RATIONAL POINTS ON SOME FANO CUBIC BUNDLES

VICTOR V. BATYREV^{∗} AND YURI TSCHINKEL^{†}

Abstract. We consider smooth Fano hypersurfaces X_{n+2} ⊂ P^{n} × P^{3} (n ≥ 1) given by a polynomial

∑_{i=0}^{3} l_{i}(x)y_{i}^{3} ∈ Q[x_{0}, …, x_{n}, y_{0}, …, y_{3}]

where l_{0}(x), …, l_{3}(x) are homogeneous linear forms in x_{0}, …, x_{n}. We obtain lower bounds for the number of F -rational points of bounded anticanonical height in arbitrary nonempty Zariski open subsets U ⊂ X_{n+2} for number fields F containing Q(\sqrt{−3}). These bounds contradict previous expectations about the distribution of F -rational points of bounded height on Fano varieties.

Points rationnels sur certains fibrés en cubiques de Fano

Résumé français. Nous considérons des hypersurfaces lisses de Fano X_{n+2} ⊂ P^{n} × P^{3} (n ≥ 1) données par un polynôme

∑_{i=0}^{3} l_{i}(x)y_{i}^{3} ∈ Q[x_{0}, …, x_{n}, y_{0}, …, y_{3}]

où l_{0}(x), …, l_{3}(x) sont des formes linéaires homogènes en x_{0}, …, x_{n}. Nous obtenons des bornes inférieures pour le nombre de points F -rationnels de hauteur bornée (relativement au diviseur anticanonique) dans des ouverts de Zariski non vides quelconques U ⊂ X_{n+2} pour tous les corps de nombres F contenant Q(\sqrt{−3}). Ces bornes vont à l’encontre de la distribution attendue des points F -rationnels de hauteur bornée sur les variétés de Fano.

Version française abrégée

Théorème. Pour tout n ≥ 1, l’hypersurface X_{n+2} a les propriétés suivantes – c’est une variété de Fano lisse contenant un ouvert de Zariski isomorphe à A^{n+2} ;

– les fibres de la projection naturelle π : X_{n+2} → P^{n} au-dessus des points fermés de l’ouvert de Zariski U_{P} ⊂ P^{n} défini par l’équation ∏_{i=0}^{3} l_{i}(x) ≠ 0 sont des surfaces cubiques diagonales lisses de P^{3} ;

∗ Supported by Deutsche Forschungsgemeinschaft † Leibniz Fellow at ENS, Paris

– le groupe de Picard de X_{n+2} sur tout corps contenant Q est isomorphe à Z ⊕ Z ;

– le fibré canonique sur X_{n+2} induit sur chaque fibre de π le fibré canonique de cette fibre.

Théorème. Soit Y une surface cubique diagonale dans P^{3} de la forme

b_{0}^{3}y_{0}^{3} + ⋯ + b_{3}^{3}y_{3}^{3} = 0

où les b_{i} sont des éléments non nuls d’un corps F contenant Q(\sqrt{−3}). Alors, il existe une constante c > 0 telle que, pour tout ouvert de Zariski non vide U ⊂ Y, le nombre de points F -rationnels de U dont la hauteur anticanonique est inférieure à B est minoré par cB(log B)^{3}.

La conjecture suivante a été inspirée par la conjecture de croissance linéaire de Manin [6] et par l’extrapolation de résultats connus (méthode du cercle, variétés de drapeaux, variétés toriques) :

Conjecture. Soit X une variété de Fano lisse sur un corps de nombres E. Alors, il existe un ouvert de Zariski U ⊂ X et une extension finie F_{0} de E, tels que pour toute extension finie F de F_{0}, le nombre de points F -rationnels contenus dans U dont la hauteur anticanonique est majorée par B est asymptotique à cB(log B)^{t−1} où t est le rang du groupe de Picard de X sur F et c une constante positive.

Cette conjecture a été raffinée par E. Peyre qui a proposé une interprétation adélique de la constante c.

Les énoncés précédents prouvent que cette conjecture est fausse pour les fibrés en cubiques de Fano X_{n+2}.

Nous sommes très reconnaissants à Antoine Chambert-Loir pour son aide lors de la préparation de la version française de ce texte.

1. Cubic bundles

Let X_{n+2} be a hypersurface in P^{n} × P^{3} (n ≥ 1) defined by the equation

P (x, y) = ∑_{i=0}^{3} l_{i}(x)y_{i}^{3} = 0

where

P (x, y) ∈ Q[x_{0}, …, x_{n}, y_{0}, …, y_{3}]

and l_{0}(x), …, l_{3}(x) are homogeneous linear forms in x_{0}, …, x_{n}. Put k = min(n + 1, 4). We shall always assume that any k forms among l_{0}(x), …, l_{3}(x) are linearly independent. It is elementary to check the following statements:

Proposition 1.1. The hypersurface X_{n+2} is a smooth Fano variety containing a Zariski open subset U_{n+2} which is isomorphic to A^{n+2}.

Proposition 1.2. Let U_{P} ⊂ P^{n} be the Zariski open subset defined by the condition

∏_{i=0}^{3} l_{i}(x) ≠ 0.

Then the fibers of the natural projection π : X_{n+2} → P^{n} over closed points of U_{P} are smooth diagonal cubic surfaces in P^{3}.

Proposition 1.3. The Picard group of X_{n+2} over an arbitrary field containing Q is isomorphic to Z ⊕ Z.

2. Heights on cubic surfaces

Let F be a number field, V al(F) the set of all valuations of F, W a projective algebraic variety over F, W (F) the set of F -rational points of W, D a very ample divisor on W, and γ = {s_{0}, …, s_{m}} a basis over F of the space of global sections Γ(W, O(D)). The height function associated with D and γ

H(W, D, γ, x) : W (F) → R_{>0}

is given by the formula

H(W, D, γ, x) = ∏_{v∈V al(F)} max_{i=0,... ,m} |s_{i}(x)|_{v},

where | · |_{v} : F_{v} → R_{>0} is the multiplier of a Haar measure on the additive group of the v-adic completion of F.

Definition 2.1. Let Z ⊂ W be a locally closed algebraic subset of W, B a positive real number. Define

N(Z, D, γ, B) := Card{x ∈ W (F) ∩ Z | H(W, D, γ, x) ≤ B}.

The following classical statement is due to A. Weil:

Theorem 2.2. Let γ′ = {s′_{0}, …, s′_{m}} be another basis in Γ(W, O(D)). Then there exist two positive constants c_{1}, c_{2} such that

c_{1} ≤ \frac{H(W, D, γ, x)}{H(W, D, γ′, x)} ≤ c_{2}

for all x ∈ W (F).

For a smooth projective variety W we denote by −K_{W} the anticanonical divisor on W.

Theorem 2.3. Let Y ⊂ P^{3} be a smooth cubic surface over F, γ = {s_{0}, s_{1}, s_{2}, s_{3}} the basis of global sections of O(−K_{Y}) corresponding to the standard homogeneous coordinates on P^{3}. Assume that Y can be obtained by blowing up of 6 F -rational points in P^{2}. Then for any nonempty Zariski open subset U ⊂ Y one has

N(U, −K_{Y} , γ, B) ≥ cB(log B)^{3}

for all B > 0 and for some positive constant c.

Proof. Assume that Y is obtained by blowing up of p_{1}, …, p_{6} ∈ P^{2}(F). By 2.2, we can assume without loss of generality that p_{1} = (1 : 0 : 0), p_{2} = (0 : 1 : 0) and p_{3} = (0 : 0 : 1). Denote by Y_{0} the Del Pezzo surface obtained by blowing up p_{1}, p_{2}, p_{3}. Let f : Y → Y_{0} be the contraction of exceptional curves C_{4}, C_{5}, C_{6} ⊂ Y lying over p_{4}, p_{5}, p_{6}.

Let V be the 10-dimensional space over F of all homogeneous polynomials of degree 3 in variables z_{0}, z_{1}, z_{2}. We identify Γ(Y, O(−K_{Y})) with the subspace in V consisting of all polynomials vanishing in p_{1}, …, p_{6}. Analogously, we identify Γ(Y_{0}, O(−K_{Y_{0}})) with the subspace in V consisting of all polynomials vanishing in p_{1}, p_{2}, p_{3}. Let γ_{0} = {s_{0}, …, s_{6}} ⊂ V the extension of the basis γ to a basis of the subspace Γ(Y_{0}, O(−K_{Y_{0}})) ⊂ V.

The surface Y_{0} is an smooth equivariant compactification of the split 2-dimensional algebraic torus over F

(G_{m})^{2} = P^{2} \ {l_{12}, l_{13}, l_{23}}

where l_{ij} denotes the projective line in P^{2} through p_{i} and p_{j}. Since Y_{0} is a smooth toric variety, the main theorem in [3] shows that the following asymptotic formula holds:

N((G_{m})^{2}, −K_{Y_{0}}, γ′_{0}, B) = c_{0}B(log B)^{3}(1 + o(1)), B → ∞, (1)

where c_{0} is some positive constant and

γ′_{0} = {z_{0}z_{1}z_{2}, z_{1}^{2}z_{2}, z_{1}z_{2}^{2}, z_{2}^{2}z_{0}, z_{2}z_{0}^{2}, z_{0}^{2}z_{1}, z_{0}z_{1}^{2}}.

Let U be any nonempty Zariski open subset in Y. We denote by U_{0} a nonempty open subset in U such that the restriction of f on U_{0} is an isomorphism and f(U_{0}) is contained in (G_{m})^{2} ⊂ Y_{0}. Since

∏_{v∈V al(F)} max_{i=0,... ,3} |s_{i}(x)|_{v} ≤ ∏_{v∈V al(F)} max_{i=0,... ,6} |s_{i}(x)|_{v}

holds for every F -rational point x ∈ U_{0}, we obtain

N(U_{0}, −K_{Y} , γ, B) ≥ N(U_{0}, −K_{Y_{0}}, γ_{0}, B) (2)

for any B > 0. By 2.2, there exists a positive constant c_{3} such that

N(U_{0}, −K_{Y_{0}}, γ_{0}, B) ≥ c_{3}N(U_{0}, −K_{Y_{0}}, γ′_{0}, B). (3)

On the other hand,

N(U_{0}, −K_{Y_{0}}, γ′_{0}, B) = N((G_{m})^{2}, −K_{Y_{0}}, γ′_{0}, B) − N(Z, −K_{Y_{0}}, γ′_{0}, B), (4)

where Z = (G_{m})^{2} \ U_{0}. Let Z_{1}, …, Z_{l} be the irreducible components of Z, \overline{Z}_{i} the closure of Z_{i} in Y_{0} (i = 1, …, l). It is known that

N(Z_{i}, −K_{Y_{0}}, γ′_{0}, B) ≤ c_{4}B^{2/(deg \overline{Z}_{i})} (5)

holds for some positive constant c_{4}, where deg \overline{Z}_{i} denotes the degree of \overline{Z}_{i} with respect to the anticanonical divisor −K_{Y_{0}}. Since every irreducible curve C ⊂ Y_{0} with deg C = 1 is a component of Y_{0} \ (G_{m})^{2}, we have deg \overline{Z}_{i} ≥ 2; i.e.,

N(Z_{i}, −K_{Y_{0}}, γ′_{0}, B) ≤ c_{4}B (6)

holds for all i = 1, …, l.

It follows from the asymptotic formula (1) combined with (2), (3) and (4) that there exists a positive constant c such that

N(U_{0}, −K_{Y} , γ, B) ≥ cB(log B)^{3}

holds for all B > 0. This yields the statement, since U_{0} is contained in U. □

Corollary 2.4. Let Y be a smooth diagonal cubic surface in P^{3} defined by the equation

a_{0}y_{0}^{3} + a_{1}y_{1}^{3} + a_{2}y_{2}^{3} + a_{3}y_{3}^{3} = 0

with coefficients a_{0}, …, a_{3} in a number field F which contains Q(\sqrt{−3}). Assume that there exist numbers b_{0}, …, b_{3} ∈ F^{∗} such that a_{i} = b_{i}^{3} (i = 0, …, 3). Then for any nonempty Zariski open subset U ⊂ Y one has

N(U, −K_{Y} , γ, B) ≥ cB(log B)^{3}

for all B > 0 and some positive constant c.

Proof. It follows from our assumptions on the coefficients a_{0}, …, a_{3} and on the field F that all 27 lines on Y are defined over F. Hence, Y can be obtained from P^{2} by blowing up of 6 F -rational points. Now the statement follows from 2.3. □

3. Rational points on X_{n+2}

We have the natural isomorphism

Γ(X_{n+2}, O(−K_{X_{n+2}})) ≅ Γ(P^{3}, O(1)) ⊗ Γ(P^{n}, O(n)).

We remark that the canonical line bundle on X_{n+2} induces the canonical line bundle in the fibers of π. Let {t_{0}, …, t_{m}} be a basis in Γ(P^{n}, O(n)) and {s_{0}, s_{1}, s_{2}, s_{3}} the standard basis in Γ(P^{3}, O(1)). Denote by γ the basis of Γ(X_{n+2}, O(−K_{X_{n+2}})) consisting of s_{i} ⊗ t_{j} (i = 0, …, 3; j = 0, …, m).

Theorem 3.1. Let n ≥ 3. Then for any nonempty Zariski open subset U ⊂ X_{n+2} and for any field F containing Q(\sqrt{−3}), one has

N(U, −K_{X_{n+2}}, γ, B) ≥ cB(log B)^{3}

for all B > 0 and some positive constant c.

Proof. Consider the projection π : U → P^{n}. Let U′ be a Zariski open subset in π(U) ∩ U_{P} such that the image of U′ under the dominant (rational) mapping

ψ : P^{n} − − → P^{3}
ψ(x_{0} : …: x_{n}) = (l_{0}(x) : …: l_{3}(x))

is Zariski open in P^{3}. Denote by φ the finite morphism

φ : P^{3} → P^{3},
φ(z_{0} : …: z_{3}) = (z_{0}^{3} : …: z_{3}^{3}).

Since P^{3}(F) is Zariski dense in P^{3}, there is a point p ∈ P^{3}(F) ∩ φ^{−1}(ψ(U′)). Since U′(F) ∩ ψ^{−1}(φ(p)) is Zariski dense in ψ^{−1}(φ(p)), there exists q ∈ U′(F) ∩ ψ^{−1}(φ(p)). Therefore, the fiber of π over q is a diagonal cubic surface Y_{q} and U ∩ Y_{q} ⊂ Y_{q} is a nonempty Zariski open subset. It remains to apply 2.4. □

Theorem 3.2. Let n = 2. Then there exists a number field F_{0} depending only on X_{n+2} such that for any nonempty Zariski open subset U ⊂ X_{n+2} for any field F containing F_{0} one has

N(U, −K_{X_{n+2}}, γ, B) ≥ cB(log B)^{3}

for all B > 0 and some positive constant c.

Proof. Let U′ be a Zariski open subset in π(U) ∩ U_{P}. We have the linear embedding

ψ : P^{2} ↪ P^{3}

defined by l_{0}(x), …, l_{3}(x). Then φ^{−1}(ψ(P^{2})) is a smooth diagonal cubic surface S ⊂ P^{3} defined over Q. Let F_{0} be a finite extension of Q(\sqrt{−3}) such that S(F_{0}) is Zariski dense in S. Then there exists a point p ∈ S(F_{0}) such that q = φ(p) is contained in U′. Therefore, the fiber of π over q is a diagonal cubic surface Y_{q}, and U ∩ Y_{q} ⊂ Y_{q} is a nonempty Zariski open subset. It remains to apply 2.4. □

Theorem 3.3. Let n = 1. Then for any nonempty Zariski open subset U ⊂ X_{n+2}, there exists a number field F_{0} (which depends on U) such that for any field F containing F_{0}, one has

N(U, −K_{X_{n+2}}, γ, B) ≥ cB(log B)^{3}

for all B > 0 and some positive constant c.

Proof. Let U′ be a Zariski open subset in π(U) ∩ U_{P}. We have the linear embedding

ψ : P^{1} ↪ P^{3}

defined by l_{0}(x), …, l_{3}(x). Then φ^{−1}(ψ(P^{1})) is an algebraic curve C ⊂ P^{3} which is a complete intesection of two diagonal cubic surfaces defined over Q. Let F_{0} be a finite extension of Q(\sqrt{−3}) such that there exists an F_{0}-rational point p ∈ C(F_{0})∩φ^{−1}(U′). Then the fiber of π over q = φ(p) is a diagonal cubic surface Y_{q} and U ∩ Y_{q} ⊂ Y_{q} is a nonempty Zariski open subset. It remains to apply 2.4. □

4. Conclusions

The following statement, inspired by the Linear Growth conjecture of Manin ([6]) and by extrapolation of known results (circle method, flag varieties, toric varieties), has been expected to be true [1, 4]:

Conjecture 4.1. Let X be a smooth Fano variety over a number field E. Then there exist a Zariski open subset U ⊂ X and a finite extension F_{0} of E such that for all number fields F containing F_{0} the following asymptotic formula holds

N(U, −K_{X}, γ, B) = cB(log B)^{t−1}(1 + o(1)), B → ∞,

where t is the rank of the Picard group of X over F.

Some lower and upper bounds for N(U, −K_{X}, γ, B) for Del Pezzo surfaces and Fano threefolds have been obtained in [6, 7].

The conjecture 4.1 was refined by E. Peyre who proposed an adelic interpretation for the constant c introducing Tamagawa numbers of Fano varieties [8]. This refined version of the conjecture has been proved for toric varieties in [2, 3].

The statements in Theorems 3.1, 3.2, 3.3 and the property 1.3 show that Conjecture 4.1 is not true for Fano cubic bundles X_{n+2} (n ≥ 1).

References

1. V.V. Batyrev, Yu.I. Manin, Sur le nombre des points rationnels de hauteur bornée des variétés algébriques, Math. Ann., 286, (1990), 27-43.

2. V.V. Batyrev, Yu. Tschinkel, Rational points of bounded height on compactifications of anisotropic tori, Int. Math. Res. Notices, 12, (1995), .

3. V.V. Batyrev, Yu. Tschinkel, Manin’s conjecture for toric varieties, Preprint IHES, (1995).

4. J. Franke, Yu.I. Manin, Yu. Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math., 95, (1989), 425-435.

5. Yu.I. Manin, Cubic forms, North-Holland, 2nd ed., (1986).

6. Yu.I. Manin, Notes on the arithmetic of Fano threefolds, Compositio Math., 85 (1993), 37-55.

7. Yu.I. Manin, Yu. Tschinkel, Rational points of bounded height on Del Pezzo surfaces, Compositio Math., 85 (1993), 315-332.

8. E. Peyre, Hauteurs et nombres de Tamagawa sur les variétés de Fano, Duke Math. J., 79 (1995), 101-218.

V.B. : Universität-GHS-Essen, Fachbereich 6, Mathematik, Universitätsstr. 3, 45141 Essen, Germany

E-mail address: victor.batyrev@uni-essen.de

Y.T. : Dept. of Mathematics, U.I.C., Chicago, (IL) 60608, U.S.A. E-mail address: yuri@math.uic.edu