ÉTALE COHOMOLOGY, LEFSCHETZ THEOREMS AND NUMBER OF POINTS OF SINGULAR VARIETIES

OVER FINITE FIELDS

SUDHIR R. GHORPADE AND GILLES LACHAUD

Dedicated to Professor Yuri Manin for his 65th birthday tiraścīno vitato raśmir es.ām^{∗}

Abstract. We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this inequality generalizes also the classical Lang-Weil inequality. Moreover, we prove the Lang-Weil inequality for affine as well as projective varieties with an explicit description and a bound for the constant appearing therein. We also prove a conjecture of Lang and Weil concerning the Picard varieties and étale cohomology spaces of projective varieties. The general inequality for complete intersections may be viewed as a more precise version of the estimates given by Hooley and Katz. The proof is primarily based on a suitable generalization of the Weak Lefschetz Theorem to singular varieties together with some Bertini-type arguments and the Grothendieck-Lefschetz Trace Formula. We also describe some auxiliary results concerning the étale cohomology spaces and Betti numbers of projective varieties over finite fields and a conjecture along with some partial results concerning the number of points of projective algebraic sets over finite fields.

Contents

Introduction 3 1. Singular Loci and Regular Flags 5 2. Weak Lefschetz Theorem for Singular Varieties 7 3. Cohomology of Complete Intersections 11 4. The Central Betti Number of Complete Intersections 13 5. Zeta Functions and the Trace Formula 15 6. Number of Points of Complete Intersections 17 7. Complete Intersections with isolated singularities 20 8. The Penultimate Betti Number 22 9. Cohomology and Albanese Varieties 26 10. A Conjecture of Lang and Weil 30 11. On the Lang-Weil Inequality 33 12. Number of Points of Algebraic Sets 36 Acknowledgements 38 References 38

Date: March 8, 2002. 1991 Mathematics Subject Classification. 11G25, 14F20, 14G15, 14M10.

∗ “Their cord was extended across” (\begin{matrix}R \\ . \end{matrix}g Veda X.129). † The first author is partially supported by a ‘Career Award’ grant from AICTE, New Delhi

and an IRCC grant from IIT Bombay.

This is an unofficial electronic version of the paper which has appeared in Moscow Mathematical Journal, Vol. 2, No. 3 (2002), pp. 589–631.

2 SUDHIR R. GHORPADE AND GILLES LACHAUD

Abbreviations 39

Introduction

This paper has roughly a threefold aim. The first is to prove the following inequality for estimating the number of points of complete intersections (in particular, hypersurfaces) which may possibly be singular:

| |X(𝔽_{q})| − π_{n} | ≤ b′_{n−s−1}(N − s − 1, d) q^{(n+s+1)/2} + C_{s}(X)q^{(n+s)/2} (1)

(cf. Theorem 6.1). Here, X denotes a complete intersection in ℙ^{N} defined over the finite field k = 𝔽_{q} of q elements, n the dimension of X, d = (d_{1}, …, d_{r}) the multidegree of X and s an integer such that dim Sing X ≤ s ≤ n − 1. Note that r = N − n. Moreover, π_{n} denotes the number of points of ℙ^{n}(𝔽_{q}), viz., π_{n} = q^{n} +q^{n−1} +⋯+1, and for any nonnegative integers j and M with M −j = r, we denote by b′_{j}(M, d) the primitive j-th Betti number of a nonsingular complete intersection in ℙ^{M} of dimension j. This primitive Betti number is explicitly given by the formula

b′_{j}(M, d) = (−1)^{j+1}(j + 1) + (−1)^{N} ∑_{c=r}^{N}(−1)^{c}\binom{N + 1}{c + 1} ∑_{ν∈M(c)} d^{ν} (2)

where M(c) denotes the set of r-tuples ν = (ν_{1}, …, ν_{r}) of positive integers such that ν_{1} + ⋯ + ν_{r} = c and d^{ν} = d_{1}^{ν_{1}} ⋯ d_{r}^{ν_{r}} for any such r-tuple ν. If we let δ = max(d_{1}, …, d_{r}) and d = d_{1} ⋯ d_{r} = deg X, then it will be seen that

b′_{j}(M, d) ≤ (−1)^{j+1}(j + 1) + d\binom{M + 1}{j}(δ + 1)^{j} ≤ \binom{M + 1}{j}(δ + 1)^{M}. (3)

Lastly, C_{s}(X) is a constant which is independent of q (or of k). We can take C_{s}(X) = 0 if s = −1, i. e., if X is nonsingular, and in general, we have

C_{s}(X) ≤ 9 × 2^{r} × (rδ + 3)^{N+1}. (4)

where δ is as above. The inequalities (1), (3) and (4) are proved in sections 6, 4 and 5 respectively.

Our second aim is to discuss and elucidate a number of results related to the conjectural statements of Lang and Weil [27]. These concern the connections between the various étale cohomology spaces (especially, the first and the penultimate) and the Picard (or the Albanese) varieties of normal projective varieties defined over 𝔽_{q}. For example, if X is any projective variety of dimension n, and P_{2n−1}^{+}(X, T) is the characteristic polynomial of the piece of maximal filtration of H^{2n−1}(X, ℚ_{ℓ}) and f_{c}(Alb_{w} X, T) is the characteristic polynomial of the Albanese-Weil variety Alb_{w} X of X, then we show that

P_{2n−1}^{+}(X, T) = q^{−g}f_{c}(Alb_{w} X, q^{n}T) (5)

where g = dim Alb_{w} X. In particular, the “(2n − 1)-th virtual Betti number” of X is twice the dimension of Alb_{w} X, and is independent of ℓ. These results are discussed in details in sections 8, 9 and 10.

The third aim is to prove an effective version of the Lang-Weil inequality [27]. Recall that the Lang-Weil inequality states that if X ⊂ ℙ^{N} is any projective variety defined over 𝔽_{q} of dimension n and degree d, then with π_{n} as above, we have

| |X(𝔽_{q})| − π_{n} | ≤ (d − 1)(d − 2)q^{n−(1/2)} + Cq^{(n−1)}, (6)

where C is a constant depending only on N, n and d. The said effective version consists in providing computable bounds for the constant C appearing in (6). For example, if X is defined by the vanishing of m homogeneous polynomials in N + 1 variables of degrees d_{1}, …, d_{m} and δ = max(d_{1}, …, d_{m}), then we show that the constant C in (6) may be chosen such that

C ≤ 9 × 2^{m} × (mδ + 3)^{N+1}. (7)

We also prove an analogue of the Lang-Weil inequality for affine varieties. These inequalities are proved in section 11. Lastly, in section 12, we describe some old, hitherto unpublished, results concerning certain general bounds for the number of points of projective algebraic sets, as well as a conjecture related to the same.

We shall now describe briefly the background to these results and some applications. For a more leisurely description of the background and an expository account of the main results of this paper, we refer to [11].

In [40], Weil proved a bound for the number of points of a nonsingular curve over a finite field 𝔽_{q}, namely that it differs from π_{1} = q + 1 by at most 2gq^{1/2}, where g is the genus of the curve. In [41], he formulated the conjectures about the number of points of varieties of arbitrary dimension. Before these conjectures became theorems, Lang and Weil [27] proved the inequality (6) in 1954. In 1974, Deligne [10] succeeded in completing the proof of Weil Conjectures by establishing the so called Riemann hypothesis for nonsingular varieties of any dimension. Using this and the Lefschetz Trace Formula, also conjectured by Weil, and proved by Grothendieck, he obtained a sharp inequality for the number of points of a nonsingular complete intersection. Later, in 1991, Hooley [18] and Katz [21] proved that if X is a complete intersection with a singular locus of dimension s, then

|X(𝔽_{q})| − π_{n} = O(q^{(n+s+1)/2}).

The inequality (1) which we prove here may be regarded as a more precise version of this estimate. In effect, we explicitly obtain the coefficient of the first term and a computable bound for the coefficient of the second term in the asymptotic expansion of the difference |X(𝔽_{q})|−π_{n}. When s = −1, i. e., when X is nonsingular, then (1) is precisely the inequality proved by Deligne [10]. On the other hand, if X is assumed normal, then we can take s = n − 2 and (1) implies the Lang-Weil inequality for normal complete intersections. The explicit formula (2) is a consequence of the work of Hirzebruch [17] and Jouanolou [20] on nonsingular complete intersections, while the computable bound (4) is obtained using some work of Katz [24] on the sums of Betti numbers. Corollaries of (1) include some results of Shparlinskiĭ and Skorobogatov [38] for complete intersections with at most isolated singularities as well as some results of Aubry and Perret [4] for singular curves. It may be remarked that the recent article [23] of Katz has a more general purpose than that of Section 6 of this article, since it provides bounds for exponential sums defined over singular varieties. However, as far as the number of points of singular varieties are concerned, the bounds presented here are more precise than those obtained by specializing the results of [23].

After proving (6), Lang and Weil [27] observed that if K is an algebraic function field of dimension n over k = 𝔽_{q}, then there is a constant γ for which (6) holds with (d − 1)(d − 2) replaced by γ, for any model X of K/k, and moreover, the smallest such constant γ is a birational invariant. They also noted that the zeros and poles of the zeta function Z(X, T) in the open disc |T | < q^{−(n−1)} are birational invariants, and that in the smaller disc |T | < q^{−(n−1/2)} there is exactly one pole of order 1 at T = q^{−n}. Then they wrote: about the behaviour of Z(X, T) for |T | ≥ q^{−(n−1/2)}, we can only make the following conjectural statements, which complement the conjectures of Weil. These statements are to the effect that when X is complete and nonsingular, the quotient

\frac{Z(X, T)(1 − q^{n}T)}{f_{c}(P, T)}

has no zeros or poles inside |T | < q^{−(n−1)} and at least one pole on |T | = q^{−(n−1)}, where P denotes the Picard variety of X. Moreover, with K/k and γ as above, we have γ = 2 dim P. Lang and Weil [27] proved that these statements are valid in the case of complete nonsingular curves, using the Riemann hypothesis for curves over finite fields. As remarked by Bombieri and Sperber [5, p. 333], some of the results conjectured by Lang and Weil are apparently known to the experts but one is unable to locate formal proofs in the literature. Some confusion is also added by the fact that there are, in fact, two notions of the Picard variety of a projective variety X. These notions differ when X is singular and for one of them, the analogue of (5) is false. Moreover, the proof in [5] of a part of the Proposition on p. 133 appears to be incomplete (in dimensions ≥ 3). In view of this, we describe in some details results such as (5) which together with the Grothendieck-Lefschetz trace formula, prove the conjecture of Lang and Weil, as well as several related results. These include the abovementioned Proposition of Bombieri and Sperber [5, p. 333], which is proved here using a different method. An equality such as (5) can also be of interest as a “motivic” result in the sense indicated in an early letter of Grothendieck [14].

For the Lang-Weil inequality (6), it is natural to try to give a proof using the trace formula. Some auxiliary results are still needed but many of these are obtained in the course of proving (1) and (5). This, then, leads to the ‘effective version’ and an affine analogue. The latter yields, for example, a version of a lower bound due to Schmidt [33] for the number of points on affine hypersurfaces. For varieties of small codimension, we obtain an improved version of the Lang-Weil inequality.

1. Singular Loci and Regular Flags

We first settle notations and terminology. We also state some preliminary results, and the proofs are omitted. Let k be a field of any characteristic p ≥ 0 and \overline{k} the algebraic closure of k. We denote by S = k[X_{0}, …, X_{N}] the graded algebra of polynomials in N + 1 variables and by ℙ^{N} = ℙ_{k}^{N} = Proj S the projective space of dimension N over k. By an algebraic variety over k we shall mean a separated scheme of finite type over k which is geometrically irreducible and reduced, i. e., geometrically integral. In this section, we use the word scheme to mean a scheme of finite type over k.

Recall that a point x in a scheme X is regular if the local ring O_{x}(X) is a regular local ring and singular otherwise. The singular locus Sing X of X is the set of singular points of X ; this is is a closed subset of X [EGA 4.2, Cor. 6.12.5, p. 166]. We denote by Reg X the complementary subset of Sing X in X. Let m ∈ ℕ with m ≤ dim X. One says that X is regular in codimension m if it satisfies the following equivalent conditions:

(i) every point x ∈ X with dim O_{x}(X) ≤ m is regular. (ii) dim X − dim Sing X ≥ m + 1. Condition (i) is called condition (R_{m}) [EGA 4.2, Déf. 5.8.2, p. 107]. A scheme is reduced if and only if it has no embedded components and satisfies condition (R_{0}) [EGA 4.2, 5.8.5, p. 108]. A scheme X of dimension n is regular if it satisfies condition (R_{n}), hence X is regular if and only if dim Sing X = −1 (with the convention dim ∅ = −1).

For any scheme X, we denote by X̄ = X ⊗_{k} \overline{k} the scheme deduced from X by base field extension from k to \overline{k}. A variety X is nonsingular if X̄ is regular. If k is perfect, then the canonical projection from X̄ to X sends Sing X̄ onto Sing X [EGA 4.2, Prop. 6.7.7, p. 148]. Hence, X̄ is regular in codimension m if and only if X has the same property, and dim Sing X̄ = dim Sing X.

Let R be a regular noetherian local ring, A = R/ℑ a quotient subring of R, and r the minimum number of generators of ℑ. Recall that A is a complete intersection in R if ht ℑ = r. A closed subscheme X of a regular noetherian scheme V over a field k is a local complete intersection at a point x ∈ X if the local ring O_{x}(X) is a complete intersection in O_{x}(V). The subscheme X of V is a local complete intersection if it is a local complete intersection at every closed point; in this case it is a local complete intersection at every point, since the set of x ∈ X such that X is a local complete intersection at x is open. A regular subscheme of V is a local complete intersection. A connected local complete intersection is Cohen-Macaulay [16, Prop. 8.23, p.186], hence equidimensional since all its connected components have the same codimension r.

Let X and Y be a pair of closed subschemes of ℙ^{N} with dim X ≥ codim Y, and x a closed point of X ∩ Y. We say that X and Y meet transversally at x if

(8) x ∈ Reg X ∩ Reg Y and dim T_{x}(X) ∩ T_{x}(Y) = dim T_{x}(X) − codim T_{x}(Y),

and that they intersect properly at x if

(9) dim O_{x}(X ∩ Y) = dim O_{x}(X) − codim O_{x}(Y).

If X and Y are equidimensional, then they intersect properly at every point of an irreducible component Z if and only if

dim Z = dim X − codim Y.

If this is fulfilled, one says that Z is a proper component of X ∩ Y or that X and Y intersect properly at Z. If every irreducible component of X ∩ Y is a proper component, one says in this case that X and Y intersect properly. For instance, if X is irreducible of dimension ≥ 1 and if Y is a hypersurface in ℙ^{N} with X_{red} ⊄ Y_{red}, then X and Y intersect properly by Krull’s Principal Ideal Theorem.

If Y is a local complete intersection, then X and Y meet transversally at x if and only if x ∈ Reg(X ∩ Y) and they intersect properly at x.

1.1. Lemma. Assume that X is equidimensional, Y is a local complete intersection, and X and Y intersect properly.

(i) If N(X, Y) is the set of closed points of Reg X ∩ Reg Y where X and Y do not meet transversally, then

Sing(X ∩ Y) = (Y ∩ Sing X) ∪ (X ∩ Sing Y) ∪ N(X, Y).

(ii) If X ∩ Y satisfies condition (R_{m}), then X and Y satisfy it as well. □

Let X be a subscheme of ℙ^{N}. We say a subscheme Z of X is a proper linear section of X if there is a linear subvariety E of ℙ^{N} properly intersecting X such that X ∩ E = Z, in such a way that dim Z = dim X − codim E. If codim E = 1, we say Z is a proper hyperplane section of X. One sees immediately from Lemma 1.1 that if X is an equidimensional subscheme of ℙ^{N}, and if there is a nonsingular proper linear section of dimension m of X (0 ≤ m ≤ dim X), then condition (R_{m}) holds for X. In fact, these two conditions are equivalent, as stated in Corollary 1.4.

From now on we assume that k = \overline{k} is algebraically closed. We state here a version of Bertini’s Theorem and some of its consequences ; an early source for this kind of results is, for instance, [42, Sec. I ]. Let ℙ̂^{N} be the variety of hyperplanes of ℙ^{N}. Let X be a closed subvariety in ℙ^{N} and 𝒰_{1}(X) be the set of H ∈ ℙ̂^{N} satisfying the following conditions:

(i) X ∩ H is a proper hyperplane section of X. (ii) X ∩ H is reduced if dim X ≥ 1. (iii) X ∩ H is irreducible if dim X ≥ 2.

(iv) dim Sing X ∩ H = \begin{cases}dim Sing X − 1 & if dim Sing X ≥ 1, \\ −1 & if dim Sing X ≤ 0. \end{cases}

(v) deg X ∩ H = deg X if dim X ≥ 1.

We need the following version of Bertini’s Theorem. A proof of this result can be obtained as a consequence of [19, Cor. 6.11, p. 89] and [39, Lemma 4.1].

1.2. Lemma. Let X be a closed subvariety in ℙ^{N}. Then 𝒰_{1}(X) contains a nonempty Zariski open set of ℙ̂^{N}. □

Let r ∈ ℕ with 0 ≤ r ≤ N. We denote by 𝔾_{r,N} the Grassmannian of linear varieties of dimension r in ℙ^{N}. Let ℱ_{r}(ℙ^{N}) be the projective variety consisting of sequences

(E_{1}, …, E_{r}) ∈ 𝔾_{N−1,N} × ⋯ × 𝔾_{N−r,N}

making up a flag of length r:

ℙ^{N} = E_{0} ⊃ E_{1} ⊃ ⋯ ⊃ E_{r}

with codim E_{m} = m for 0 ≤ m ≤ r, so that E_{m} is a hyperplane of E_{m−1}. Let X be a subvariety of ℙ^{N} and (E_{1}, …, E_{r}) a flag in ℱ_{r}(ℙ^{N}). We associate to these data a descending chain of schemes X_{0} ⊃ X_{1} ⊃ ⋯ ⊃ X_{r} defined by

X = X_{0},  X_{m} = X_{m−1} ∩ E_{m}  (1 ≤ m ≤ r).

Note that X_{m} = X ∩ E_{m} for 0 ≤ m ≤ r. We say that (E_{1}, …, E_{r}) is a regular flag of length r for X if the following conditions hold for 1 ≤ m ≤ r:

(i) X_{m} is a proper linear section of X (and hence, dim X_{m} = dim X − m if dim X ≥ m and X_{m} is empty otherwise).

(ii) X_{m} is reduced if dim X ≥ m. (iii) X_{m} is irreducible if dim X ≥ m + 1.

(iv) dim Sing X_{m} = \begin{cases}dim Sing X − m & if dim Sing X ≥ m, \\ −1 & if dim Sing X ≤ m − 1. \end{cases}

(v) deg X_{m} = deg X if dim X ≥ m. If dim X ≥ r, we denote by 𝒰_{r}(X) the set of E ∈ 𝔾_{N−r,N} such that there exists a regular flag (E_{1}, …, E_{r}) for X with E_{r} = E.

By lemma 1.2 and induction we get:

1.3. Proposition. Let X be a closed subvariety in ℙ^{N}. Then 𝒰_{r}(X) contains a nonempty Zariski open set of 𝔾_{N−r,N}. □

1.4. Corollary. Let X be a closed subvariety in ℙ^{N} of dimension n, and let s ∈ ℕ with 0 ≤ s ≤ n − 2. Then the following conditions are equivalent:

(i) There is a proper linear section of codimension s + 1 of X which is a nonsingular variety.

(ii) dim Sing X ≤ s. □ In particular, X is regular in codimension one if and only if there is a nonsingular proper linear section Y of dimension 1 of X.

Following Weil [42, p. 118], we call Y a typical curve on X if X and Y satisfy the above conditions.

2. Weak Lefschetz Theorem for Singular Varieties

We assume now that k is a perfect field of characteristic p ≥ 0. Let ℓ ≠ p be a prime number, and denote by ℚ_{ℓ} the field of ℓ-adic numbers. By H^{i}(X̄, ℚ_{ℓ}) we denote the étale ℓ-adic cohomology space of X̄ and by H_{c}^{i}(X̄, ℚ_{ℓ}) the corresponding cohomology spaces with compact support. We refer to the book of Milne [30] and to the survey of Katz [22] for the definitions and the fundamental theorems on this theory. These are finite dimensional vector spaces over ℚ_{ℓ}, and they vanish for i < 0 as well as for i > 2 dim X. If X is proper the two cohomology spaces coincide. The ℓ-adic Betti numbers are

b_{i,ℓ}(X̄) = dim H_{c}^{i}(X̄, ℚ_{ℓ}) (0 ≤ i ≤ 2n).

If X is a nonsingular projective variety, these numbers are independent of the choice of ℓ [22, p. 27], and in this case we set b_{i}(X̄) = b_{i,ℓ}(X̄). It is conjectured that this is true for any separated scheme X of finite type.

These spaces are endowed with an action of the Galois group g = Gal(\overline{k}/k). We call a map of such spaces g-equivariant if it commutes with the action of g. For c ∈ ℤ, we can consider the Tate twist by c [30, pp. 163-164]. Accordingly, by

H^{i}(X̄, ℚ_{ℓ}(c)) = H^{i}(X̄, ℚ_{ℓ}) ⊗ ℚ_{ℓ}(c)

we shall denote the corresponding twisted copy of H^{i}(X̄, ℚ_{ℓ}).

In this section, we shall prove a generalization to singular varieties of the classical Weak Lefschetz Theorem for cohomology spaces of high degree (cf. [30, Thm. 7.1, p. 253], [20, Thm. 7.1, p. 318]). It seems worthwhile to first review the case of nonsingular varieties, which is discussed below.

2.1. Theorem. Let X be an irreducible projective subscheme of dimension n, and Y a proper linear section of codimension r in X which is a nonsingular variety. Then for each i ≥ n + r, the closed immersion ι : Y ⟶ X induces a canonical g-equivariant linear map

ι_{∗} : H^{i−2r}(Ȳ , ℚ_{ℓ}(−r)) ⟶ H^{i}(X̄, ℚ_{ℓ}),

called the Gysin map, which is an isomorphism for i ≥ n + r + 1 and a surjection for i = n + r.

Proof. There is a diagram

Y ⟶^{ι} X ⟵ U

where U = X − Y. The corresponding long exact sequence in cohomology with support in Y [30, Prop. 1.25, p. 92] will be as follows.

⋯ ⟶ H^{i−1}(Ū, ℚ_{ℓ}) ⟶ Hi ; Ȳ (X̄, ℚ_{ℓ}) ⟶ H^{i}(X̄, ℚ_{ℓ}) ⟶ H^{i}(Ū, ℚ_{ℓ}) ⟶ ⋯

Since U is a scheme of finite type of dimension n which is the union of r affine schemes, we deduce from Lefschetz Theorem on the cohomological dimension of affine schemes [30, Thm. 7.2, p. 253] that H^{i}(Ū, ℚ_{ℓ}) = 0 if i ≥ n + r. Thus, the preceding exact sequence induces a surjection

Hn+r ; Ȳ(X̄, ℚ_{ℓ}) ⟶ H^{n+r}(X̄, ℚ_{ℓ}) ⟶ 0,

and isomorphisms

Hi ; Ȳ (X̄, ℚ_{ℓ}) ⟶^{∼} H^{i}(X̄, ℚ_{ℓ}) (i ≥ n + r + 1).

The cohomology groups with support in Ȳ can be calculated by excision in an étale neighbourhood of Ȳ [30, Prop. 1.27, p. 92]. Let X′ = Reg X be the smooth locus of X. Since Y is nonsingular, we find E ∩ Sing X = ∅ by Lemma 1.1(i), and hence, Y ⊂ X′, that is, X is nonsingular in a neighbourhood of Y. Thus we obtain isomorphisms

Hi ; Ȳ (X̄, ℚ_{ℓ}) ⟶^{∼} Hi ; Ȳ (X̄′, ℚ_{ℓ}), for all i ≥ 0.

Now (Y, X′) is a smooth pair of k-varieties of codimension r as defined in [30, VI.5, p. 241]. By the Cohomological Purity Theorem [30, Thm. 5.1, p. 241], there are canonical isomorphisms

H^{i−2r}(Ȳ , ℚ_{ℓ}(−r)) ⟶^{∼} H_{Y}^{i} (X̄′, ℚ_{ℓ}), for all i ≥ 0.

This yields the desired results.

2.2. Corollary. Let X be a subvariety of dimension n of ℙ^{N} with dim Sing X ≤ s and let Y be a proper linear section of codimension s + 1 ≤ n − 1 of X which is a nonsingular variety. Then:

(i) b_{i,ℓ}(X̄) = b_{i−2s−2}(Ȳ) if i ≥ n + s + 2. (ii) b_{n+s+1,ℓ}(X̄) ≤ b_{n−s−1}(Ȳ).

2.3. Remark. In view of Corollary 1.4, relation (i) implies that the Betti numbers b_{i,ℓ}(X̄) are independent of ℓ for i ≥ n + s + 2.

Now let X be a irreducible closed subscheme of dimension n of ℙ^{N}. If 0 ≤ r ≤ n, let

ℙ^{N} = E_{0} ⊃ E_{1} ⊃ ⋯ ⊃ E_{r}

be a flag in ℱ_{r}(ℙ^{N}) and

X = X_{0},  X_{m} = X_{m−1} ∩ E_{m}  (1 ≤ m ≤ r)

the associated chain of schemes. We say that (E_{1}, …, E_{r}) is a semi-regular flag of length r for X if the schemes X_{1}, …, X_{r−1} are irreducible. We denote by 𝒱_{r}(X) the set of E ∈ 𝔾_{N−r,N} such that there exists a semi-regular flag for X with E_{r} = E. Since 𝒰_{r}(X) ⊂ 𝒱_{r}(X), the set 𝒱_{r}(X) contains a nonempty Zariski open set in 𝔾_{N−r,N}. A semi-regular pair is a couple (X, Y) where X is an irreducible closed subscheme of ℙ^{N} and Y is a proper linear section Y = X ∩ E of codimension r in X, with E ∈ 𝒱_{r}(X). Hence, if r = 1, a semi-regular pair is just a couple (X, X ∩H) where X is irreducible and X ∩ H is a proper hyperplane section of X.

The generalization to singular varieties of Theorem 2.1 that we had alluded to in the beginning of this section is the following.

2.4. Theorem. (General Weak Lefschetz Theorem, high degrees). Let (X, Y) be a semi-regular pair with dim X = n, Y of codimension r in X and dim Sing Y = σ. Then for each i ≥ n + r + σ + 1 there is a canonical g-equivariant linear map

ι_{∗} : H^{i−2r}(Ȳ , ℚ_{ℓ}(−r)) ⟶ H^{i}(X̄, ℚ_{ℓ})

which is an isomorphism for i ≥ n+r +σ +2 and a surjection for i = n+r +σ +1. If X and Y are nonsingular, and if there is a regular flag (E_{1}, …, E_{r}) for X with E_{r} ∩ X = Y, then ι_{∗} is the Gysin map induced by the immersion ι : Y ⟶ X.

We call ι_{∗} the Gysin map, since it generalizes the classical one when X and Y are nonsingular.

Proof. For hyperplane sections, i. e. if r = 1, this is a result of Skorobogatov [39, Cor. 2.2]. Namely, he proved that if X ∩ H is a proper hyperplane section of X and if

α = dim X + dim Sing(X ∩ H),

then, for each i ≥ 0, there is a g-equivariant linear map

H^{α+i}(X̄ ∩ H̄, ℚ_{ℓ}(−1)) ⟶ H^{α+i+2}(X̄, ℚ_{ℓ}) (10)

which is a surjection for i = 0 and an isomorphism for i > 0. Moreover he proved also that if X and X ∩ H are nonsingular then ι_{∗} is the Gysin map. In the general case we proceed by iteration. Let (E_{1}, …, E_{r}) be a semi-regular flag of ℱ_{r}(ℙ^{N}) such that Y = X ∩ E_{r} and let

X = X_{0},  X_{m} = X_{m−1} ∩ E_{m}  (1 ≤ m ≤ r).

be the associated chain of schemes. First,

dim X_{m} = n − m for 1 ≤ m ≤ r. (11)

In fact, let η_{m} = dim X_{m−1} − dim X_{m}. Then 0 ≤ η_{m} ≤ 1 by Krull’s Principal Ideal Theorem. But

η_{1} + ⋯ + η_{r} = dim X_{0} − dim X_{r} = r.

Hence η_{m} = 1 for 1 ≤ m ≤ r, which proves (11). This relation implies that X_{m} is a proper hyperplane section of X_{m−1}. Since X_{m−1} is irreducible by hypothesis, the couple (X_{m−1}, X_{m}) is a semi-regular pair for 1 ≤ m ≤ r and we can apply the Theorem in the case of codimension one. From Lemma 1.1 we deduce

dim X_{m} − dim Sing X_{m} ≥ dim Y − dim Sing Y = n − r − σ,

and hence, dim Sing X_{m} ≤ r − m + σ. Then

dim X_{m−1} + dim Sing X_{m} ≤ α(m),

where α(m) = n + r − 2m + σ + 1. We observe that α(m − 1) = α(m) + 2. Hence from (10) we get a map

H^{α(m)+i}(X̄_{m}, ℚ_{ℓ}(−m)) ⟶ H^{α(m−1)+i}(X̄_{m−1}, ℚ_{ℓ}(−m + 1))

which is a surjection for i = 0 and an isomorphism for i > 0. The composition of these maps gives a map

ι_{∗} : H^{α(r)+i}(Ȳ , ℚ_{ℓ}(−r)) ⟶ H^{α(0)+i}(X̄, ℚ_{ℓ})

which the same properties. Since

α(r) = n − r + σ + 1, α(0) = n + r + σ + 1,

Substituting j = α(0) + i = α(r) + 2r + i, we get

ι_{∗} : H^{j−2r}(Ȳ , ℚ_{ℓ}(−r)) ⟶ H^{j}(X̄, ℚ_{ℓ}),

fulfilling the required properties. Now recall from [30, Prop. 6.5(b), p. 250] that if

X_{2} ⟶^{ι_{2}} X_{1} ⟶^{ι_{1}} X_{0}

is a so-called smooth triple over k, then the Gysin map for ι_{1} ∘ ι_{2} is the composition of the Gysin maps for ι_{1} and ι_{2}, which proves the last assertion of the Theorem.

The following Proposition gives a criterion for a pair to be semi-regular.

2.5. Proposition. If X is a irreducible closed subscheme of ℙ^{N} and if Y is a proper linear section of X of dimension ≥ 1 which is regular in codimension one, then (X, Y) is a semi-regular pair. Moreover Y is irreducible.

Proof. As in the proof of Theorem 2.4, we first prove the proposition if Y is a regular hyperplane section of X. In that case (X, Y) is a semi-regular pair, as we pointed out, and the only statement to prove is that Y is irreducible. If Y is regular in codimension one, then

σ = dim Sing Y ≤ dim Y − 2 ≤ n − 3.

This implies that 2n ≥ n + σ + 3 and we can take i = 2n in Theorem 2.4 in order to get an isomorphism

H^{2n−2}(Ȳ , ℚ_{ℓ}(−1)) ⟶^{∼} H^{2n}(X̄, ℚ_{ℓ}).

Now for any scheme X of dimension n, the dimension of H^{2n}(X̄, ℚ_{ℓ}) is equal to the number of irreducible components of X of dimension n, as can easily be deduced from the Mayer-Vietoris sequence [30, Ex. 2.24, p. 110]. Since Y is a proper linear section, it is equidimensional and the irreducibility of Y follows from the irreducibility of X. The general case follows by iteration.

Observe that Theorem 2.1 is an immediate consequence of Theorem 2.4 and Proposition 2.5.

2.6. Remark (Weak Lefschetz Theorem, low degrees). We would like to point out the following result, although we shall not use it. Assume that the resolution of singularities is possible, that is, the condition (R_{n,p}) stated in section 7 below, holds. Let X be a closed subscheme of dimension n in ℙ^{N}, and let there be given a diagram

Y ⟶^{ι} X ⟵ U

where ι is the closed immersion of a proper linear section Y of X of codimension r, and where U = X − Y. If U is a local complete intersection, then the canonical g-equivariant linear map

ι^{∗} : H^{i}(X̄, ℚ_{ℓ}) ⟶ H^{i}(Ȳ , ℚ_{ℓ})

is an isomorphism if i ≤ n − r − 1 and an injection if i = n − r.

This theorem is a consequence of the Global Lefschetz Theorem of Grothendieck for low degrees [13, Cor. 5.7 p. 280]. Notice that no hypotheses of regularity are put on Y in that statement. Moreover, if U is nonsingular, the conclusions of the theorem are valid without assuming condition (R_{n,p}) [20, Thm. 7.1, p. 318].

2.7. Remark (Poincaré Duality). By combining Weak Lefschetz Theorem for high and low degrees, we get a weak version of Poincaré Duality for singular varieties. Namely, assume that (R_{n,p}) holds and let X be a closed subvariety of dimension n in ℙ^{N} which is a local complete intersection such that dim Sing X ≤ s. Then, for 0 ≤ i ≤ n − s − 2 there is a nondegenerate pairing

H^{i}(X̄, ℚ_{ℓ}) × H^{2n−i}(X̄, ℚ_{ℓ}(n)) ⟶ ℚ_{ℓ}.

Furthermore, if we denote this pairing by (ξ, η), then

(g.ξ, g.η) = (ξ, η) for every g ∈ g.

In particular,

b_{i,ℓ}(X̄) = b_{2n−i}(X̄) for 0 ≤ i ≤ n − s − 2,

and these numbers are independent of ℓ.

3. Cohomology of Complete Intersections

Let k be a field. A closed subscheme X of ℙ^{N} of codimension r is a complete intersection if X is the closed subscheme determined by an ideal ℑ generated by r homogeneous polynomials f_{1}, …, f_{r}.

A complete intersection is a local complete intersection, and in particular X is Cohen-Macaulay and equidimensional. Moreover, if dim X ≥ 1, then X is connected, hence X is integral if it is regular in codimension one.

The multidegree d = (d_{1}, …, d_{r}) of the system f_{1}, …, f_{r}, usually labelled so that d_{1} ≥ ⋯ ≥ d_{r}, depends only on ℑ and not of the chosen system of generators f_{1}, …, f_{r}, since the Hilbert series of the homogeneous coordinate ring of X equals

H(T) = \frac{(1 − T^{d_{1}})(1 − T^{d_{2}}) ⋯ (1 − T^{d_{r}})}{(1 − T)^{N+1}}

[32, Ex. 7.15, p. 350] or [7, Prop. 6, p. AC VIII.50]. This implies

deg X = d_{1} ⋯ d_{r}.

Now assume that k is separably closed. First of all, recall that if X = ℙ^{n}, and if 0 ≤ i ≤ 2n, then [30, Ex. 5.6, p. 245]:

(12) H^{i}(X, ℚ_{ℓ}) = \begin{cases}ℚ_{ℓ}(−i/2) & if i is even \\ 0 & if i is odd. \end{cases} Consequently, dim H^{i}(ℙ^{n}, ℚ_{ℓ}) = ε_{i} for 0 ≤ i ≤ 2n, where we set

ε_{i} = \begin{cases}1 & if i is even \\ 0 & if i is odd. \end{cases}

Now let X be a nonsingular projective subvariety of ℙ^{N} of dimension n ≥ 1. For any i ≥ 0, the image of the canonical morphism H^{i}(ℙ^{N}, ℚ_{ℓ}) ⟶ H^{i}(X, ℚ_{ℓ}) is isomorphic to H^{i}(ℙ^{n}, ℚ_{ℓ}). Hence, one obtains a short exact sequence

0 ⟶ H^{i}(ℙ^{n}, ℚ_{ℓ}) ⟶ H^{i}(X, ℚ_{ℓ}) ⟶ P^{i}(X, ℚ_{ℓ}) ⟶ 0 (13)

where P^{i}(X, ℚ_{ℓ}) is the cokernel of the canonical morphism, called the primitive part of H^{i}(X, ℚ_{ℓ}). If i is even, the image of H^{i}(ℙ^{n}, ℚ_{ℓ}) is the one-dimensional vector space generated by the cup-power of order i/2 of the cohomology class of a hyperplane section. Hence, if we define the primitive i-th Betti number of a nonsingular projective variety X over k as

b′_{i,ℓ}(X) = dim P^{i}(X, ℚ_{ℓ}),

then by (12) and (13), we have

b_{i,ℓ}(X) = b′_{i,ℓ}(X) + ε_{i}.

3.1. Proposition. Let X be a nonsingular complete intersection in ℙ^{N} of codimension r, of multidegree d, and let dim X = n = N − r.

(i) If i ≠ n, then P^{i}(X, ℚ_{ℓ}) = 0 for 0 ≤ i ≤ 2n. Consequently, X satisfies (12) for these values of i.

(ii) The n-th Betti number of X depends only on n, N and d.

Proof. Statement (i) is easily proved by induction on r, if we use the Veronese embedding, Weak Lefschetz Theorem 2.4 and Poincaré Duality in the case of nonsingular varieties. Statement (ii) follows from Theorem 4.1 below.

In view of Proposition 3.1, we denote by b′_{n}(N, d) the primitive n-th Betti number of any nonsingular complete intersection in ℙ^{N} of codimension r = N − n and of multidegree d. It will be described explicitly in Section 4.

The cohomology in lower degrees of general (possibly singular) complete intersections can be calculated with the help of the following simple result.

3.2. Proposition. Let X be a closed subscheme of ℙ^{N} defined by the vanishing of r forms. Then the g-equivariant restriction map

ι^{∗} : H^{i}(ℙ^{N}, ℚ_{ℓ}) ⟶ H^{i}(X, ℚ_{ℓ})

is an isomorphism for i ≤ N − r − 1, and is injective for i = N − r.

Proof. The scheme U = ℙ^{N} − X is a scheme of finite type of dimension N which is the union of r affine open sets. From Affine Lefschetz Theorem [30, Thm. 7.2, p. 253], we deduce

H^{i}(U, ℚ_{ℓ}) = 0 if i ≥ N + r.

Since U is smooth, by Poincaré Duality we get

H_{c}^{i}(U, ℚ_{ℓ}) = 0 if i ≤ N − r.

Now, the excision long exact sequence in compact cohomology [30, Rem. 1.30, p. 94] gives:

⋯ ⟶ H_{c}^{i}(U, ℚ_{ℓ}) ⟶ H^{i}(ℙ^{N}, ℚ_{ℓ}) ⟶ H^{i}(X, ℚ_{ℓ}) ⟶ H_{c}^{i+1}(U, ℚ_{ℓ}) ⟶ ⋯ (14)

from which the result follows.

3.3. Remark. If a closed subscheme X of ℙ^{N} of dimension n is defined by the vanishing of r forms, then N − r ≤ n, and it can occur that N − r < n. However, we can prove that if condition (R_{n,p}) stated in section 7 holds true, then the map ι^{∗} in Proposition 3.2 is an isomorphism for i ≤ n − 1, and is injective for i = n. The proof uses the Weak Lefschetz Theorem for low degrees quoted in Remark 2.6. However, we shall not need this result.

We shall now study the cohomology in higher degrees of general complete intersections.

3.4. Proposition. Let X be a complete intersection in ℙ^{N} of dimension n ≥ 1 and multidegree d with dim Sing X ≤ s. Then:

(i) Relation (12) holds, and hence, b′_{i}(X) = 0, if n + s + 2 ≤ i ≤ 2n. (ii) b_{n+s+1,ℓ}(X) ≤ b_{n−s−1}(N − s − 1, d). (iii) Relation (12) holds, and hence, b′_{i}(X) = 0, if 0 ≤ i ≤ n − 1.

Proof. Thanks to Proposition 1.3, there are regular flags for X; let (E_{1}, …, E_{s+1}) be one of them, and let Y = X ∩E_{s+1}. Then Y is a nonsingular variety by definition of regular flags, and a complete intersection. Applying the Weak Lefschetz Theorem 2.1, we deduce that the Gysin map

ι_{∗} : H^{i−2s−2}(Y, ℚ_{ℓ}(−s − 1)) ⟶ H^{i}(X, ℚ_{ℓ})

is an isomorphism for i ≥ n+s+2 and a surjection for i = n+s+1, which proves (i) and (ii) in view of Proposition 3.1. Assertion (iii) follows from Proposition 3.2.

3.5. Remarks. (i) The case s = 0 of this theorem is a result of I.E. Shparlinskiĭ and A.N. Skorobogatov [38, Thm. 2.3], but the proof of (a) and (b) in that Theorem is unclear to us: in any case their proof use results which are valid only if the characteristic is 0 or if one assumes the resolution of singularities (condition (R_{n,p}) in section 7 below). (ii) In [21, proof of Thm. 1], N. Katz proves (i) with the arguments used here to prove Proposition 7.1 below.

3.6. Remark. For further reference, we note that if s ≥ 0 and if i = n + s + 1 is even, then H^{i}(X, ℚ_{ℓ}) contains a subspace isomorphic to ℚ_{ℓ}(−i/2). In fact, with the notations of the proof of Proposition 3.4, the map

ι_{∗} : H^{n−s−1}(Y, ℚ_{ℓ}(−s − 1)) ⟶ H^{n+s+1}(X, ℚ_{ℓ})

is a surjection.

4. The Central Betti Number of Complete Intersections

We now state a well-known consequence of the Riemann-Roch-Hirzebruch Theorem [17, Satz 2.4, p. 136], [20, Cor. 7.5]; see [3, Cor 4.18] for an alternative proof. If ν = (ν_{1}, …, ν_{r}) ∈ ℕ^{r}, and if d = (d_{1}, …, d_{r}) ∈ (ℕ^{×})^{r}, we define

d^{ν} = d_{1}^{ν_{1}} … d_{r}^{ν_{r}}.

If c ≥ 1, let

M(c) = {(ν_{1}, …, ν_{r}) ∈ ℕ^{r} | ν_{1} + ⋯ + ν_{r} = c and ν_{i} ≥ 1 for 1 ≤ i ≤ r} .

4.1. Theorem. The primitive n-th Betti number of any nonsingular complete intersection in ℙ^{N} of codimension r = N − n and of multidegree d is

b′_{n}(N, d) = (−1)^{N−r+1}(N − r + 1) + (−1)^{N} ∑_{c=r}^{N}(−1)^{c}\binom{N + 1}{c + 1} ∑_{ν∈M(c)} d^{ν}. □

We now give estimates on the primitive n-th Betti number of a nonsingular complete intersection.

4.2. Proposition. Let d = (d_{1}, …, d_{r}) ∈ (ℕ^{×})^{r} and b′_{n}(N, d) be the n^{th} primitive Betti number of a nonsingular complete intersection in ℙ^{N} of dimension n = N − r and multidegree d. Let δ = max(d_{1}, …, d_{r}) and d = deg X = d_{1} ⋯ d_{r}. Then, for r ≥ 1, we have

b′_{n}(N, d) ≤ (−1)^{n+1}(n + 1) + d\binom{N + 1}{n}(δ + 1)^{n}.

In particular,

b′_{n}(N, d) ≤ \binom{N + 1}{n}(δ + 1)^{N}.

Proof. Observe that for any c ≥ r and ν ∈ M(c), we have

d^{ν} ≤ dδ^{c−r} and |M(c)| = \binom{c − 1}{r − 1}.

Hence, by Theorem 4.1,

b′_{n}(N, d) − (−1)^{n+1}(n + 1) ≤ ∑_{c=r}^{N} \binom{N + 1}{c + 1}\binom{c − 1}{r − 1}dδ^{c−r}.

Now, for c ≥ r ≥ 1, we have

\binom{N + 1}{c + 1} = \frac{N(N + 1)}{c(c + 1)}\binom{N − 1}{c − 1} ≤ \frac{N(N + 1)}{r(r + 1)}\binom{N − 1}{c − 1},

and moreover,

∑_{c=r}^{N} \binom{N − 1}{c − 1}\binom{c − 1}{r − 1}δ^{c−r} = \binom{N − 1}{r − 1} ∑_{c=r}^{N} \binom{n}{c − r}δ^{c−r} = \binom{N − 1}{r − 1}(δ + 1)^{n}.

It follows that

b′_{n}(N, d) − (−1)^{n+1}(n + 1) ≤ d \frac{N(N + 1)}{r(r + 1)}\binom{N − 1}{r − 1}(δ + 1)^{n} = d\binom{N + 1}{r + 1}(δ + 1)^{n}.

This proves the first assertion. The second assertion is trivially satisfied if n = 0, while if n ≥ 1, then

\binom{N+1}{n} ≥ N + 1 ≥ n + 1,

and since d = d_{1} ⋯ d_{r} < (δ + 1)^{r}, we have d(δ + 1)^{n} ≤ (δ + 1)^{N} − 1 so that

(−1)^{n+1}(n + 1) + d\binom{N+1}{n}(δ + 1)^{n} ≤ (n + 1) + (δ + 1)^{N} \binom{N+1}{n} − \binom{N+1}{n}
≤ \binom{N+1}{n}(δ + 1)^{N},

and the second assertion is thereby proved.

4.3. Examples. The calculations in the examples below are left to the reader.

(i) The primitive n-th Betti number of nonsingular hypersurfaces of dimension n and degree d is equal to:

b′_{n}(N, d) = \frac{d − 1}{d}((d − 1)^{N} − (−1)^{N}) ≤ (d − 1)^{N} − ε_{N}.

In particular, if N = 2 (plane curves), then b_{1}(d) = (d−1)(d−2), whereas if N = 3 (surfaces in 3-space), then

b′_{2}(3, d) = (d − 1)((d − 1)(d − 2) + 1) = d^{3} − 4d^{2} + 6d − 3.

(ii) The Betti number of a nonsingular curve which is a complete intersection of r = N − 1 forms in ℙ^{N} of multidegree d = (d_{1}, …, d_{r}) is equal to

b_{1}(N, d) = b′_{1}(N, d) = (d_{1} ⋯ d_{r})(d_{1} + ⋯ + d_{r} − N − 1) + 2.

Now observe that if r and d_{1}, …, d_{r} are any positive integers, then

d_{1} + ⋯ + d_{r} ≤ (d_{1} ⋯ d_{r}) + r − 1,

and the equality holds if and only if either r = 1 or r > 1 and at least r − 1 among the numbers d_{i} are equal to 1.

To see this, we proceed by induction on r. The case r = 1 is trivial and if r = 2, then the assertion follows easily from the identity

d_{1}d_{2} + 1 − (d_{1} + d_{2}) = (d_{1} − 1)(d_{2} − 1).

The inductive step follows readily using the assertion for r = 2.

For the nonsingular curve above, if we let d = d_{1} ⋯ d_{r}, then d is its degree and

b_{1}(N, d) = d(d_{1} + ⋯ + d_{r} − r − 2) + 2 ≤ d(d − 3) + 2 = (d − 1)(d − 2).

Since for a nonsingular plane curve of degree d the first Betti number is, as noted in (i), always equal to (d − 1)(d − 2), it follows that in general,

b_{1}(N, d) ≤ (d − 1)(d − 2)

and the equality holds if and only if the corresponding curve is a nonsingular plane curve. (iii) The Betti number of a nonsingular surface which is a complete intersection of r = N − 2 forms in ℙ^{N} of multidegree d = (d_{1}, …, d_{r}) is equal to

b_{2}(N, d) = b′_{2}(N, d) + 1 = d (\binom{r + 3}{2} − (r + 3) ∑_{1≤i≤r} d_{i} + ∑_{1≤i,j≤r} d_{i}d_{j}) − 2,

where d = d_{1}d_{2} ⋯ d_{r} is the degree of the surface. (iv) The primitive n-th Betti number of a complete intersection defined by r = 2 forms of the same degree d is equal to

b′_{n}(N, (d, d)) = (N − 1)(d − 1)^{N} + 2 \frac{d − 1}{d}((d − 1)^{N−1} + (−1)^{N}).

5. Zeta Functions and the Trace Formula

In this section k = 𝔽_{q} is the finite field with q elements. We denote by k_{r} the subfield of \overline{k} which is of degree r over k. Let X be a separated scheme of finite type defined over the field k. We denote by X̄ = X ⊗_{k} \overline{k} the scheme deduced from X by base field extension from k to \overline{k} ; it remains unchanged if we replace k by one of its extensions. The zeta function of X is

Z(X, T) = exp ∑_{r=1}^{∞} \frac{T^{r}}{r} |X(k_{r})|, (15)

where T is an indeterminate. From the work of Dwork and Grothendieck (see, for example, the Grothendieck-Lefschetz Trace Formula below), we know that the function Z(X, T) is a rational function of T. This means that there are two families of complex numbers (α_{i})_{i∈I} and (β_{j})_{j∈J}, where I and J are finite sets, such that

Z(X, T) = \frac{∏_{j∈J}(1 − β_{j}T)}{∏_{i∈I}(1 − α_{i}T)}. (16)

We assume that the fraction in the right-hand side of the above equality is irreducible. Thus, the family (α_{i})_{i∈I} (resp. (β_{j})_{j∈J}) is exactly the family of poles (resp. of zeroes) of Z(X, T), each number being enumerated a number of times equal to its multiplicity. We call the members of the families (α_{i})_{i∈I} and (β_{j})_{j∈J} the characteristic roots of Z(X, T). The degree deg Z(X, T) of Z(X, T) is the degree of its numerator minus the degree of its denominator; the total degree tot.deg Z(X, T) of Z(X, T) is the sum of the degrees of its numerator and of its denominator. In the usual way, from the above expression of Z(X, T), we deduce

|X(k_{r})| = ∑_{i∈I} α_{i}^{r} − ∑_{j∈J} β_{j}^{r}.

In order to simplify the notation, we write now

H_{c}^{i}(X̄) = H_{c}^{i}(X̄, ℚ_{ℓ}),

where ℓ is a prime number other than p; in the case of proper subschemes, we may use H^{i}(X̄) instead. If we denote by φ the element of g given by φ(x) = x^{q}, then the geometric Frobenius element F of g is defined to be the inverse of φ. In the twisted space ℚ_{ℓ}(c), we have

F.x = q^{−c} x for x ∈ ℚ_{ℓ}(c).

The geometric Frobenius element F canonically induces on H_{c}^{i}(X̄) an endomorphism denoted by F | H_{c}^{i}(X̄). A number α ∈ \overline{ℚ_{ℓ}} is pure of weight r if α is an algebraic integer and if |ι(α)| = q^{r/2} for any embedding ι of \overline{ℚ_{ℓ}} into ℂ.

We recall the following fundamental results.

Theorem. Let X be a separated scheme of finite type over k, of dimension n. Then the Grothendieck-Lefschetz Trace Formula holds:

|X(k_{r})| = ∑_{i=0}^{2n}(−1)^{i} Tr(F^{r} | H_{c}^{i}(X̄)), (17)

and Deligne’s Main Theorem holds:

(18) The eigenvalues of F | H_{c}^{i}(X̄) are pure of weight ≤ i.

See, for instance, [30, Thm. 13.4, p. 292] for the Grothendieck-Lefschetz Trace Formula, and [10, Thm. 1, p. 314] for Deligne’s Main Theorem.

The Trace Formula (17) is equivalent to the equality

Z(X, T) = \frac{P_{1,ℓ}(X, T) ⋯ P_{2n−1,ℓ}(X, T)}{P_{0,ℓ}(X, T) ⋯ P_{2n,ℓ}(X, T)}, (19)

where P_{i,ℓ}(X, T) = det(1 − T F | H_{c}^{i}(X̄)). We can write

P_{i,ℓ}(X, T) = ∏_{j=1}^{b_{i,ℓ}}(1 − ω_{ij,ℓ}T), (20)

where ω_{ij,ℓ} ∈ \overline{ℚ_{ℓ}}, and b_{i,ℓ} = b_{i,ℓ}(X̄) = deg P_{i,ℓ}(X, T). The numbers ω_{ij,ℓ} are called the reciprocal roots of P_{i,ℓ}(X, T). The Trace Formula (17) may be written as

|X(k)| = ∑_{i=0}^{2n} (−1)^{i} ∑_{j=1}^{b_{i,ℓ}} ω_{ij,ℓ},

with the convention that if b_{i,ℓ} = 0, then the value of the corresponding sum is zero. The compact étale ℓ-adic Euler-Poincaré characteristic of X̄ is

χ_{ℓ}(X̄) = ∑_{i=0}^{2n}(−1)^{i} b_{i,ℓ}(X̄).

On the other hand, let us define

σ_{ℓ}(X̄) = ∑_{i=0}^{2n} b_{i,ℓ}(X̄).

The equality between the right-hand sides of (16) and (19) imply

deg Z(X, T) = χ_{ℓ}(X̄).

Hence, the compact étale Euler-Poincaré characteristic of X̄ is independent of ℓ, and the number deg Z(X, T) depends only on X̄. In the same way we find

tot.deg Z(X, T) ≤ σ_{ℓ}(X̄) .

Here, the two sides may be different because of a possibility of cancellations occurring in the right-hand side of (19).

If k′ is any field, let us say that a projective scheme X defined over k′ is of type (m, N, d) if X is a closed subscheme in ℙ_{k′}^{N} which can be defined, schemetheoretically, by the vanishing of a system of m nonzero forms with coefficients in k′, of multidegree d = (d_{1}, …, d_{m}).

We now state a result of N. Katz [24, Cor. of Th. 3], whose proof is based on a result of Adolphson and Sperber [2, Th. 5.27]. If one checks the majorations in the proof of [24, loc. cit.], the number 9 appearing therein can be replaced by the number 8 in the inequality below.

Theorem (Katz’s Inequality). Let X be a closed subscheme in ℙ^{N} defined over an algebraically closed field, and of type (m, N, d). If d = (d_{1}, …, d_{m}), then let δ = max(d_{1}, …, d_{m}). We have

σ_{ℓ}(X̄) ≤ 8 × 2^{m} × (mδ + 3)^{N+1}. □ (21)

We would like to compare the number of points of X and that of the projective space of dimension equal to that of X. Hence, if dim X = n, we introduce the rational function

\frac{Z(X, T)}{Z(ℙ^{n}, T)} = exp ∑_{r=1}^{∞} \frac{T^{r}}{r} ( |X(k_{r})| − |ℙ^{n}(k_{r})| ) . (22)

where

Z(ℙ^{n}, T) = \frac{1}{(1 − T) … (1 − q^{n}T)}.

Since τ(X) ≤ σ_{ℓ}(X̄) + n, from Katz’s Inequality (21) we deduce :

5.1. Proposition. Given any projective scheme X of dimension n defined over k, let

τ(X) = tot.deg \frac{Z(X, T)}{Z(ℙ^{n}, T)} .

Also, given any nonnegative integers m, N, and d = (d_{1}, …, d_{m}) ∈ (ℕ^{×})^{m}, let

τ_{k}(m, N, d) = sup_{X} τ(X),

where the supremum is over projective schemes X defined over k, and of type (m, N, d). If δ = max(d_{1}, …, d_{m}), then

τ_{k}(m, N, d) ≤ 9 × 2^{m} × (mδ + 3)^{N+1},

and, in particular, τ_{k}(m, N, d) is bounded by a constant independent of the field k. □

6. Number of Points of Complete Intersections

In this section k = 𝔽_{q}. We now state our main Theorem on the number of points of complete intersections. The number b′_{n}(N, d) is defined in Theorem 4.1.

6.1. Theorem. Let X be an irreducible complete intersection of dimension n in ℙ_{k}^{N}, defined by r = N − n equations, with multidegree d = (d_{1}, …, d_{r}), and choose an integer s such that dim Sing X ≤ s ≤ n − 1. Then

| |X(k)| − π_{n} | ≤ b′_{n−s−1}(N − s − 1, d) q^{(n+s+1)/2} + C_{s}(X)q^{(n+s)/2},

where C_{s}(X) is a constant independent of k. If X is nonsingular, then C_{−1}(X) = 0. If s ≥ 0, then

C_{s}(X) = ∑_{i=n}^{n+s} b_{i,ℓ}(X̄) + ε_{i}

and upon letting δ = max(d_{1}, …, d_{r}), we have

C_{s}(X) ≤ τ(X) ≤ τ(r, N, d) ≤ 9 × 2^{r} × (rδ + 3)^{N+1}.

Proof. Equality (19) implies

\frac{Z(X, T)}{Z(ℙ^{n}, T)} = \frac{P_{1,ℓ}(X, T) ⋯ P_{2n−1,ℓ}(X, T)(1 − T) … (1 − q^{n}T)}{P_{0,ℓ}(X, T) ⋯ P_{2n,ℓ}(X, T)},

and therefore, in view of (20) and (22), we get

|X(k)| − π_{n} = ∑_{i=0}^{2n} (−ε_{i} q^{i/2} + (−1)^{i} ∑_{j=1}^{b_{i}} ω_{ij,ℓ}) (23)

where it may be recalled that ε_{i} = 1 if i is even and 0 otherwise. Moreover, when all the cancellations have been performed, the number of terms of the right-hand side of (23) is at most equal to the total degree τ(X) of the rational fraction above. Now from Proposition 3.4(i) and (iii) we have

P_{i,ℓ}(X, T) = 1 − ε_{i}q^{i/2}T if i ∉ [n, n + s + 1] .

Hence, these polynomials cancel if Z(X, T)/Z(ℙ^{n}, T) is in irreducible form. Accordingly, from (23) we deduce

|X(k)| − π_{n} = ∑_{i=n}^{n+s+1} (−ε_{i} q^{i/2} + (−1)^{i} ∑_{j=1}^{b_{i}} ω_{ij,ℓ}),

and this gives

| |X(k)| − π_{n} | ≤ A + B,

where A = 0 if s = n − 1 and

A = | −ε_{n+s+1} q^{(n+s+1)/2} + (−1)^{n+s+1} ∑_{j=1}^{b_{n+s+1}} ω_{(n+s+1)j,ℓ} | if s < n − 1,

while

B = | ∑_{i=n}^{n+s} (−ε_{i} q^{i/2} + (−1)^{i} ∑_{j=1}^{b_{i}} ω_{ij,ℓ}) | if s ≤ n − 1.

Now by Deligne’s Main Theorem (18), the numbers ω_{ij,ℓ} are pure of weight ≤ i/2, and thus

B ≤ C_{s}(X) q^{(n+s)/2} where C_{s}(X) = ∑_{i=n}^{n+s} b_{i,ℓ}(X̄) + ε_{i} .

Clearly, C_{s}(X) is independent of k and is at most equal to the number of terms in the right-hand side of (23). Hence by Proposition 5.1,

C_{s}(X) ≤ τ(X) ≤ τ(r, N, d) ≤ 9 × 2^{r} × (rδ + 3)^{N+1}.

Next, if s = n − 1, then A = 0. Suppose s ≤ n − 2. If n + s + 1 is odd (in particular, if s = n − 2), then by Proposition 3.4(ii).

A = | ∑_{j=1}^{b_{n+s+1}} ω_{(n+s+1)j,ℓ} | ≤ b_{n−s−1}(N − s − 1, d) q^{(n+s+1)/2}.

On the other hand, if 0 ≤ s ≤ n − 3 and if n + s + 1 is even, then

A = | ∑_{j=1}^{b_{n+s+1}} ω_{(n+s+1)j,ℓ} − q^{(n+s+1)/2} |.

But in view of Remark 3.6, H^{n+s+1}(X, ℚ_{ℓ}) contains a subspace which is isomorphic to ℚ_{ℓ}(−(n+s+1)/2). Thus q^{(n+s+1)/2} is an eigenvalue of the highest possible weight of F | H_{c}^{i}(X̄), and hence it is a reciprocal root of P_{n+s+1}(X, T). It follows that

A ≤ (b_{n−s−1}(N − s − 1, d) − 1) q^{(n+s+1)/2}.

Thus in any case, A ≤ b_{n−s−1}(N − s − 1, d) q^{(n+s+1)/2}. The case when X is nonsingular follows similarly using Proposition 3.1.

The case where X is nonsingular is Deligne’s Theorem [9, Thm. 8.1]. In the opposite, since we always have s ≤ n − 1, Theorem 6.1 implies the following weak version of the Lang-Weil inequality for complete intersections:

|X(k)| − π_{n} = O (q^{n−(1/2)}) .

We shall obtain a much better result in Theorem 11.1. Nevertheless, the following Corollary shows that we can obtain the Lang-Weil inequality (in fact, a stronger result) as soon as some mild regularity conditions are satisfied.

If X is regular in codimension one, i. e., if dim Sing X ≤ n − 2, then, as stated in the beginning of Section 3, X is integral, and so the hypothesis that X is irreducible is automatically fulfilled. Moreover, notice that for a complete intersection X in ℙ^{N}, Serre’s Criterion of Normality [EGA 4.2, Thm. 5.8.6, p. 108] implies that X is normal if and only if it is regular in codimension one.

6.2. Corollary. If X is a normal complete intersection of dimension n in ℙ_{k}^{N} with multidegree d, then

| |X(k)| − π_{n} | ≤ b′_{1}(N − n + 1, d) q^{n−(1/2)} + C_{n−2}(X)q^{n−1},

where C_{n−2}(X) is as in Theorem 6.1. Moreover, if d = deg X, then

b′_{1}(N − n + 1, d) ≤ (d − 1)(d − 2),

with equality holding if and only if X is a hypersurface.

Proof. Follows from Theorem 6.1 with s = n − 2 and the observations in Example 4.3 (ii).

6.3. Remark. It is also worthwhile to write down explicitly the particular case of a complete intersection X of dimension n in ℙ_{k}^{N} regular in codimension 2. Namely, if dim Sing X ≤ n − 3, then

| |X(k)| − π_{n} | ≤ b′_{2}(N − n − 2, d) q^{n−1} + C_{n−3}(X)q^{n−(3/2)}.

7. Complete Intersections with isolated singularities

We now prove an inequality for the central Betti number of complete intersections with only isolated singularities. Unfortunately, the proof of this result depends on the following condition:

(R_{n,p}) The resolution of singularities holds in characteristic p for excellent local rings of dimension at most n.

This condition means the following. Let A be an excellent local ring of equal characteristic p of dimension ≤ n and X = Spec A. Let U be a regular open subscheme of X, and S = X − U. Then, there is a commutative diagram

Ũ ⟶ X̃ ⟵ S̃
↓  ↓^{π}  ↓
U ⟶ X ⟵ S

where X̃ is a regular scheme and where π is a proper morphism which is a birational isomorphism, and an isomorphism when restricted to Ũ. Moreover S̃ = X̃ − Ũ is a divisor with normal crossings, that is, a family of regular schemes of pure codimension 1 such that any subfamily intersects properly.

Recall that the resolution of singularities holds in characteristic 0, and that (R_{2,p}) holds for any p. For details, see the book [1] by Abhyankar and the survey [29] by Lipman.

7.1. Proposition. Assume that (R_{n,p}) holds. Let k be a separably closed field, and X be a complete intersection in ℙ^{N} of dimension n ≥ 1 with dim Sing X = 0. Then b_{n,ℓ}(X) ≤ b_{n}(N, d).

Proof. Assume that X is defined by the vanishing of the system of forms (f_{1}, …, f_{r}). Since k is separably closed, there exists a nonsingular complete intersection Y of ℙ^{N} of codimension r of multidegree d. Let (g_{1}, …, g_{r}) be a system defining Y. The one-parameter family of polynomials

T g_{j}(X_{0}, …, X_{N}) + (1 − T)f_{j}(X_{0}, …, X_{N}), (1 ≤ j ≤ r)

define a scheme Z coming with a morphism

π_{0} : Z ⟶ 𝔸_{k}^{1}.

The morphism π_{0} is proper and flat. Let S = Spec k{T } where k{T } is the strict henselization of k[T] at the ideal generated by T [30, p. 38]. Then S is the spectrum of a discrete valuation ring with separably closed residual field, and S has two points: the closed one s, with residual field k and the generic one η (such a scheme is called a trait strictement local in French). Let Z = Z ×_{k} S the scheme obtained by base change :

Z ⟶ Z
π↓  ↓^{π_{0}}
S ⟶ 𝔸_{k}^{1}

Then π is a proper and flat morphism, and the closed fiber Z_{s} = π^{−1}(s) is isomorphic to X. If η̄ is a geometric point of S mapping to η, the geometric fiber Zη̄ = Z × κ(η̄) is a nonsingular complete intersection of codimension r. We thus get a diagram

Zη̄ ⟶ Z ⟵ X
↓  ^{π}↓  ↓
η̄ ⟶ S ⟵ s

By [8, Eq. 2.6.2, p. 9], there is a g-equivariant long exact sequence of cohomology

⋯ ⟶ ϕ_{gl}^{i−1} ⟶ H^{i}(Z_{s}, ℚ_{ℓ}) ⟶^{sp^{i}} H^{i}(Zη̄, ℚ_{ℓ}) ⟶ ϕ_{gl}^{i} ⟶ …

where ϕ_{gl}^{i} is the space of global vanishing cycles, and sp^{i} is the specialization morphism. The Theorem on Sheaves of Vanishing Cycles for cohomology of low degree [8, Thm. 4.5 and Var. 4.8] states that ϕ_{gl}^{i} = 0 for i ≤ n − 1. Hence sp^{i} is an isomorphism for i ≤ n − 1 and an injection for i = n which proves the required result, with the following warning: according to [8, 4.4, p. 14], the proof of the Theorem on Sheaves of Vanishing Cycles relies on (R_{n,p}).

¿From now on let k = 𝔽_{q}. The following corollary is essentially a result of I.E. Shparlinskiĭ and A.N. Skorobogatov [38]. However, as noted in Remark 3.5, one needs to assume (R_{n,p}) for proving such a result.

7.2. Corollary. Assume that (R_{n,p}) holds. If X is a complete intersection of dimension n in ℙ_{k}^{N} with multidegree d with only isolated singularities, then

| |X(k)| − π_{n} | ≤ b′_{n−1}(N − 1, d) q^{(n+1)/2} + (b_{n}(N, d) + ε_{n}) q^{n/2}.

Proof. Follows from Theorem 6.1 with s = 0 and Proposition 7.1 with k = 𝔽_{q}.

For hypersurfaces, this implies a worse but a particularly simple inequality.

7.3. Corollary. Assume that (R_{n,p}) holds. If X is a hypersurface in ℙ_{k}^{n+1} of degree d with only isolated singularities, then

| |X(k)| − π_{n} | ≤ (d − 1)^{n+1} q^{(n+1)/2}.

Proof. Follows from Corollary 7.2 and the inequality in Example 4.3 (i).

The following result may be thought of as an analogue of the Weil inequality for certain singular curves and it is a more precise version of a result by Aubry and Perret [4, Cor. 2.5].

7.4. Corollary. If X is an irreducible curve in ℙ_{k}^{N}, then

| |X(k)| − (q + 1) | ≤ b_{1}(X̄)\sqrt{q}.

Moreover, if X is a complete intersection with multidegree d, and if d denotes the degree of X, then

b_{1}(X̄) ≤ b_{1}(N, d) ≤ (d − 1)(d − 2),

and the last inequality is an equality if and only if X is isomorphic to a nonsingular plane curve.

Proof. The first statement follows from the Trace Formula (17) and Deligne’s Main Theorem (18). In the second statement, the first inequality follows from Proposition 7.1 since (R_{n,p}) is true for n = 1. The last assertion is a consequence of the observations in Example 4.3 (ii).

7.5. Example. We work out here a simple case which will be used later on in Example 9.2. Let C be a nonsingular plane curve of genus g and degree d, defined over k, and let X be the projective cone in ℙ_{k}^{3} over C [16, Ex. 2.10, p. 13]. Thus X is a surface which is a complete intersection, and with exactly one singular point, namely, the vertex of the cone. Then

|X(k_{m})| = q^{m}|C(k_{m})| + 1.

If α_{1}, …, α_{2g} are the roots of F in H^{1}(C), then

|X(k_{m})| = q^{m}(q^{m} + 1 − ∑_{j=1}^{2g} α_{j}^{m}) + 1 = q^{2m} − ∑_{j=1}^{2g}(qα_{j})^{m} + q^{m} + 1, (24)

and therefore

| |X(k)| − π_{2} | ≤ (d − 1)(d − 2) q^{3/2}.

Observe that the equality can occur if C has the maximum number of points allowed by Weil’s inequality. From (24) and the definition (15) of the zeta function, we get

Z(X, T) = \frac{P_{1}(C, qT)}{(1 − q^{2}T)(1 − qT)(1 − T)}.

We compare this with the expression (19) of the zeta function. We note that X is irreducible, that the eigenvalues of the Frobenius in H^{3}(X) are pure of weight 3, and that those of H^{1}(X) are pure of weight ≤ 1. Hence

P_{1}(X, T) = 1, P_{2}(X, T) = (1 − qT), P_{3}(X, T) = P_{1}(C, qT).

By looking at the degrees of these polynomials, we find

b_{1}(X) = 0, b_{2}(X) = 1, b_{3}(X) = 2g(C).

8. The Penultimate Betti Number

Let X be a separated scheme of finite type over k = 𝔽_{q} of dimension n. Denote by H_{+}^{2n−1}(X̄) the subspace of H_{c}^{2n−1}(X̄) generated by the (generalized) eigenvectors of the Frobenius endomorphism whose eigenvalues are pure of weight exactly equal to 2n − 1. This subspace is the component of maximal weight 2n − 1 in the increasing filtration of H_{c}^{2n−1}(X̄) induced by the weight. Define

P_{2n−1}^{+}(X, T) = det(1 − T F | H_{+}^{2n−1}(X̄)) ∈ ℤ_{ℓ}[T].

The 2n − 1-th virtual Betti number of Serre [22, p. 28] is

b_{2n−1}^{+}(X) = deg P_{2n−1}^{+}(X, T) = dim H_{+}^{2n−1}(X̄).

Recall that two schemes are birationally equivalent if they have isomorphic dense open sets.

8.1. Proposition. Let X be a separated scheme of dimension n defined over k. (i) The polynomial P_{2n−1}^{+}(X, t) has coefficients independent of ℓ. (ii) The space H_{+}^{2n−1}(X̄) is a birational invariant. More precisely, if U is a dense

open set in X, then the open immersion j : U ⟶ X induces an isomorphism

j^{∗} : H_{+}^{2n−1}(Ū) ⟶ H_{+}^{2n−1}(X̄).

(iii) The polynomial P_{2n−1}^{+}(X, T) and the virtual Betti number b_{2n−1}^{+}(X) are birational invariants.

Assertion (iii) is reminiscent of [27, Cor. 6].

Proof. Assertion (i) can be easily checked by the following formula, a particular case of Jensen’s formula. In the complex plane, let γ be the oriented boundary of an annulus

r′ ≤ |w| ≤ r, with q^{−n} < r′ < q^{−n+(1/2)} < r < q^{−n+1}.

If t is a complex number with |t| > q^{−n+1}, then

P_{2n−1}^{+}(X, t) = exp \frac{1}{2iπ} ∫_{γ} log(1 − w^{−1}t) \frac{Z′(X, w)}{Z(X, w)} dw.

Let us prove (ii). Set Z = X−U and consider the long exact sequence of cohomology with compact support [30, Rem. 1.30, p. 94]:

⋯ ⟶ H_{c}^{2n−2}(Z̄) ⟶^{i} H_{c}^{2n−1}(Ū) ⟶^{j_{∗}} H_{c}^{2n−1}(X̄) ⟶ H_{c}^{2n−1}(Z̄) ⟶ …

and recall that the homomorphisms of this exact sequence are g-equivariant. Now H_{c}^{2n−1}(Z̄) = 0 since dim Z ≤ n − 1. Hence we get an exact sequence

0 ⟶ H_{c}^{2n−2}(Z̄)/ ker i ⟶^{ĩ} H_{c}^{2n−1}(Ū) ⟶^{j_{∗}} H_{c}^{2n−1}(X̄) ⟶ 0

The dimension of H_{c}^{2n−2}(Z̄) is equal to the number of irreducible components of Z̄ of dimension n − 1, and all the eigenvalues of the Frobenius automorphism in this space are pure of weight 2n − 2. Hence the eigenvalues of the Frobenius in Im ĩ are also pure of weight 2n − 2. Thus H_{+}^{2n−1}(Ū) ∩ Im ĩ = 0 and the restriction of j_{∗} to H_{+}^{2n−1}(Ū) is an isomorphism. Assertion (iii) is a direct consequence of (ii).

The following elementary result will be needed below.

8.2. Lemma. Let A and B be two finite sets of complex numbers included in the circle |z| = M. If, for some λ with 0 < λ < M and for every integer s sufficiently large,

∑_{β∈B} β^{s} − ∑_{α∈A} α^{s} = O(λ^{s}),

then A = B.

Proof. Suppose A ≠ B. By interchanging A and B if necessary, we can suppose that there is an element β_{0} ∈ B \ A. Then the rational function

R(z) = ∑_{α∈A} \frac{1}{1 − αz} − ∑_{β∈B} \frac{1}{1 − βz}

is holomorphic for |z| < M^{−1} and admits the pole β_{0}^{−1} on the circle |z| = M^{−1}. But the Taylor series of R(z) at the origin is

R(z) = ∑_{s=0}^{∞} (∑_{α∈A} α^{s} − ∑_{β∈B} β^{s}) z^{s},

and the radius of convergence of this series is ≥ λ^{−1} > M^{−1}.

8.3. Lemma. Let X be an irreducible scheme of dimension n defined over k. The following are equivalent :

(i) There is a constant C such that

| |X(𝔽_{q^{s}})| − q^{ns} | ≤ Cq^{s(n−1)} for any s ≥ 1.

(ii) We have H_{+}^{2n−1}(X̄) = 0.

Proof. Let A be the set of eigenvalues of the Frobenius automorphism in H_{+}^{2n−1}(X̄). By the Trace Formula (17) and Deligne’s Main Theorem (18),

| |X(F_{q^{s}})| − cq^{ns} − ∑_{α∈A} α^{s} | ≤ C′(X)q^{s(n−1)} for any s ≥ 1, (25)

where c = dim H_{c}^{2n}(X̄) is the number of irreducible components of X of dimension n and where

C′(X) = ∑_{i=0}^{2n−2} b_{i,l}(X̄)

is independent of q. Here c = 1 since X is irreducible. Now assume that (i) holds. From Formula (25) above, we deduce

| ∑_{α∈A} α^{s} | ≤ (C′(X) + C)q^{s(n−1)} for any s ≥ 1.

So Lemma 8.2 implies A = ∅, and hence, H_{+}^{2n−1}(X̄) = 0. The converse implication is an immediate consequence of (25).

8.4. Lemma. Let k be a separably closed field, and X a projective curve in ℙ_{k}^{N}. Then we have the following.

(i) If p_{a}(X) is the arithmetic genus of X, then

b_{1}(X) ≤ 2p_{a}(X).

(ii) If d denotes the degree of X, and if X̃ is a nonsingular projective curve birationally equivalent to X, then

b_{1}^{+}(X) = 2g(X̃) ≤ (d − 1)(d − 2).

During the proof of the Lemma, we shall make use of the following standard construction, when X is a curve.

8.5. Remark. Let X be a closed subvariety of dimension n in ℙ_{k}^{N} distinct from the whole space. It is easy to see that there is a linear subvariety E of codimension n + 2 in ℙ_{k}^{N} disjoint from X such that the projection π with center E gives rise to a diagram

X ⟶^{i} ℙ_{k}^{N} − E
↓^{π_{X}}  ↓^{π}
X′ ⟶^{i′} ℙ_{k}^{n+1}

such that X′ is an irreducible hypersurface with deg X′ = deg X, and where the restriction π_{X} is a finite birational morphism.

Proof of Lemma 8.4. Let π : X̃ ⟶ X be a normalization of X, let S = Sing X and S̃ = π^{−1}(S). Moreover let

d(X) = ∑_{x̃∈S̃} deg x̃ − ∑_{x∈S} deg x,

where deg x = [k(x) : k] is the degree of the residual field of the local ring O_{x}(X) of x ∈ X. By [4, Thm. 2.1] for instance,

P_{1}(X, T) = P_{1}(X̃, T) ∏_{j=1}^{d(X)}(1 − ω_{j}T),

where the numbers ω_{j} are roots of unity. This implies

b_{1}^{+}(X) = g(X̃), b_{1}(X) = b_{1}(X̃) + d(X),

since, as is well-known, b_{1}(X̃) = 2g(X̃), where g(X̃) is the geometric genus of X̃. Let

δ(X) = p_{a}(X) − g(X̃).

Then d(X) ≤ δ(X) [4, Sec. 2]. Hence

b_{1}(X) ≤ 2g(X̃) + d(X) ≤ 2g(X̃) + 2δ(X) ≤ 2p_{a}(X),

and this proves (i). If π : X ⟶ X′ is a finite birational morphism to a plane curve of degree d as constructed in Remark 8.5, then

b_{1}^{+}(X) = b_{1}^{+}(X′) ≤ b_{1}(X′) ≤ 2p_{a}(X′),

where the first equality follows from Proposition 8.1(iii), and the second inequality from assertion (i). Now

2p_{a}(X′) = (d − 1)(d − 2),

since X′ is a plane curve of degree d ([16, Ex. I.7.2, p. 54]).

From now on, suppose k = 𝔽_{q}. If X is a separated scheme of finite type over k, we say that the space H_{c}^{i}(X̄) is pure of weight i if all the eigenvalues of the Frobenius automorphism in this space are pure of weight i.

8.6. Proposition. Let X be a closed subvariety over k of dimension n in ℙ_{k}^{N} which is regular in codimension one. Then:

(i) The space H^{2n−1}(X̄) is pure of weight 2n − 1. (ii) If Y is a typical curve on X defined over k, then P_{2n−1}(X, T) divides P_{1}(Y, q^{n−1}T).

Proof. By passing to finite extension k′ of k, we can find a typical curve Y defined over k′. In that case the Gysin map

ι_{∗} : H^{1}(Ȳ , ℚ_{ℓ}(1 − n)) ⟶ H^{2n−1}(X̄, ℚ_{ℓ})

is a surjection by Corollary 2.1, and all the eigenvalues are pure of the same weight, since ι_{∗} is g-equivariant. This proves (i), and also that P_{2n−1}(X, T) divides P_{1}(Y, q^{n−1}T) if k = k′, which proves (ii).

In view of (i), a natural question is then to ask under which conditions the space H^{i}(X̄) is pure of weight i. The following proposition summarizes the results on this topic that we can state.

8.7. Proposition. Let X be a projective variety of dimension n and assume that dim Sing X ≤ s.

(i) The space H^{i}(X̄) is pure of weight i if i ≥ n + s + 1.

Assume now that (R_{n,p}) holds, and that X is a complete intersection with only isolated singularities. Then:

(ii) The space H^{n}(X̄) is pure of weight n.

Proof. By Corollary 1.4, we can find a nonsingular proper linear section Y of X of codimension s + 1. Since the Gysin maps are g-equivariant, Corollary 2.1 implies (i). Finally, as in the proof of Proposition 7.1, we deduce from the Theorem on Sheaves of Vanishing Cycles a g-equivariant exact sequence

0 = ϕ_{gl}^{n−s−1} ⟶ H^{n−s}(X̄) ⟶ H^{n−s}(Zη̄) ⟶ …

where Zη̄ is a nonsingular complete intersection. This implies (ii).

8.8. Remark. If X is a complete intersection with only isolated singularities, then the spaces H^{n}(X̄) and H^{n+1}(X̄) are the only ones for which the non-primitive part is nonzero. Hence, provided (R_{n,p}) holds, Proposition 8.7 shows that

(iii) H^{i}(X̄) is pure of weight i for 0 ≤ i ≤ 2n.

Thus for this kind of singular varieties, the situation is the same as for nonsingular varieties. It is worthwhile recalling that if X is locally the quotient of a nonsingular variety by a finite group, then (iii) holds without assuming (R_{n,p}), by [10, Rem. 3.3.11, p. 383].

9. Cohomology and Albanese Varieties

We begin this section with a brief outline of the construction of certain abelian varieties associated to a variety, namely the Albanese and Picard varieties. Later we shall discuss their relation with some étale cohomology spaces. For the general theory of abelian varieties, we refer to [28] and [31].

Let X be a variety defined over a field k and assume for simplicity that X has a k-rational nonsingular point x_{0} (by enlarging the base field if necessary). We say that a rational map g from X to an abelian variety B is admissible if g is defined at x_{0} and if g(x_{0}) = 0. An Albanese-Weil variety (resp. an Albanese-Serre variety) of X is an abelian variety A defined over k equipped with an admissible rational map (resp. an admissible morphism) f from X to A satisfying the following universal property:

(Alb) Any admissible rational map (resp. any admissible morphism) g from X to an abelian variety B factors uniquely as g = φ_{∘}f for some homomorphism φ : A ⟶ B of abelian varieties defined over k :

[figure: g; \begin{matrix}X \\ f & ↓ \\ A & ⟶^{φ} & B \end{matrix}]

Assume that A exists. If U is an open subset of X containing x_{0} where f is defined, then the smallest abelian subvariety containing f(U) is equal to A. The abelian variety A is uniquely determined up to isomorphism. Thus, the canonical map f and the homomorphism φ are uniquely determined.

The Albanese-Serre variety Alb_{s} X, together with a canonical morphism

f_{s} : X ⟶ Alb_{s} X

exists for any variety X [34, Thm. 5].

Let X be a variety, and let ι : X̃ ⟶ X be any birational morphism to X from a nonsingular variety X̃ (take for instance X̃ = Reg X). Since any rational map of a variety into an abelian variety is defined at every nonsingular point [28, Thm. 2, p. 20], any admissible rational map of X into an abelian variety B induces a morphism of X̃ into B, and factors through the Albanese-Serre variety Alb_{s} X̃ :

[figure: f_{w}; \begin{matrix}X̃ & ⟶^{ι} & X \\ f̃_{s}↓ & & ↓^{g} \\ Alb_{s} X̃ & ⟶^{φ} & B \end{matrix}]

This implies that we can take Alb_{w} X = Alb_{s} X̃, if we define the canonical map as f_{w} = f̃_{s} _{∘}ι^{−1}. Hence, the Albanese-Weil variety Alb_{w} X, together with a canonical map

f_{w} : X ⟶ Alb_{w} X

exists for any variety X, and two birationally equivalent varieties have the same Albanese-Weil variety. These two results have been proved by Weil [28, Thm. 11, p. 41], and [28, p. 152].

We recall now the following result [34, Th. 6].

9.1. Proposition (Serre). Let X be a projective variety.

(i) The canonical map f_{s} : X ⟶ Alb_{s} X factors uniquely as f_{s} = ν_{∘}f_{w} where ν is a surjective homomorphism of abelian varieties defined over k:

[figure: f_{s}; \begin{matrix}X \\ f_{w} & ↓ \\ Alb_{w} X & ⟶^{ν} & Alb_{s} X \end{matrix}]

(ii) If X is normal, then ker ν is connected, and ν induces an isomorphism

(Alb_{w} X)/ ker ν ⟶^{∼} Alb_{s} X.

(iii) if X is nonsingular, then ν is an isomorphism.

9.2. Example. Notice that Alb_{s} X is not a birational invariant and moreover that the inequality dim Alb_{s} X < dim Alb_{w} X can occur. For instance, let C be a nonsingular plane curve of genus g, defined over k, and let X be the normal projective cone in ℙ_{k}^{3} over C, as in Example 7.5. Since X is an hypersurface, Alb_{s} X is trivial by Remark 9.3. On the other hand, X is birationally equivalent to the nonsingular projective surface X̃ = C × ℙ^{1}. Since any rational map from 𝔸^{1} to an abelian variety is constant, the abelian variety Alb_{w} X = Alb_{w} X̃ is equal to the Jacobian Jac C = Alb_{w} C of C, an abelian variety of dimension g.

Let X be a normal projective variety defined over k. The Picard-Serre variety Pic_{s} X of X is the dual abelian variety of Alb_{s} X. The abelian variety Pic_{s} X should not be confused with Pic_{w} X, the Picard-Weil variety of X [28, p. 114], [37], which is the dual abelian variety of Alb_{w} X [28, Thm. 1, p. 148].

One can also define the Picard-Serre variety from the Picard scheme \underline{Pic}_{X/k} of X [12], [6, Ch. 8], which is a separated commutative group scheme locally of finite type over k. Its identity component \underline{Pic}_{X/k}^{0} is an abelian scheme defined over k, and Pic_{s} X = (\underline{Pic}_{X/k}^{0})_{red} [12, Thm. 3.3(iii), p. 237].

9.3. Remark (complete intersections). The Zariski tangent space at the origin of \underline{Pic}_{X/k} is the coherent cohomology group H^{1}(X, O_{X}) [12, p. 236], and hence,

dim \underline{Pic}_{X/k} ≤ dim H^{1}(X, O_{X}).

For instance, if X is a projective normal complete intersection of dimension ≥ 2, a theorem of Serre [16, Ex. 5.5, p. 231] asserts that H^{1}(X, O_{X}) = 0; hence, Pic_{s} X and Alb_{s} X are trivial for such a scheme.

If φ : Y ⟶ X is a rational map defined over k, and if f(Y) is reduced, then there exists one and only one homomorphism φ_{∗} : Alb_{w} Y ⟶ Alb_{w} X defined over k such that the following diagram is commutative :

Y ⟶^{φ} X
g̃_{w}↓  f̃_{w}↓
Alb_{w} Y ⟶^{φ_{∗}} Alb_{w} X

We use this construction in the following situation. Recall that the set 𝒰_{r}(X) has been defined in section 1.

9.4. Proposition. Let X be a projective variety of dimension n embedded in ℙ^{N}. If 1 ≤ r ≤ n − 1, and if E ∈ 𝒰_{r}(X), let Y = X ∩ E be the corresponding linear section of dimension n − r, and let ι : Y ⟶ X be the canonical closed immersion.

(i) If n − r ≥ 2, the set of E ∈ 𝒰_{r}(X) such that ι_{∗} is a purely inseparable isogeny contains a nonempty open set of 𝔾_{N−r,N}.

(ii) The set of E ∈ 𝒰_{n−1}(X) such that ι_{∗} is surjective contains a nonempty open set of 𝔾_{N−n−1,N}. If E belongs to this set, then Y = X ∩ E is a curve with

dim Alb_{w} X ≤ dim Jac Y.

(iii) If deg X = d, and Y = X ∩ E is a curve as in (ii) and if Ỹ is a nonsingular projective curve birationally equivalent to Y, then

dim Alb_{w} X ≤ g(Ỹ) ≤ \frac{(d − 1)(d − 2)}{2}.

Proof. The results (i) and (ii) are classical. For instance, assertion (i) follows from induction using Chow’s Theorem, viz., Thm. 5, Ch. VIII, p. 210 in Lang’s book [28] while assertion (ii) is stated on p. 43, § 3, Ch. II in the same book. See also Theorem 11 and its proof in [34, p. 159] for very simple arguments to show that ι_{∗} is surjective. The inequalities in (iii) are immediate consequences of (ii) and of Lemma 8.4(ii).

9.5. Remarks. (i) Let X be a projective variety regular in codimension one, and Y a typical curve on X. A theorem of Weil [42, Cor. 1 to Thm. 7] states that the homomorphism

ι^{∗} : Pic_{w} X ⟶ Jac Y

induced by ι has a finite kernel, which implies (ii) by duality in this case. (ii) Up to isogeny, any abelian variety A appears as the Albanese-Weil variety of a surface. To see this, it suffices to take a suitable linear section.

Let A be an abelian variety defined over k, of dimension g. For each integer m ≥ 1, let A_{m} denote the group of elements a ∈ A(\overline{k}) such that ma = 0. Let ℓ be a prime number different from the characteristic of k. The ℓ-adic Tate module T_{ℓ}(A) of A is the projective limit of the groups A_{ℓ^{n}}, with respect to the maps induces by multiplication by ℓ; this is a free ℤ_{ℓ}-module of rank 2g, and the group g operates on T_{ℓ}(A). The tensor product

V_{ℓ}(A) = T_{ℓ}(A) ⊗_{ℤ_{ℓ}} ℚ_{ℓ}

is a vector space of dimension 2g over ℚ_{ℓ}.

We recall the following result [25, Lem. 5] and [30, Cor. 4.19, p. 131], which gives a description in purely algebraic terms of H^{1}(X̄, ℚ_{ℓ}) when X is a normal projective variety.

9.6. Proposition. Let X be a normal projective variety defined over k. Then, there is a g-equivariant isomorphism

h_{X} : V_{ℓ}(Pic_{s} X)(−1) ⟶^{∼} H^{1}(X̄, ℚ_{ℓ}).

In particular, b_{1,ℓ}(X̄) = 2 dim Pic_{s} X is independent of ℓ. □

9.7. Remarks. (i) If X is a normal complete intersection, then Proposition 9.6 and Example 9.3 imply H^{1}(X̄, ℚ_{ℓ}) = 0, in accordance with Proposition 3.4(iii). (ii) If X is a normal projective variety, we get from Proposition 9.6 a g-equivariant isomorphism

H^{1}(Alb_{s} X̄, ℚ_{ℓ}) ⟶^{∼} H^{1}(X̄, ℚ_{ℓ}).

9.8. Proposition. Let X be a normal projective variety of dimension n ≥ 2 defined over k which is regular in codimension 2. Then there is a g-equivariant isomorphism

j_{X} : V_{ℓ}(Alb_{w} X) ⟶^{∼} H^{2n−1}(X̄, ℚ_{ℓ}(n)).

Proof. Since X is regular in codimension 2, we deduce from Corollary 1.4 that there are (lots of) typical surfaces on X, i. e., nonsingular proper linear sections of dimension 2 of X. For such a typical surface Y = X ∩ E, the closed immersion ι : Y ⟶ X induces a homomorphism ι_{∗} : Alb_{w} Y ⟶ Alb_{w} X. By Proposition 9.4(i), the set of linear varieties E such that ι_{∗} is a purely inseparable isogeny contains a nonempty open subset of the Grassmannian 𝔾_{N−n+2,N}. If E is chosen in that way, we get a g-equivariant isomorphism

V_{ℓ}(ι_{∗}) : V_{ℓ}(Alb_{w} Y) ⟶^{∼} V_{ℓ}(Alb_{w} X).

Since Y is nonsingular, we get from Poincaré Duality Theorem for nonsingular varieties [30, Cor. 11.2, p. 276] a g-equivariant nondegenerate pairing

H^{1}(Ȳ , ℚ_{ℓ}) × H^{3}(Ȳ , ℚ_{ℓ}(2)) ⟶ ℚ_{ℓ},

from which we deduce a g-equivariant isomorphism

ψ : Hom(H^{1}(Ȳ , ℚ_{ℓ}), ℚ_{ℓ}) ⟶ H^{3}(Ȳ , ℚ_{ℓ}(2)).

Since (X, Y) is a semi-regular pair with Y nonsingular, from Corollary 2.1 we know that the Gysin map

ι_{∗} : H^{3}(Ȳ , ℚ_{ℓ}(2 − n)) ⟶ H^{2n−1}(X̄, ℚ_{ℓ})

is an isomorphism. Now the isomorphism j_{X} is defined as the isomorphism making the following diagram commutative:

Hom(V_{ℓ}(Pic_{s} Y)(−1), ℚ_{ℓ}) ⟶_{∼}^{ϖ} V_{ℓ}(Alb_{w} Y) ⟶_{∼}^{V_{ℓ}(ι_{∗})} V_{ℓ}(Alb_{w} X)
th_{Y} ↓^{∼}  ^{j_{X}} ↓
Hom(H^{1}(Ȳ , ℚ_{ℓ}), ℚ_{ℓ}) ⟶_{∼}^{ψ} H^{3}(Ȳ , ℚ_{ℓ})(2) ⟶_{∼}^{ι_{∗}} H^{2n−1}(X̄, ℚ_{ℓ}(n))

Here ϖ is defined by the Weil pairing, and ^{t}h_{Y} is the transpose of the map h_{Y} defined in Proposition 9.6.

9.9. Remark. Let X be a complete intersection of dimension ≥ 2 which is regular in codimension 2. Then Proposition 9.8 implies that the Albanese-Weil variety Alb_{w} X is trivial, since Proposition 3.4(i) implies that H^{2n−1}(X̄, ℚ_{ℓ}) = 0.

The following result is a weak form of Poincaré Duality between the first and the penultimate cohomology spaces of some singular varieties.

9.10. Corollary. Let X be a normal projective variety of dimension n ≥ 2 defined over k regular in codimension 2. Then there is a g-equivariant injective linear map

H^{1}(X̄, ℚ_{ℓ}) ⟶ Hom(H^{2n−1}(X̄, ℚ_{ℓ}(n)), ℚ_{ℓ}).

Proof. Proposition 9.6 furnishes an isomorphism

h_{X}^{−1} : H^{1}(X̄, ℚ_{ℓ}) ⟶^{∼} V_{ℓ}(Pic_{s} X)(−1).

¿From the surjective map ν : Alb_{w} X ⟶ Alb_{s} X defined in Proposition 9.1, we get by duality a homomorphism with finite kernel ^{t}ν : Pic_{s} X ⟶ Pic_{w} X generating an injective homomorphism

V_{ℓ}(^{t}ν) : V_{ℓ}(Pic_{s} X) ⟶ V_{ℓ}(Pic_{w} X).

Now the Weil pairing induces an isomorphism

V_{ℓ}(Pic_{w} X)(−1) ⟶ Hom(V_{ℓ}(Alb_{w} X), ℚ_{ℓ})

and Proposition 9.8 gives an isomorphism

Hom(V_{ℓ}(Alb_{w} X), ℚ_{ℓ}) ⟶ Hom(H^{2n−1}(X̄, ℚ_{ℓ}(n)), ℚ_{ℓ}).

The required linear map is the combination of all the preceding maps.

10. A Conjecture of Lang and Weil

We assume now that k = 𝔽_{q} is a finite field. In order to state the results of this section, we introduce a weak form of the resolution of singularities for a variety X of dimension ≥ 2, which is of course implied by (R_{n,p}).

(RS2) X is birationally equivalent to a normal projective variety X̃ defined over k, which is regular in codimension 2.

As a special case of Abhyankar’s results [1], (RS2) is valid in any characteristic p > 0 if dim X ≤ 3 (except perhaps when dim X = 3 and p = 2, 3, 5).

The following gives a description in purely algebraic terms of the birational invariant H_{+}^{2n−1}(X̄, ℚ_{ℓ}).

10.1. Theorem. Let X be a variety of dimension n ≥ 2 defined over k satisfying (RS2). Then there is a g-equivariant isomorphism

j_{X} : V_{ℓ}(Alb_{w} X) ⟶^{∼} H_{+}^{2n−1}(X̄, ℚ_{ℓ}(n)),

and hence b_{2n−1}^{+}(X) = 2 dim Alb_{w} X. In particular this number is even.

Proof. Take X̃ birationally equivalent to X as in (RS2). Then Alb_{w} X̃ = Alb_{w} X since the Albanese-Weil variety is a birational invariant, and

H^{2n−1}(X̃ ⊗ k̄, ℚ_{ℓ}) = H_{+}^{2n−1}(X ⊗ k̄, ℚ_{ℓ})

by Propositions 8.1(ii) and 8.7(i). Now apply Proposition 9.8 to X̃.

10.2. Remark. The preceding result can be interpreted in the category of motives over k (cf. [35], [M]): this is what we mean by a “description in purely algebraic terms” of a cohomological property. Let us denote by h(X) the motive of a variety X of dimension n defined over k, and by h^{i}(X) the (mixed) component of h(X) corresponding to cohomology of degree i. One can identify the category of abelian varieties up to isogenies with the category of pure motives of weight −1. Denote by 𝕃 = h^{2}(ℙ^{1}) be the Lefschetz motive, which is pure of weight 2. Proposition 9.6 implies that if X is normal, then

h^{1}(X) = Pic_{s}(X) ⊗ 𝕃.

Now, denote by h_{+}^{i}(X) the part of h^{i}(X) which is pure of weight i. Theorem 10.1 means that if X satisfies (RS2), then

h_{+}^{2n−1}(X) = Alb_{w}(X) ⊗ 𝕃^{n}.

These results are classical if X is nonsingular.

10.3. Remark. Similarly, Theorem 10.1 is in accordance with the following conjectural statement of Grothendieck [14, p. 343] in the case i = n : “ In odd dimensions, the piece of maximal filtration of H^{2i−1}(X, ℤ_{ℓ}(i)) is also the greatest “abelian piece”, and corresponds to the Tate module of the intermediate Jacobian J^{i}(X) (defined by the cycles algebraically equivalent to 0 of codimension i on X). ” Notice that the group of cycles algebraically equivalent to 0 of codimension n on X maps in a natural way to the Albanese variety of X.

10.4. Remark. In the first part of their Proposition in [5, p. 333], Bombieri and Sperber state Theorem 10.1 without any assumption about resolution of singularities, but they give a proof only if dim X ≤ 2, in which case (RS2) holds in any characteristic, by Abhyankar’s results.

For any separated scheme X of finite type over k, the group g operates on X̄ and the Frobenius morphism of X̄ is the automorphism corresponding to the geometric element φ ∈ g. If A is an abelian variety defined over k, then φ is an endomorphism of A. It induces an endomorphism T_{ℓ}(φ) of T_{ℓ}(A), and

deg(n.1_{A} − φ) = f_{c}(A, n),

where

f_{c}(A, T) = det(T − T_{ℓ}(φ)).

The polynomial f_{c}(A, T) is a monic polynomial of degree 2g with coefficients in ℤ, called the characteristic polynomial of A. Moreover, if dim A = g,

f_{c}(A, T) = ∏_{j=1}^{2g}(T − α_{j}),

where the characteristic roots α_{j} are pure of weight one [28, p. 139], [31, p. 203-206]. The constant term of f_{c}(A, T) is equal to

deg(φ) = det T_{ℓ}(φ) = ∏_{j=1}^{2g} α_{j} = q^{g}. (26)

The trace of φ is the unique rational integer Tr(φ) such that

f_{c}(A, T) ≡ T^{2g} − Tr(φ) T^{2g−1} (mod T^{2g−2}).

In order to state the next results, we need to introduce some conventions. If X is a separated scheme of finite type over k, we call M the following set of conditions about H_{c}^{i}(X̄, ℚ_{ℓ}):

– The action of the Frobenius morphism F in H_{c}^{i}(X̄, ℚ_{ℓ}) is diagonalizable.

– The space H_{c}^{i}(X̄, ℚ_{ℓ}) is pure of weight i.

– The polynomial P_{i}(X, T) has coefficients in ℤ which are independent of ℓ. Furthermore, for any polynomial f of degree d, we write its reciprocal polynomial as f^{∨}(T) = T^{d}f(T^{−1}).

10.5. Corollary. Let X be a normal projective variety defined over k. Then:

(i) We have

P_{1}(X, T) = f_{c}^{∨}(Pic_{s} X, T).

(ii) If φ is the Frobenius endomorphism of Pic_{s} X, then

Tr(F | H^{1}(X̄, ℚ_{ℓ})) = Tr(φ).

(iii) Conditions M hold for H^{1}(X̄, ℚ_{ℓ}).

Proof. Let A be an abelian variety of dimension g defined over k with

f_{c}(A, T) = ∏_{j=1}^{2g}(T − α_{j}).

Since the arithmetic Frobenius F is the inverse of φ,

det(T − V_{ℓ}(F) | V_{ℓ}(A)) = ∏_{j=1}^{2g}(T − α_{j}^{−1}).

Since the map α ↦ qα^{−1} is a permutation of the characteristic roots, we have

det(T − V_{ℓ}(F) | V_{ℓ}(A)(−1)) = ∏_{j=1}^{2g}(T − qα_{j}^{−1}) = f_{c}(A, T),

which implies

det(1 − T V_{ℓ}(F) | V_{ℓ}(A)(−1)) = f_{c}^{∨}(A, T).

If we apply the preceding equality to A = Pic_{s} X, we get (i) with the help of Proposition 9.6. Now (ii) is an immediate consequence of (i) by looking at the coefficient of T^{2g−1}. As stated above, the polynomial f_{c}(A, T) belongs to ℤ[T], and its roots are pure of weight 1. By [31, Prop., p. 203], the automorphism V_{ℓ}(φ) of V_{ℓ}(A) is diagonalizable, and hence, (iii) follows again from Proposition 9.6.

10.6. Corollary. If X is a normal projective surface defined over k, then the polynomials P_{i}(X, T) are independent of ℓ for 0 ≤ i ≤ 4.

Proof. The polynomial P_{3}(X, T) = P_{3}^{+}(X, T) is independent of ℓ by Proposition 8.1(i). Moreover P_{1}(X, T) is independent of ℓ by Corollary 10.5. Hence we have proved that these polynomials are independent of ℓ for all but one value of i, namely i = 2. Following an observation of Katz, the last one must also be independent of ℓ, since

Z(X, T) = \frac{P_{1}(X, T)P_{3}(X, T)}{P_{0}(X, T)P_{2}(X, T)P_{4}(X, T)} ,

and the zeta function Z(X, T) is independent of ℓ.

The following result gives an explicit description of the birational invariants

P_{2n−1}^{+}(X, T), b_{2n−1}^{+}(X), Tr(F | H_{+}^{2n−1}(X̄, ℚ_{ℓ}))

in purely algebraic terms.

10.7. Theorem. Let X be a variety of dimension n ≥ 2 defined over k. (i) If g = dim Alb_{w} X, then

P_{2n−1}^{+}(X, T) = q^{−g}f_{c}(Alb_{w} X, q^{n}T).

In particular, b_{2n−1}^{+}(X) = 2g. (ii) If φ is the Frobenius endomorphism of Alb_{w} X, then

Tr(F | H_{+}^{2n−1}(X̄, ℚ_{ℓ})) = q^{n−1} Tr(φ).

10.8. Remark. Serge Lang and André Weil have conjectured [27, p. 826-827] that the equality

P_{2n−1}^{+}(X, T) = q^{−g}f_{c}(Pic_{w} X, q^{n}T)

holds if X is a variety defined over k, provided X is complete and nonsingular. If X is only assumed to be normal, then Pic_{w} X and Alb_{w} X are isogenous, and hence, f_{c}(Alb_{w} X, T) = f_{c}(Pic_{w} X, T). Thus the Lang-Weil Conjecture is a particular case of Theorem 10.7(i).

Example 9.2 shows that we cannot replace Pic_{w} X by Pic_{s} X in the statement of Theorem 10.7, even if X is projectively normal.

The full proof of this theorem will be given in section 11. We first prove:

10.9. Proposition. If X satisfies (RS2), then the conclusions of Theorem 10.7 hold true. Moreover (iii) Conditions M hold for H_{+}^{2n−1}(X̄, ℚ_{ℓ}).

Proof. Let A be an abelian variety of dimension g defined over k. Then

f_{c}(A, T) = ∏_{j=1}^{2g}(T − α_{j}) = q^{g} ∏_{j=1}^{2g}(1 − α_{j}^{−1}T), (27)

the last equality coming from (26). Now

det(T − V_{ℓ}(F) | V_{ℓ}(A)(−n)) = ∏_{j=1}^{2g}(T − q^{n}α_{j}^{−1}).

Hence

det(1 − T V_{ℓ}(F) | V_{ℓ}(A)(−n)) = ∏_{j=1}^{2g}(1 − q^{n}α_{j}^{−1}T), (28)

and we get from (27)

det(1 − T V_{ℓ}(F) | V_{ℓ}(A)(−n)) = q^{−g}f_{c}(A, q^{n}T).

Hence, (i) follows from Proposition 10.1. From (28) we deduce that β is an eigenvalue of F in H^{2n−1}(X̄, ℚ_{ℓ}) if and only if q^{n}/β is among the characteristic roots of φ. Now since α_{j}\overline{α}_{j} = q, we get:

Tr(F | H^{2n−1}(X̄, ℚ_{ℓ})) = ∑_{j=0}^{2g} \frac{q^{n}}{α_{j}} = q^{n−1} ∑_{j=0}^{2g} \overline{α}_{j} = q^{n−1} Tr(φ).

Finally (iii) follows from Theorem 10.1 as in the proof of Corollary 10.5.

We end this section by stating a weak form of the functional equation relating the polynomials P_{1}(X, T) and P_{2n−1}^{+}(X, T) when X is nonsingular.

10.10. Corollary. Let X be a normal projective variety of dimension n ≥ 2, defined over k. If g = dim Alb_{w} X, then

q^{−g}P_{1}^{∨}(X, q^{n}T) divides P_{2n−1}^{+}(X, T).

Proof. By Corollary 10.5(i) and Theorem 10.7(i), we know that

P_{1}^{∨}(X, T) = f_{c}(Pic_{s} X, T), P_{2n−1}^{+}(X, T) = q^{−g}f_{c}(Alb_{w} X, q^{n}T),

and Proposition 9.1(i) implies that f_{c}(Pic_{s} X, T) divides f_{c}(Alb_{w} X, T).

11. On the Lang-Weil Inequality

We now state the classical Lang-Weil inequality, except that we give an explicit bound for the remainder.

11.1. Theorem. Let X be a projective algebraic subvariety in ℙ_{k}^{N} of dimension n and of degree d defined over the field k with q^{n} elements. Then

| |X(k)| − π_{n} | ≤ (d − 1)(d − 2)q^{n−(1/2)} + C_{+}(X) q^{n−1},

where C_{+}(X) depends only on X̄, and

C_{+}(X) ≤ 9 × 2^{m} × (mδ + 3)^{N+1},

if X is of type (m, N, d), with d = (d_{1}, …, d_{m}) and δ = max(d_{1}, …, d_{m}). In particular, C_{+}(X) is bounded by a quantity which is independent of the field k.

In order to prove this, we need a preliminary result.

11.2. Proposition. Let X be a projective algebraic subvariety in ℙ_{k}^{N} of dimension n defined over k. Then

| |X(k)| − π_{n} − Tr(F | H_{+}^{2n−1}(X̄, ℚ_{ℓ})) | ≤ C_{+}(X) q^{n−1},

where C_{+}(X) is as in Theorem 11.1.

Proof. Denote by H_{−}^{2n−1}(X̄) the subspace of H^{2n−1}(X̄) corresponding to eigenvalues of the Frobenius endomorphism of weight strictly smaller than 2n − 1 and define

C_{+}(X) = dim H_{−}^{2n−1}(X̄) + ∑_{i=0}^{2n−2} b_{i,l}(X̄) + ε_{i}.

Clearly, C_{+}(X) depends only on X̄, and by Proposition 5.1,

C_{+}(X) ≤ τ(X) ≤ τ_{k}(m, N, d) ≤ 9 × 2^{m} × (mδ + 3)^{N+1},

with the notations introduced therein. Then the Trace Formula (17) and Deligne’s Main Theorem (18) imply the required result.

Proof of Theorem 11.1. By Theorem 10.7(i) and Proposition 9.4(iii),

b_{2n−1}^{+}(X) = 2 dim Alb_{w} X ≤ g(Ỹ) ≤ (d − 1)(d − 2),

if Y is a suitably chosen linear section of X of dimension 1, and if Ỹ is a nonsingular curve birationally equivalent to Y. Since all the eigenvalues of the Frobenius automorphism in H_{+}^{2n−1}(X̄, ℚ_{ℓ}) are pure of weight 2n − 1 by definition, we obtain by Proposition 11.2:

| |X(k)| − π_{n} | ≤ 2g(Ỹ)q^{n−(1/2)} + C_{+}(X) q^{n−1},

which is better than the desired inequality.

11.3. Remark. In exactly the same way, applying [24, Th. 1], one establishes that if X is a closed algebraic subvariety in 𝔸_{k}^{N}, of dimension n, of degree d and of type (m, N, d), defined over k, then

| |X(k)| − q^{n} | ≤ (d − 1)(d − 2)q^{n−(1/2)} + C_{+}(X) q^{n−1},

where C_{+}(X) depends only on X̄ and is not greater than 6 × 2^{m} × (mδ + 3)^{N+1}.

11.4. Remark. As a consequence of Remark 11.3, we easily obtain the following version of a lower bound due to W. Schmidt [33] for the number of points of affine hypersurfaces. If f ∈ k[T_{1}, …, T_{N}] is an absolutely irreducible polynomial of degree d, and if X is the hypersurface in 𝔸_{k}^{N} with equation f(T_{1}, …, T_{N}) = 0, then

|X(k)| ≥ q^{N−1} − (d − 1)(d − 2)q^{N−(3/2)} − 12(d + 3)^{N+1}q^{N−2}.

It may be noted that with the help of Schmidt’s bound, one is able to replace 12(d + 3)^{N+1} by a much better constant, namely 6d^{2}, but his bound is only valid for large values of q.

Proposition 11.2 gives the second term in the asymptotic expansion of |X(k_{n})| when n is large. From Theorem 10.7, we deduce immediately the following precise inequality, which involves only purely algebraic terms.

11.5. Corollary. Let X be a projective variety of dimension n ≥ 2 defined over k, and let φ be the Frobenius endomorphism of Alb_{w} X. Then

| |X(k)| − π_{n} + q^{n−1} Tr(φ) | ≤ C_{+}(X) q^{n−1},

where C_{+}(X) is as in Theorem 11.1.

11.6. Corollary. With hypotheses as above, the following are equivalent: (i) There is a constant C such that, for every r ≥ 1,

| |X(𝔽_{q^{r}})| − q^{nr} | ≤ Cq^{r(n−1)}.

(ii) The Albanese-Weil variety of X is trivial.

Proof. By Lemma 8.3 we know that (i) holds if and only if H_{+}^{2n−1}(X̄) = 0, and this last condition is equivalent to (ii) by Theorem 10.7.

For instance, rational varieties, and, by Remark 6.3 or 9.9, complete intersections regular in codimension 2 satisfy the conditions of the preceding Lemma.

It remains to prove Theorem 10.7. For that purpose, we need the following non-effective estimate [5, p. 333].

11.7. Lemma (Bombieri and Sperber). Let X be a projective variety of dimension n ≥ 2 defined over k, and let φ as above. Then

|X(k)| = π_{n} − q^{n−1} Tr(φ) + O(q^{n−1}).

Proof. By induction on n = dim X. If dim X = 2, then (RS2) is satisfied and

| |X(k)| − π_{2} + q Tr(φ) | ≤ C_{+}(X) q,

by Proposition 11.2 and Theorem 10.9(ii). Suppose that n ≥ 3 and that the lemma is true for any projective variety of dimension n − 1. We can assume that X is a projective algebraic subvariety in ℙ_{k}^{N} not contained in any hyperplane. In view of Lemma 1.2, if q is sufficiently large, there is a linear subvariety E of ℙ_{k}^{N}, of codimension 2, defined over k, such that dim X ∩ E = n − 2. For any u ∈ ℙ_{k}^{1}, the reciprocal image of u by the projection

π : ℙ_{k}^{N} − E ⟶ ℙ_{k}^{1}

is a hyperplane H_{u} containing E, and Y_{u} = X ∩ H_{u} is a projective algebraic set of dimension ≤ n − 1 and of degree ≤ deg X. Now we have

X(k) = ⋃_{u∈ ℙ^{1}(k)} Y_{u}(k),

hence

|X(k)| = ∑_{u∈ ℙ^{1}(k)} |Y_{u}(k)| + O(q^{n−1}) ;

in fact the error term is bounded by (q + 1)|X(k) ∩ E(k)|, and can be estimated by Proposition 12.1 below. Let S be the set of u ∈ ℙ_{k}^{1} such that Y_{u} is not a subvariety of dimension n − 1, or such that the canonical morphism

λ : Alb_{w} Y_{u} ⟶ Alb_{w} X

is not a purely inseparable isogeny. By Lemma 1.2 and Proposition 9.4(i), the set S is finite and applying Proposition 12.1 again, we get

∑_{u∈S} |Y_{u}(k)| = O(q^{n−1}),

hence,

|X(k)| = ∑_{u∉S} |Y_{u}(k)| + O(q^{n−1}).

Let φ_{u} be the Frobenius endomorphism of Alb_{w} Y_{u}. By the induction hypothesis,

|Y_{u}(k)| = π_{n−1} − q^{n−2} Tr(φ_{u}) + O(q^{n−2}).

But if u ∉ S then Tr(φ_{u}) = Tr(φ) by hypothesis. The two preceding relations then imply

|X(k)| = (q + 1 − |S|)(π_{n−1} − q^{n−2} Tr(φ)) + O(q^{n−1}),

and the lemma is proved.

Proof of Theorem 10.7. ¿From Proposition 11.2 and from Lemma 11.7 we get

Tr(F | H_{+}^{2n−1}(X̄, ℚ_{ℓ})) − q^{n−1} Tr(φ) = O(q^{n−1})

But the eigenvalues of F in H_{+}^{2n−1}(X̄, ℚ_{ℓ}) and the numbers q^{n−1}α_{j}, where (α_{j}) is the family of characteristic roots of Alb_{w} X, are pure of weight 2n − 1. From Lemma 8.2 we deduce that these two families are identical, and

Tr(F | H_{+}^{2n−1}(X̄, ℚ_{ℓ})) = q^{n−1} Tr(φ),

from which the result follow.

One can also improve the Lang-Weil inequality when the varieties are of small codimension:

11.8. Corollary. Let X be a projective subvariety in ℙ_{k}^{N}, and assume

dim Sing X ≤ s, codim X ≤ dim X − s − 1.

If dim X = n, then

| |X(k)| − π_{n} | ≤ C_{N+s}(X̄) q^{(N+s)/2},

where C_{N+s}(X̄) is as in Theorem 6.1.

Proof. ¿From Barth’s Theorem [15, Thm. 6.1, p. 146] and Theorem 2.4, we deduce that (12) holds if i ≥ N + s + 1. Now apply the Trace Formula (17) and Deligne’s Main Theorem (18).

12. Number of Points of Algebraic Sets

In most applications, it is useful to have at one’s disposal some bounds on general algebraic sets. If X ⊂ ℙ^{N} is a projective algebraic set defined over k we define the dimension (resp. the degree) of X as the maximum (resp. the sum) of the dimensions (resp. of the degrees) of the k-irreducible components of X. The following statement is a quantitative version of Lemma 1 of Lang-Weil [27] and generalizes Proposition 2.3 of [26].

12.1. Proposition. If X ⊂ ℙ^{N} is a projective algebraic set defined over k of dimension n and of degree d, then

|X(k)| ≤ dπ_{n}.

Proof. By induction on n. Recall that an algebraic set X ⊂ ℙ^{N} is nondegenerate in ℙ^{N} if X is not included in any hyperplane, i. e., if the linear subvariety generated by X is equal to ℙ^{N}. If n = 0 then |X(k)| ≤ d. Assume now that n ≥ 1. We first prove the desired inequality when X is k-irreducible. Let E be the linear subvariety in ℙ^{N}, defined over k, generated by X; set m = dim E ≥ n and identify E with ℙ^{m}. Then X is a nondegenerate k-irreducible subset in ℙ^{m}. Let

T = {(x, H) ∈ X(k) × (ℙ^{m})^{∗}(k) | x ∈ X(k) ∩ H(k)} .

We get a diagram made up of the two projections

T
p_{1} ↙  ↘^{p_{2}}
X(k)  (ℙ^{m})^{∗}(k)

If x ∈ X(k) then p_{1}^{−1}(x) is in bijection with the set of hyperplanes H ∈ (ℙ^{m})^{∗}(k) with x ∈ H(k); hence |p_{1}^{−1}(x)| = π_{m−1} and

|T (k)| = π_{m−1}|X(k)|. (29)

On the other hand, if H ∈ (ℙ^{m})^{∗}(k), then p_{2}^{−1}(H) is in bijection with to X(k) ∩ H(k), hence

|T (k)| = ∑_{H}|X(k) ∩ H(k)|, (30)

where H runs over the whole of (ℙ^{m})^{∗}(k). Since X is a nondegenerate k-irreducible subset in ℙ^{m}, every H ∈ (ℙ^{m})^{∗}(k) properly intersects X and the hyperplane section X ∩ H is of dimension ≤ n − 1. Moreover, such a hyperplane section is of degree d by Bézout’s Theorem. Thus by the induction hypothesis,

|X(k) ∩ H(k)| ≤ dπ_{n−1},

and from (29) and (30) we deduce

|X(k)| ≤ \frac{π_{m}}{π_{m−1}} dπ_{n−1}.

Now it is easy to check that if m ≥ n, then

q ≤ \frac{π_{m}}{π_{m−1}} ≤ \frac{π_{n}}{π_{n−1}} ≤ q + 1,

so the desired inequality is proved when X is irreducible of dimension n. In the general case, let

X = Y_{1} ∪ ⋯ ∪ Y_{s}

be the irredundant decomposition of X in k-irreducible components, in such a way that

dim Y_{i} ≤ n, d_{1} + ⋯ + d_{s} = d, (deg Y_{i} = d_{i}).

Then

|X(k)| ≤ |Y_{1}(k)| + ⋯ + |Y_{s}(k)| ≤ (d_{1} + ⋯ + d_{s})π_{n} = dπ_{n}.

We take this opportunity to report the following conjecture on the number of points of complete intersections of small codimension.

12.2. Conjecture (Lachaud). If X ⊂ ℙ_{k}^{N} is a projective algebraic set defined over k of dimension n ≥ N/2 and of degree d ≤ q + 1 which is a complete intersection, then

|X(k)| ≤ dπ_{n} − (d − 1)π_{2n−N} = d(π_{n} − π_{2n−N}) + π_{2n−N}.

12.3. Remark. The preceding conjecture is true in the following cases:

(i) X is of codimension 1. (ii) X is a union of linear varieties of the same dimension.

Assertion (i) is Serre’s inequality [36]: if X is an hypersurface of dimension n and of degree d ≤ q + 1, then

|X(k)| ≤ dq^{n} + π_{n−1}.

Now assume that X is the union of d linear varieties G_{1}, …, G_{d} of dimension n ≥ N/2. We prove (ii) by induction on d. Write G_{i}(k) = G_{i} (1 ≤ i ≤ d) for brevity. If d = 1 then

|G_{1}| = π_{n} = (π_{n} − π_{2N−n}) + π_{2N−n},

and the assertion is true. Now if G_{1} and G_{2} are two linear varieties of dimension n, then dim G_{1} ∩ G_{2} ≥ 2n − N. Hence for d > 1,

|G_{d} ∩ (G_{1} ∪ ⋯ ∪ G_{d−1})| ≥ π_{2n−N}.

Now note that

|G_{1} ∪ ⋯ ∪ G_{d}| = |G_{1} ∪ ⋯ ∪ G_{d−1}| + |G_{d}| − |G_{d} ∩ (G_{1} ∪ ⋯ ∪ G_{d−1})|.

If we apply the induction hypothesis we get

|G_{1} ∪ ⋯ ∪ G_{d}| ≤ (d − 1)(π_{n} − π_{2n−N}) + π_{2n−N} + π_{n} − π_{2n−N}
= d(π_{n} − π_{2n−N}) + π_{2n−N},

which proves the desired inequality.

Acknowledgements

This research was partly supported by the Indo-French Mathematical Research Program of the Centre National de la Recherche Scientifique (CNRS) of France and the National Board for Higher Mathematics (NBHM) of India, and we thank these organisations for their support. We would also like to express our warm gratitude to Jean-Pierre Serre for his interest, and to Alexeï Skorobogatov for his valuable suggestions. Thanks are also due to the referees for pointing out some corrections in earlier versions of this paper and making suggestions for improvements.

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Abbreviations

[EGA 4.2] Grothendieck, A., Dieudonné, J., Éléments de Géométrie Algébrique, Ch. IV, Seconde Partie, Étude locale des schémas et des morphismes de schémas, Publ. Math. I.H.E.S. 24 (1965).

[SGA 2] Cohomologie Locale des Faisceaux Cohérents et Théorèmes de Lefschetz Locaux et

Globaux, par A. Grothendieck, North-Holland, Amsterdam, 1968.

[SGA 5] Cohomologie ℓ-adique et Fonctions L, par A. Grothendieck, Lect. Notes in Math.

589, Springer-Verlag, Berlin, 1977.

[SGA 7.1] Groupes de Monodromie en Géométrie Algébrique, Vol. 1, dirigé par A. Grothen-

dieck, Lect. Notes in Math. 288, Springer-Verlag, Berlin, 1972.

[M] Jannsen, U., Kleiman, S., Serre, J.-P., Motives, Seattle 1991, Proc. Symp. Pure Math. 55, Amer. Math. Soc., Providence, 1994.

S.R.G.: Department of Mathematics

Indian Institute of Technology, Bombay

Powai, Mumbai 400076 - INDIA

   E-mail address: srg@math.iitb.ac.in

G.L.: Équipe “Arithmétique et Théorie de l’Information”

Institut de Mathématiques de Luminy

Luminy Case 907, 13288 Marseille Cedex 9 - FRANCE

   E-mail address: lachaud@iml.univ-mrs.fr