Theorem 1.1. Let Z be a closed subscheme of X such that Z∩U = ∅, Σ afinite set of closed points of Z, σ0 ∈ Γ(Z,OZ(m0)), m0 divisible by p. In theabove situation there is a closed subset F of U of codimension ≥ s such thatfor every m divisible by p and sufficiently large there is σ ∈ Γ(X,O(m))that satisfies

Vol. 11, 2001 ON SPACE FILLING CURVES AND ALBANESE VARIETIES 1193

(i) V (π∇(σ)) has underlying set F ;

(ii) ∃ϕ ∈ Γ(Z,O(

m−m0p

)), ϕ invertible at Σ, σ|Z = σ0 ϕ

p.

The theorem is analogous to classical general position statements (cf.[B, §8]). The proof is by induction on s. In the proof X is fixed but Z,U, Ecan change.

Proof for s = 1. We take a finite set A of closed points of U that meets allthe irreducible components of U . For a ∈ A let [a] = Spec (OU,a/m

2U,a) be

the first infinitesimal neighborhood. π∇ gives a surjective map for p | mda(m) : O[a](m) → E|a(m) .

For m ∈ pN large enough we can find (using [H, III Proposition 5.3])ϕ ∈ Γ

(Z,O(

m−m0p

))that is invertible at Σ and extend σ0 ϕ

p to a sectionσ1 of OX(m) that vanishes on [a] for every a ∈ A.

Also we can choose for every a ∈ A an element of O[a](m) not inKer da(m) and extend these to a section σ2 of IX,Z(m). Let σ = σ1 + σ2.This satisfies the assertion of the theorem except that the underlying setof V (π∇(σ)) depends on m. To make it independent of m we make theabove construction for some m and for b ∈ N large enough we can findR1 ∈ Γ(X,O(b)) and R2 ∈ Γ(X,O(b + 1)) that are invertible on Σ ∪ A. Ifm′ satisfies

m′ −m

p> b (b + 1)

we can write (m′ −m)/p = α b + β(b + 1), α, β > 0 in N. Take σm′ =Rpα

1 Rpβ2 σ, F = V (π∇(σ)) ∪ [V (R1 R2) ∩ U ].

Proof for s > 1.

Claim 1.2. There are sections σi ∈ Γ(X,O(mi)), mi ∈ pN, 1 ≤ i ≤ s,with the following properties:

(i) codimU V (π∇(σ1)) ≥ s− 1.

(ii) for 2 ≤ i ≤ s, the closed subset of U where the “values” of π∇(σ1),π∇(σ2), . . . , π∇(σi) (that are well defined up to scalar in the samevector space) are linearly dependent is of codimension ≥ s− i + 1.

(iii) σ1|Z is of the form σ0 ϕp with ϕ invertible on Σ and σi|Z = 0 for i > 1.

(iv) π∇(σ2)|ξ = 0 for every maximal point (= a generic point of an irre-ducible component) ξ of V (π∇(σ1)).

Proof. The σi’s are constructed inductively. σ1 exists by the inductionhypothesis for s − 1. Let U1 = U − V (π∇(σ1)), E1 = E|U1/〈π∇(σ1)〉. LetB be a finite set of closed points of U − U1 that meets all its irreduciblecomponents. To construct σ2 we apply the induction hypothesis for s − 1

1194 O. GABBER GAFA

with U replaced by U1, E replaced by E1, Z replaced by Z ∪ ( ⋃b∈B [b]

),

m0 = 0, Σ replaced by B and σ0 chosen to vanish on Z and for every b ∈ Bto be not in Ker db. If s ≥ 3 we apply the induction hypothesis for s − 2with U replaced by U2 = the locus where the values of π∇(σ1), π∇(σ2) arelinearly independent, E replaced by E2 = (E|U2)/〈π∇(σ1), π∇(σ2)〉, m0 = 0,Σ empty, σ0 = 0, etc.

For 1 ≤ i ≤ s let Di = x ∈ U | π∇(σi)(x) lies in the subspace ofE|x spanned by π∇(σj)(x) for j < i, with the same abuses of languageas above. D1 is closed and Di is constructible. For a subset J ⊂ s =1, 2, . . . , s let

DJ =⋂j∈J

Dj , with D∅ = U .

Claim 1.3. ∀ J , codimU (DJ) ≥ |J |.Proof. Case 1. J = ∅: clear.

Case 2. J = ∅ and 1 /∈ J : Let i ≥ 2 be the smallest element of J . ThenDJ ⊂ Di ⊂ U − Ui and by 1.2(ii) codimU (U − Ui) ≥ s− i + 1 ≥ |J |.

Case 3. 1 ∈ J and J = s: Then DJ ⊂ D1 and by 1.2(i) it has codimen-sion ≥ s− 1 ≥ |J |.

Case 4. J = s: Then DJ ⊂ D1 ∩D2 = (D1 ∩ D2) ∪ (D1 ∩ (D2 −D2)).We estimate each term. By 1.2(iv) the maximal points of D1 are not in D2,hence the closure of D1 ∩ D2 has irreducible components that are strictlycontained in irreducible components of D1 so codimU (D1 ∩D2) ≥ s. Alsowe already know that D2 is of codimension ≥ s − 1, and the irreduciblecomponents of the closure of D2 −D2 are contained strictly in irreduciblecomponents of D2, so they are of codimension ≥ s.

Claim 1.4.i (1 ≤ i ≤ s). One can find d2, . . . , di ∈ N and jR1 ∈Γ(X,O(dj)), jR2 ∈ Γ(X,O(dj + 1)), 2 ≤ j ≤ i, such that jR1 and jR2

vanish on Dj (with the reduced scheme structure) and if we set

iDj =

V (jR1) ∪ V (jR2) for 2 ≤ j ≤ i

Dj otherwise

and iDJ = ∩j∈J

iDj then

∀ J codimU (iDJ ) ≥ |J | .Claim 1.4.1 is just Claim 1.3. We prove 1.4.i inductively so we have to

show how to pass from 1.4.i to 1.4.i + 1.For every J ⊂ s−i + 1 and every irreducible component C of iDJ of

codimension |J | we choose a closed point pC ∈ C−Di+1 which exists since


C ∩ Di+1 has larger codimension. Take di+1 large enough so that one canchoose i+1R1 and i+1R2 that vanish on Di+1 and have non-zero values atevery pC .

In the same way one shows

Claim 1.5. There is d1 ∈ N, 1R1 ∈ Γ(X,O(d1)), 1R2 ∈ Γ(X,O(d1 + 1))that are invertible at Σ and satisfy

∀ J ⊂ s− 1 , codimU

(V (1R1 · 1R2) ∩ sDJ

) ≥ |J | + 1 .

We now prove Theorem 1.1. If m is divisible by p and m − mi >pdi(di + 1) for every i write

m−mi

p= αi di + βi (di + 1) , αi, βi ∈ N>0

and define

σ = t1 + t2 + · · · + ts , ti = iRpαi1 · iR

pβi2 σi ∈ Γ

(X,O(m)

).

Condition (ii) holds since ti|Z = 0 for i > 1 and t1|Z has the required formby construction.

For every x ∈ U and 1 ≤ i ≤ s, π∇(ti)(x) is either 0 or not in thespan of the previous ones since the construction is designed to ensure that.Hence the set theoretic zero locus of π∇(σ) is the intersection of the zeroloci of the π∇(ti)’s, i.e. it is

F :=[D1 ∪ V (1R1 · 1R2|U )

] ∩ ⋂2≤i≤s

V (iR1 · iR2) ,

which has the required codimension estimate by 1.4.s and 1.5.

Corollary 1.6. If X is a projective scheme of pure dimension r > 0 withan ample line bundle O(1) over a finite field k of characteristic p and A,Bare disjoint finite sets of closed points of X, then for m divisible by p andsufficiently large ∃h ∈ H0(X,O(m)) such that A ⊂ V (h), B ∩ V (h) = ∅and Ω ∩ V (h) is a smooth divisor in Ω where Ω is the smooth locus of X.

Proof. Apply Theorem 1.1 with Z the union of B,A−Ω and [a] for a ∈ A∩Ω,Σ the underlying set of Z, m0 = 0, U = Ω − Z, s = r, E = Ω1

U , σ0 = 1 atpoints of B, σ0 = 0 at points of A−Ω, σ0 a non-zero element of ma/m

2a at

points of A ∩ Ω.Then F in the conclusion of the theorem is a finite set of closed points

of U . We take m large enough so that the conclusion of the theorem holdsand

H1(X, IA IB IF (m/p)

)= 0 .

1196 O. GABBER GAFA

Then we can choose a section g ∈ H0(X,O(m/p)) which vanishes on A∪Band has specified values at points of F that are chosen so that h = σ + gp

is invertible at F .This answers [K, Question 10] and (partially) [K, Question 13].

Corollary 1.7. Let X be a projective scheme over a field k, U an opensmooth subscheme of X of pure dimension r, 0 ≤ s ≤ r, O(1) an ample linebundle on X, A a finite set of closed points of U whose residue fields areall separable over k. Then there are m1, . . . ,ms > 0 and hi ∈ Γ(X,O(mi))such that A ⊂ V (h1, . . . , hs), V (h1, . . . , hs) ∩ U is smooth of pure di-mension r − s, V (h1, . . . , hs) ∩ (X − U) is either empty or of dimension≤ dim(X − U) − s.

Proof. It suffices to treat the case s = 1. If k is finite choose a closedpoint on each irreducible component of X − U and apply Corollary 1.6. Ifk is infinite this follows from the Bertini smoothness theorem (cf. [H, IITheorem 8.18]) on X ′, where π : X ′ → X is the blowing-up with center A;note that if E is the exceptional divisor on X ′ then for m large enoughπ∗O(m)(−E) is very ample on X ′.

Recall the following for later use:

Lemma 1.8. Let X be an irreducible projective variety of dimension r overan algebraically closed field k, 0 ≤ s ≤ r− 1, L1, . . . , Ls ample line bundleson X, σi ∈ Γ(X,Li). Then V (σ1, . . . , σs) is connected.

If s = 1 see [H, III Corollary 7.9] and [G, p. 180]. In general one canreplace the σi’s by powers and assume each Li is very ample. Consider theincidence correspondence Z ⊂ X × P where P =

∏i P(H0(X,Li)∨). By

Zariski’s connectedness theorem [GD, III Corollaire (4.3.7)] it suffices toknow that the generic fiber of Z → P is geometrically irreducible, whichfollows from a repeated application of the Bertini irreducibility theorem [Z,Theorem I.6.3].

2 Albanese Varieties

All schemes below are of finite type over a ground field or over Z. Let X bean algebraic scheme over a field k. We can “spread it out” to a morphismof finite type π : X → Spec(R) where R is a finitely generated subring of kand X ⊗R k X.

We say that X satisfies condition (∗) iff for some (equivalently, for every)spread-out as above there is an open dense U ⊂ Spec(R) such that ∀ y ∈ U ,


κ(y) ∼→ O(π−1(y)).

One can show that condition (∗) is equivalent to the following: Γ(X,OX)= k and X has a spread-out π that is cohomologically flat in dimension 0in the sense of [GD, III (7.8.1)].

If X is a proper geometrically integral k-scheme (or an algebraic space)and Z ⊂ X a subspace of codimension ≥ 2 then X−Z satisfies (∗) (extendedif needed to algebraic spaces). If f : X → Y is schematically dominant [GD,IV 11.10] (i.e. OY → f∗OX) and X satisfies (∗) then so does Y . [For thisone has to know that the property of a morphism of being schematicallydominant is constructible in the sense of [GD, IV (9.2.1), (9.2.2)], whichfollows from the lemma of generic flatness in the form [M, Theorem 24.1].]

Lemma 2.1. A k group scheme G that satisfies (∗) is an abelian variety.

Proof. By assumption G is connected, of finite type, and the global regularfunctions on G are constants. This implies that G is smooth, becauseotherwise over the perfect closure of k the homogeneous space G/Gred wouldgive non-zero nilpotent functions on G.

Similarly it follows that G is commutative: consider the adjoint actionsG → Aut (OG,e/m

n).By Chevalley’s theorem [BoLR, 9.2 Theorem 1] we have, after replac-

ing k by a finite purely inseparable extension, an affine smooth connectedalgebraic group L ⊂ G such that A = G/L is an abelian variety. This hasa spread-out to an exact sequence of smooth group schemes

0 → L → G → A → 0

over Spec(R), with L affine with connected fibers and A an abelian scheme.It follows that G is an abelian variety iff dim(L) = 0 iff the fibre of G oversome point of Spec(R) is an abelian variety. This reduces the Lemma tothe case k = Fp. Then Ext1(A,Ga) and Ext1(A,Gm) PicoA are torsion,so by the known structure of L Ext1(A,L) is torsion, so for some m > 0the multiplication by m mA : A → A lifts to a morphism m : A → G.m(A) is an abelian subvariety of G and G/m(A) L/L ∩ m(A) is affine,hence G = m(A) as desired.

Let V be a geometrically reduced geometrically connected k-scheme, Aa commutative algebraic group and f : V → A a morphism. Recall (cf.[S, §1]) that we have an algebraic subgroup A′ of A such that A′(k) is thegroup generated by the differences f(x)− f(y), x, y ∈ V (k), and we have asurjective morphism V 2n → A′. In op.cit. the case k = k and V irreducibleis treated; in general A′

kis the sum of the subgroups generated by the

1198 O. GABBER GAFA

restrictions of fk to the irreducible components of Vk and one verifies thatit descends to k.Remark 2.2. If V satisfies (∗) one verifies that A′ satisfies (∗), hence it isan abelian variety. This proves a fact considered as likely in [S, Remarqueon p. 10].

In a similar way one can discuss the question of the field of definition ofthe Albanese variety. Let V be a smooth geometrically integral variety overa field k. We follow the definition in Lang’s book [L, p. 45]: an admissiblecouple (A, f) is an abelian variety A and a morphism f : V × V → A thatvanishes on the diagonal, and a k-Albanese variety of V is an initial object(Albk(V ), f) in the category of admissible couples.

If K is an extension of k we have a canonical homomorphism IKk :

AlbK(V ) → Albk(V )K making the diagram commute. It is a radicialisogeny [L, II Theorem 12], it is an isomorphism if K is separable over k,but it is not an isomorphism in general. A case when IK

k is not an iso-morphism is when dimV = 1, V is birational to a regular non-smoothprojective curve over k but VK is birational to a smooth projective curveof genus > 0 over K. IK

k is classically known to be an isomorphism for Vthe smooth locus of a geometrically normal projective variety [L, VIII §2].This is generalized byLemma 2.3. If k = H0(V,OV ) then IK

k is an isomorphism.

Proof. It suffices to consider the case that K/k is a finite purely inseparableextension of fields of characteristic p > 0. Consider the Weil restriction[BoLR, 7.6]

R = RK/k(AlbK(V )) .It is a smooth commutative group scheme over k equipped by constructionwith a homomorphism

h : R⊗kK → AlbK(V )

which as a morphism of functors on the category of commutative k-algebrasis

A → (AlbK(V )

(K ⊗

kA

) → AlbK(V )(A))

(where K⊗kA is a K-algebra via the first factor). h is smooth and surjectiveand Ker (h) is a successive extension of Ga’s. If pm Ker (h) = 0 then pm RK

is an abelian subvariety of RK that descends to an abelian subvariety pm Rof R and the quotient R/pmR is affine. By the definition of Weil restrictionthe canonical admissible map f : VK ×VK → AlbK(V ) gives g : V ×V → R


with f = hgK . The assumption on V implies that the composite V ×V →R → R/pm R is constant, hence it is 0 since g vanishes on the diagonal.We get an admissible map V × V → pm R. Since h1 : (pm R)K → AlbK(V )is an isogeny the universal property of (AlbK(V ), f) implies that h1 is anisomorphism, thus the conclusion by [L, II Proposition 9].

Proposition 2.4. In the situation of Corollary 1.7 assume that U isgeometrically irreducible and dim(X − U) < s < r, and that h1, . . . , hs

satisfy the conclusion of the Corollary, Y = V (h1, . . . , hs), and m is suchthat (Ω1

U )∨(m) is generated by its global sections. Then

(i) Alb(Y ) → Alb(U) is surjective with connected kernel.(ii) If (∀ i) mi > m then H0(U,Ω1) → H0(Y,Ω1

Y ) is injective.(iii) If char(k) = 0 or (∀ i) mi > m then Alb(Y ) → Alb(U) is smooth.

Proof. We may assume that k is algebraically closed, choose a base pointin Y (k), and consider the canonical maps to the Albanese that map thebase point to 0. We have a commutative square

Yi−−→ U

Alb(Y )Alb(i)−−−−→ Alb(U) .

(i) Surjectivity holds by [S, Theoreme 11]. If N is a finite abelian groupthen by a classical result H1

et(Alb(U), N) classifies the extensions of Alb(U)by N . The maximality of U → Alb(U) implies that H1

et(Alb(U), N) →H1

et(U,N) is injective (cf. [S, Theoreme 10]). The form 1.8 of Bertini’stheorem applied to coverings of U implies that i induces a surjection onthe algebraic fundamental groups and H1

et(U,N) → H1et(Y,N) is injective.

Hence (by the diagram) H1et(Alb(U), N) → H1

et(Alb(Y ), N) is injectivewhich yields (i).

(ii) Let I denote the ideal sheaf defining Y in U . Note thatKer (Ω1

U ⊗OUOY → Ω1

Y ) I/I2 ⊕OY (−mi)has no non-zero global sections and that ∀n > 0, In Ω1

U/In+1 Ω1U embeds

in a direct sum of copies of Sn(I/I2)(m) that has no non-zero global sec-tions. Hence a 1-form on U that vanishes on Y must vanish on the formalcompletion of U along Y , so it is 0. Statement (ii) is a substitute for theweak Lefschetz theorem of [Be], which applies for projective U .

(iii) If char(k) = 0 then Ker Alb(i) must be smooth. If (∀ i) mi >m then by (ii) and the fact (consequence of [L, VIII Theorems 3, 4]; [S,Theoreme 3]) that H0(Alb(U),Ω1) → H0(U,Ω1) is injective we concludethat Alb(i)∗ is injective on global 1-forms.

1200 O. GABBER GAFA

Corollary 2.5. If A is an abelian variety over a field k then the aboveresults show that there is a smooth geometrically integral curve C on thedual abelian variety A that defines an exact sequence of abelian varieties0 → B → J → A → 0 where J is the Jacobian of C. As J is autodual thetransposed sequence embeds A in J .

References

[B] H. Bass, K-theory and stable algebra, Publ. Math. IHES 22 (1964), 5–60.[Be] P. Berthelot, Sur le “Theoreme de Lefschetz faible” en cohomologie

cristalline, C.R. Acad. Sci. t. 277 (1973), 955–958.[BoLR] S. Bosch, W. Lutkebohmert, M. Raynaud, Neron Models, Springer-

Verlag, 1990.[G] A. Grothendieck, Cohomologie locale des faisceaux coherents et

Theoremes de Lefschetz locaux et globaux (SGA2), North-Holland, 1968.[GD] A. Grothendieck, J. Dieudonne, Elements de geometrie algebrique,

Publ. Math. IHES, 1960-1967.[H] R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.[K] N.M. Katz, Space filling curves over finite fields, Math. Res. Letters 6

(1999), 613–624.[L] S. Lang, Abelian Varieties, Interscience publishers, 1959.[M] H. Matsumura, Commutative ring theory, Cambridge University Press,

1986.[P] B. Poonen, Bertini theorems over finite fields, preprint.[S] J.-P. Serre, Morphismes universels et variete d’Albanese, Seminaire

C. Chevalley 1958/59.[Z] O. Zariski, Introduction to the Problem of Minimal Models in the Theory

of Algebraic Surfaces, Publ. Math. Soc. Japan 4 (1958).

Ofer Gabber, CNRS, IHES, 35 route de Chartres, F-91440 Bures-sur-Yvette,France

Submitted: July 2001

on space filling curves and albanese varieties

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