Étale cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields
aa r X i v : . [ m a t h . AG ] A ug ´ETALE COHOMOLOGY, LEFSCHETZ THEOREMSAND NUMBER OF POINTS OF SINGULAR VARIETIESOVER FINITE FIELDS SUDHIR R. GHORPADE AND GILLES LACHAUD
Dedicated to Professor Yuri Manin for his 65th birthday ∗ : Abstract.
We prove a general inequality for estimating the number of pointsof arbitrary complete intersections over a finite field. This extends a resultof Deligne for nonsingular complete intersections. For normal complete in-tersections, this inequality generalizes also the classical Lang-Weil inequality.Moreover, we prove the Lang-Weil inequality for affine as well as projectivevarieties with an explicit description and a bound for the constant appearingtherein. We also prove a conjecture of Lang and Weil concerning the Picardvarieties and ´etale cohomology spaces of projective varieties. The general in-equality for complete intersections may be viewed as a more precise versionof the estimates given by Hooley and Katz. The proof is primarily based ona suitable generalization of the Weak Lefschetz Theorem to singular varietiestogether with some Bertini-type arguments and the Grothendieck-LefschetzTrace Formula. We also describe some auxiliary results concerning the ´etalecohomology spaces and Betti numbers of projective varieties over finite fieldsand a conjecture along with some partial results concerning the number ofpoints of projective algebraic sets over finite fields.
Date : August 6, 2008. This is a corrected, revised and updated version of the paper publishedin
Mosc. Math. J. (2002), 589–631. MR 1988974 (2004d:11049a).1991 Mathematics Subject Classification.
Key words and phrases.
Hyperplane Sections, ´Etale Cohomology, Weak Lefschetz Theorems,Complete Intersections, Varieties over Finite Fields, Trace Formula, Betti numbers, Zeta Func-tions, Motives, Lang-Weil Inequality, Albanese Variety. ∗ “ tira´sc¯ıno vitato ra´smir es.¯am ” (Their cord was extended across) (R. g Veda X.129). † The first author is partially supported by a ‘Career Award’ grant from AICTE, New Delhiand an IRCC grant from IIT Bombay.
Contents
Introduction 31. Singular Loci and Regular Flags 52. Weak Lefschetz Theorem for Singular Varieties 73. Cohomology of Complete Intersections 114. The Central Betti Number of Complete Intersections 135. Zeta Functions and the Trace Formula 156. Number of Points of Complete Intersections 177. Complete Intersections with isolated singularities 198. The Penultimate Betti Number 229. Cohomology and Albanese Varieties 2610. A Conjecture of Lang and Weil 3111. On the Lang-Weil Inequality 3512. Number of Points of Algebraic Sets 38Acknowledgements 40References 40Abbreviations 41
INGULAR VARIETIES OVER FINITE FIELDS 3
Introduction
This paper has roughly a threefold aim. The first is to prove the following in-equality for estimating the number of points of complete intersections (in particular,hypersurfaces) which may possibly be singular:(1) (cid:12)(cid:12)(cid:12) | X ( F q ) | − π n (cid:12)(cid:12)(cid:12) ≤ b ′ n − s − ( N − s − , d ) q ( n + s +1) / + C s ( X ) q ( n + s ) / (cf. Theorem 6.1). Here, X denotes a complete intersection in P N defined overthe finite field k = F q of q elements, n the dimension of X , d = ( d , . . . , d r )the multidegree of X and s an integer such that dim Sing X ≤ s ≤ n −
1. Notethat r = N − n . Moreover, π n denotes the number of points of P n ( F q ), viz., π n = q n + q n − + · · · + 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 completeintersection in P M of dimension j . This primitive Betti number is explicitly givenby the formula(2) b ′ j ( M, d ) = ( − j +1 ( j + 1) + ( − N N X c = r ( − c (cid:18) N + 1 c + 1 (cid:19) X ν ∈ M ( c ) d ν where M ( c ) denotes the set of r -tuples ν = ( ν , . . . , ν r ) of positive integers suchthat ν + · · · + ν r = c and d ν = d ν · · · d rν r for any such r -tuple ν . If we let δ = max( d , . . . , d r ) and d = d · · · d r = deg X , then it will be seen that(3) b ′ j ( M, d ) ≤ ( − j +1 ( j + 1) + d (cid:18) M + 1 j (cid:19) ( δ + 1) j ≤ (cid:18) M + 1 j (cid:19) ( δ + 1) M . Lastly, C s ( X ) is a constant which is independent of q (or of k ). We can take C s ( X ) = 0 if s = − i. e. , if X is nonsingular, and in general, we have(4) C s ( X ) ≤ × r × ( rδ + 3) N +1 . where δ is as above. The inequalities (1), (3) and (4) are proved in sections 6, 4and 5 respectively.Our second aim is to discuss and elucidate a number of results related to the con-jectural statements of Lang and Weil [27]. These concern the connections betweenthe various ´etale cohomology spaces (especially, the first and the penultimate) andthe 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 +2 n − ( X, T ) is thecharacteristic polynomial of the piece of maximal filtration of H n − ( 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(5) P +2 n − ( X, T ) = q − g f c (Alb w X, q n T )where g = dim Alb w X . In particular, the “(2 n − X is twice the dimension of Alb w X , and is independent of ℓ . These results arediscussed 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 varietydefined over F q of dimension n and degree d , then with π n as above, we have(6) (cid:12)(cid:12)(cid:12) | X ( F q ) | − π n (cid:12)(cid:12)(cid:12) ≤ ( d − d − q n − (1 / + Cq ( n − , where C is a constant depending only on N , n and d . The said effective versionconsists in providing computable bounds for the constant C appearing in (6). Forexample, if X is defined by the vanishing of m homogeneous polynomials in N + 1 INGULAR VARIETIES OVER FINITE FIELDS 4 variables of degrees d , . . . , d m and δ = max( d , . . . , d m ), then we show that theconstant C in (6) may be chosen such that(7) C ≤ × m × ( mδ + 3) N +1 . We also prove an analogue of the Lang-Weil inequality for affine varieties. Theseinequalities are proved in section 11. Lastly, in section 12, we describe some old,hitherto unpublished, results concerning certain general bounds for the number ofpoints 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 applica-tions. For a more leisurely description of the background and an expository accountof the main results of this paper, we refer to [11].In [41], Weil proved a bound for the number of points of a nonsingular curveover a finite field F q , namely that it differs from π = q + 1 by at most 2 gq / ,where g is the genus of the curve. In [42], he formulated the conjectures aboutthe number of points of varieties of arbitrary dimension. Before these conjecturesbecame theorems, Lang and Weil [27] proved the inequality (6) in 1954. In 1974,Deligne [10] succeeded in completing the proof of Weil Conjectures by establishingthe so called Riemann hypothesis for nonsingular varieties of any dimension. Usingthis and the Lefschetz Trace Formula, also conjectured by Weil, and proved byGrothendieck, he obtained a sharp inequality for the number of points of a nonsin-gular complete intersection. Later, in 1991, Hooley [18] and Katz [21] proved thatif X is a complete intersection with a singular locus of dimension s , then | X ( F q ) | − π n = O ( q ( n + s +1) / ) . The inequality (1) which we prove here may be regarded as a more precise versionof this estimate. In effect, we explicitly obtain the coefficient of the first term and acomputable bound for the coefficient of the second term in the asymptotic expansionof the difference | X ( F q ) |− π n . When s = − i. e. , when X is nonsingular, then (1) isprecisely the inequality proved by Deligne [10]. On the other hand, if X is assumednormal, then we can take s = n − K is an algebraic functionfield of dimension n over k = F q , then there is a constant γ for which (6) holdswith ( d − d −
2) replaced by γ , for any model X of K/k , and moreover, thesmallest such constant γ is a birational invariant. They also noted that the zerosand poles of the zeta function Z ( X, T ) in the open disc | T | < q − ( n − are birationalinvariants, and that in the smaller disc | T | < q − ( n − / there is exactly one poleof order 1 at T = q − n . Then they wrote: about the behaviour of Z ( X, T ) for | T | ≥ q − ( n − / , we can only make the following conjectural statements, whichcomplement the conjectures of Weil . These statements are to the effect that when X is complete and nonsingular, the quotient Z ( X, T )(1 − q n T ) f c ( P, T ) INGULAR VARIETIES OVER FINITE FIELDS 5 has no zeros or poles inside | T | < q − ( n − and at least one pole on | T | = q − ( n − ,where P denotes the Picard variety of X . Moreover, with K/k and γ as above, wehave γ = 2 dim P . Lang and Weil [27] proved that these statements are valid in thecase of complete nonsingular curves, using the Riemann hypothesis for curves overfinite fields. As remarked by Bombieri and Sperber [5, p. 333], some of the resultsconjectured by Lang and Weil are apparently known to the experts but one is unableto locate formal proofs in the literature. Some confusion is also added by the factthat 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) isfalse. Moreover, the proof in [5] of a part of the Proposition on p. 133 appears to beincomplete (in dimensions ≥ 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 ≥ k thealgebraic closure of k . We denote by S = k [ X , . . . , X N ] the graded algebra ofpolynomials in N + 1 variables and by P N = P Nk = Proj S the projective space ofdimension N over k . By an algebraic variety over k we mean a separated scheme offinite type over k which is geometrically irreducible and reduced, i. e. , geometricallyintegral, and by a curve we mean an algebraic variety of dimension one. In thispaper, 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 aregular local ring and singular otherwise. The singular locus Sing X of X is the setof 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 thefollowing 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, D´ef. 5.8.2, p. 107]. A scheme isreduced if and only if it has no embedded components and satisfies condition ( R )[EGA 4.2, 5.8.5, p. 108]. A scheme X of dimension n is regular if it satisfies condi-tion ( R n ), hence X is regular if and only if dim Sing X = − ∅ = − X , we denote by ¯ X = X ⊗ k k the scheme deduced from X bybase field extension from k to k . A variety X is nonsingular if ¯ X is regular. If k is perfect, then the canonical projection from ¯ X to X sends Sing ¯ X onto Sing X [EGA 4.2, Prop. 6.7.7, p. 148]. Hence, ¯ X is regular in codimension m if and onlyif X has the same property, and dim Sing ¯ X = 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 INGULAR VARIETIES OVER FINITE FIELDS 6 in R if ht I = r . A closed subscheme X of a regular noetherian scheme V overa 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 completeintersection if it is a local complete intersection at every closed point; in this caseit 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 localcomplete intersection. A connected local complete intersection is Cohen-Macaulay[16, Prop. 8.23, p.186], hence equidimensional since all its connected componentshave 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 anirreducible component Z if and only ifdim 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 propercomponent, one says in this case that X and Y intersect properly . For instance, if X is irreducible of dimension ≥ 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 ifand 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 donot 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 linearsection of X if there is a linear subvariety E of P N properly intersecting X suchthat 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 Lemma1.1 that if X is an equidimensional subscheme of P N , and if there is a nonsingularproper 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 = k is algebraically closed . We state here aversion of Bertini’s Theorem and some of its consequences ; an early source for thiskind of results is, for instance, [43, Sec. I ]. Let b P N be the variety of hyperplanes of P N . Let X be a closed subvariety in P N and U ( X ) be the set of H ∈ b P N satisfyingthe following conditions:(i) X ∩ H is a proper hyperplane section of X .(ii) X ∩ H is reduced if dim X ≥ X ∩ H is irreducible if dim X ≥ X ∩ H = ( dim Sing X − X ≥ , − X ≤ . (v) deg X ∩ H = deg X if dim X ≥ INGULAR VARIETIES OVER FINITE FIELDS 7
We need the following version of Bertini’s Theorem. A proof of this result canbe obtained as a consequence of [19, Cor. 6.11, p. 89] and [40, Lemma 4.1].1.2.
Lemma.
Let X be a closed subvariety in P N . Then U ( X ) contains a non-empty Zariski open set of b P N . Let r ∈ N with 0 ≤ r ≤ N . We denote by G r,N the Grassmannian of linearvarieties of dimension r in P N . Let F r ( P N ) be the projective variety consisting ofsequences ( E , . . . , E r ) ∈ G N − ,N × · · · × G N − r,N making up a flag of length r : P N = E ⊃ E ⊃ · · · ⊃ E r with codim E m = m for 0 ≤ m ≤ r , so that E m is a hyperplane of E m − . Let X bea subvariety of P N and ( E , . . . , E r ) a flag in F r ( P N ). We associate to these dataa descending chain of schemes X ⊃ X ⊃ · · · ⊃ X r defined by X = X , X m = X m − ∩ E m (1 ≤ m ≤ r ) . Note that X m = X ∩ E m for 0 ≤ m ≤ r . We say that ( E , . . . , 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 ifdim 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 = ( dim Sing X − m if dim Sing X ≥ m, − X ≤ m − . (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 aregular flag ( E , . . . , 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 anonempty Zariski open set of G N − r,N . Corollary.
Let X be a closed subvariety in P N of dimension n , and let s ∈ N with ≤ s ≤ n − . Then the following conditions are equivalent: (i) There is a proper linear section of codimension s + 1 of X which is anonsingular variety. (ii) dim Sing X ≤ s . In particular, X is regular in codimension one if and only if there is a nonsingularproper linear section Y of dimension of X . (cid:3) Following Weil [43, p. 118], we call Y a typical curve on X if X and Y satisfythe 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 aprime number, and denote by Q ℓ the field of ℓ -adic numbers. Given an algebraicvariety X defined over k , by H i ( ¯ X, Q ℓ ) we denote the ´etale ℓ -adic cohomology space of ¯ X and by H ic ( ¯ X, 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 definitionsand the fundamental theorems on this theory. These are finite dimensional vector INGULAR VARIETIES OVER FINITE FIELDS 8 spaces over Q ℓ , and they vanish for i < i > X . If X is properthe two cohomology spaces coincide. The ℓ -adic Betti numbers of X are b i,ℓ ( ¯ X ) = dim H ic ( ¯ X, Q ℓ ) (0 ≤ i ≤ n ) . If X is a nonsingular projective variety, these numbers are independent of the choiceof ℓ [22, p. 27], and in this case we set b i ( ¯ X ) = b i,ℓ ( ¯ X ). It is conjectured that thisis true for any separated scheme X of finite type.These spaces are endowed with an action of the Galois group g = Gal( k/k ). Wecall 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 ( ¯ X, Q ℓ ( c )) = H i ( ¯ X, Q ℓ ) ⊗ Q ℓ ( c )we shall denote the corresponding twisted copy of H i ( ¯ X, 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 ofnonsingular varieties, which is discussed below.2.1.
Theorem.
Let X be an irreducible projective scheme of dimension n , and Y aproper linear section of codimension r in X which is a nonsingular variety. Assumethat both X and Y are defined over k . Then for each i ≥ n + r , the closed immersion ι : Y −→ X induces a canonical g -equivariant linear map ι ∗ : H i − r ( ¯ Y , Q ℓ ( − r )) −→ H i ( ¯ X, Q ℓ ) , called the Gysin map , which is an isomorphism for i ≥ n + r + 1 and a surjectionfor i = n + r .Proof. There is a diagram Y ι −−−−→ X ←−−−− U where U = X \ Y . The corresponding long exact sequence in cohomology withsupport in Y [30, Prop. 1.25, p. 92] will be as follows. · · · −→ H i − ( ¯ U , Q ℓ ) −→ H i ¯ Y ( ¯ X, Q ℓ ) −→ H i ( ¯ X, Q ℓ ) −→ H i ( ¯ U , Q ℓ ) −→ · · · Since U is a scheme of finite type of dimension n which is the union of r affineschemes, we deduce from Lefschetz Theorem on the cohomological dimension ofaffine schemes [30, Thm. 7.2, p. 253] that H i ( ¯ U , Q ℓ ) = 0 if i ≥ n + r . Thus, thepreceding exact sequence induces a surjection H n + r ¯ Y ( ¯ X, Q ℓ ) −→ H n + r ( ¯ X, Q ℓ ) −→ , and isomorphisms H i ¯ Y ( ¯ X, Q ℓ ) ∼ −→ H i ( ¯ X, Q ℓ ) ( i ≥ n + r + 1) . The cohomology groups with support in ¯ Y can be calculated by excision in an´etale neighbourhood of ¯ Y [30, Prop. 1.27, p. 92]. Let X ′ = Reg X be the smoothlocus of X . Since Y is nonsingular, we find E ∩ Sing X = ∅ by Lemma 1.1(i), andhence, Y ⊂ X ′ , that is, X is nonsingular in a neighbourhood of Y . Thus we obtainisomorphisms H i ¯ Y ( ¯ X, Q ℓ ) ∼ −→ H i ¯ Y ( ¯ X ′ , Q ℓ ) , for all i ≥ . 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 arecanonical isomorphisms H i − r ( ¯ Y , Q ℓ ( − r )) ∼ −→ H iY ( ¯ X ′ , Q ℓ ) , for all i ≥ . This yields the desired results. (cid:3)
INGULAR VARIETIES OVER FINITE FIELDS 9
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 − of X which is anonsingular variety. Then: (i) b i,ℓ ( ¯ X ) = b i − s − ( ¯ Y ) if i ≥ n + s + 2 . (ii) b n + s +1 ,ℓ ( ¯ X ) ≤ b n − s − ( ¯ Y ) . (cid:3) Remark.
In view of Corollary 1.4, relation (i) implies that the Betti numbers b i,ℓ ( ¯ X ) 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 ⊃ E ⊃ · · · ⊃ E r be a flag in F r ( P N ) and X = X , X m = X m − ∩ E m (1 ≤ m ≤ r )the associated chain of schemes. We say that ( E , . . . , E r ) is a semi-regular flag oflength r for X if the schemes X , . . . , X r − are irreducible. We denote by V r ( X ) theset 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 closedsubscheme 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 toin the beginning of this section is the following.2.4. Theorem. (General Weak Lefschetz Theorem, high degrees) . Let ( X, Y ) be asemi-regular pair with dim X = n , Y of codimension r in X and dim Sing Y = σ .Assume that both X and Y are defined over k . Then for each i ≥ n + r + σ + 1 there is a canonical g -equivariant linear map ι ∗ : H i − r ( ¯ Y , Q ℓ ( − r )) −→ H i ( ¯ X, 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 , . . . , E r ) defined over k 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 [40,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(10) H α + i ( ¯ X ∩ ¯ H, Q ℓ ( − −→ H α + i +2 ( ¯ X, Q ℓ )which is a surjection for i = 0 and an isomorphism for i >
0. Moreover he provedalso that if X and X ∩ H are nonsingular then ι ∗ is the Gysin map. In the generalcase we proceed by iteration. Let ( E , . . . , E r ) be a semi-regular flag of F r ( P N )such that Y = X ∩ E r and let X = X , X m = X m − ∩ E m (1 ≤ m ≤ r ) . be the associated chain of schemes. First,(11) dim X m = n − m for 1 ≤ m ≤ r. INGULAR VARIETIES OVER FINITE FIELDS 10
In fact, let η m = dim X m − − dim X m . Then 0 ≤ η m ≤ η + · · · + η r = dim X − 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 − . Since X m − is irreducible by hypothesis,the couple ( X m − , X m ) is a semi-regular pair for 1 ≤ m ≤ r and we can apply theTheorem in the case of codimension one. From Lemma 1.1 we deducedim X m − dim Sing X m ≥ dim Y − dim Sing Y = n − r − σ, and hence, dim Sing X m ≤ r − m + σ . Thendim X m − + dim Sing X m ≤ α ( m ) , where α ( m ) = n + r − m + σ + 1. We observe that α ( m −
1) = α ( m ) + 2. Hencefrom (10) we get a map H α ( m )+ i ( ¯ X m , Q ℓ ( − m )) −→ H α ( m − i ( ¯ X m − , Q ℓ ( − m + 1))which is a surjection for i = 0 and an isomorphism for i >
0. The composition ofthese maps gives a map ι ∗ : H α ( r )+ i ( ¯ Y , Q ℓ ( − r )) −→ H α (0)+ i ( ¯ X, 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 ) + 2 r + i , we get ι ∗ : H j − r ( ¯ Y , Q ℓ ( − r )) −→ H j ( ¯ X, Q ℓ ) , fulfilling the required properties. Now recall from [30, Prop. 6.5(b), p. 250] that if X ι −→ X ι −→ X is a so-called smooth triple over k , then the Gysin map for ι ◦ ι is the compositionof the Gysin maps for ι and ι , which proves the last assertion of the Theorem. (cid:3) The following Proposition gives a criterion for a pair to be semi-regular.2.5.
Proposition.
Let X be an irreducible closed subscheme of P N and Y a properlinear section of X of dimension ≥ which is regular in codimension one. Assumethat both X and Y are defined over k . 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 aregular hyperplane section of X . In that case ( X, Y ) is a semi-regular pair, as wepointed out, and the only statement to prove is that Y is irreducible. If Y is regularin codimension one, then σ = dim Sing Y ≤ dim Y − ≤ n − . This implies that 2 n ≥ n + σ + 3 and we can take i = 2 n in Theorem 2.4 in orderto get an isomorphism H n − ( ¯ Y , Q ℓ ( − ∼ −→ H n ( ¯ X, Q ℓ ) . Now for any scheme X of dimension n , the dimension of H n ( ¯ X, Q ℓ ) is equal to thenumber of irreducible components of X of dimension n , as can easily be deducedfrom the Mayer-Vietoris sequence [30, Ex. 2.24, p. 110]. Since Y is a properlinear section, it is equidimensional and the irreducibility of Y follows from theirreducibility of X . The general case follows by iteration. (cid:3) Observe that Theorem 2.1 is an immediate consequence of Theorem 2.4 andProposition 2.5.
INGULAR VARIETIES OVER FINITE FIELDS 11
Remark (Weak Lefschetz Theorem, low degrees) . We would like to point outthe following result, although we shall not use it. Assume that the resolution ofsingularities 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 defined over k , and letthere 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 defined over k , and where U = X \ Y . If U is a local complete intersection, thenthe canonical g -equivariant linear map ι ∗ : H i ( ¯ X, Q ℓ ) −→ H i ( ¯ Y , Q ℓ )is an isomorphism if i ≤ n − r − i = n − r .This theorem is a consequence of the Global Lefschetz Theorem of Grothendieckfor low degrees [13, Cor. 5.7 p. 280]. Notice that no hypotheses of regularity areput on Y in that statement. Moreover, if U is nonsingular, the conclusions of thetheorem are valid without assuming condition ( R n , p ) [20, Thm. 7.1, p. 318].2.7. Remark (Poincar´e Duality) . By combining Weak Lefschetz Theorem for highand low degrees, we get a weak version of Poincar´e Duality for singular varieties.Namely, assume that ( R n , p ) holds and let X be a closed subvariety of dimension n in P N defined over k which is a local complete intersection such that dim Sing X ≤ s .Assume that there is a proper linear section of codimension s + 1 of X defined over k . Then, for 0 ≤ i ≤ n − s − H i ( ¯ X, Q ℓ ) × H n − i ( ¯ X, 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 ( ¯ X ) = b n − i ( ¯ X ) for 0 ≤ i ≤ n − s − , and these numbers are independent of ℓ .3. Cohomology of Complete Intersections
Let k be a perfect field. A closed subscheme X of P N = P Nk of codimension r is a complete intersection if X is the closed subscheme determined by an ideal I generated by r homogeneous polynomials f , . . . , f r .A complete intersection is a local complete intersection, and in particular X isCohen-Macaulay and equidimensional. Moreover, if dim X ≥
1, then X is con-nected, hence X is integral if it is regular in codimension one.The multidegree d = ( d , . . . , d r ) of the system f , . . . , f r , usually labelled sothat d ≥ · · · ≥ d r , depends only on I and not of the chosen system of generators f , . . . , f r , since the Hilbert series of the homogeneous coordinate ring of X equals H ( T ) = (1 − T d )(1 − T d ) · · · (1 − T d r )(1 − T ) N +1 [32, Ex. 7.15, p. 350] or [7, Prop. 6, p. AC VIII.50]. This impliesdeg X = d · · · d r . First of all, recall that if X = P n , and if 0 ≤ i ≤ n , then [30, Ex. 5.6, p. 245]:(12) H i ( ¯ X, Q ℓ ) = ( Q ℓ ( − i/
2) if i is even0 if i is odd . INGULAR VARIETIES OVER FINITE FIELDS 12
Consequently, dim H i ( P n ¯ k , Q ℓ ) = ε i for 0 ≤ i ≤ n , where we set ε i = ( i is even0 if i is odd . Now let X be a nonsingular projective subvariety of P N of dimension n ≥
1. Forany i ≥
0, the image of the canonical morphism H i ( P N ¯ k , Q ℓ ) −→ H i ( ¯ X, Q ℓ ) isisomorphic to H i ( P n ¯ k , Q ℓ ). Hence, one obtains a short exact sequence(13) 0 −→ H i ( P n ¯ k , Q ℓ ) −→ H i ( ¯ X, Q ℓ ) −→ P i ( ¯ X, Q ℓ ) −→ P i ( ¯ X, Q ℓ ) is the cokernel of the canonical morphism, called the primitivepart of H i ( ¯ X, Q ℓ ). If i is even, the image of H i ( P n ¯ k , Q ℓ ) is the one-dimensionalvector space generated by the cup-power of order i/ 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 . Proposition.
Let X be a nonsingular complete intersection in P N of codimen-sion r , of multidegree d , and let dim X = n = N − r . (i) If i = n , then P i ( ¯ X, Q ℓ ) = 0 for ≤ i ≤ n . 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 Veroneseembedding, Weak Lefschetz Theorem 2.4 and Poincar´e Duality in the case of non-singular varieties. Statement (ii) follows from Theorem 4.1 below. (cid:3) In view of Proposition 3.1, we denote by b ′ n ( N, d ) the primitive n -th Betti numberof any nonsingular complete intersection in P N of codimension r = N − n and ofmultidegree d . It will be described explicitly in Section 4.The cohomology in lower degrees of general (possibly singular) complete inter-sections 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 defined over k . Then the g -equivariant restriction map ι ∗ : H i ( P N ¯ k , Q ℓ ) −→ H i ( X, Q ℓ ) is an isomorphism for i ≤ N − r − , and is injective for i = N − r .Proof. The scheme U = P N \ X is a scheme of finite type of dimension N which isthe 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 Poincar´e Duality we get H ic ( ¯ U , Q ℓ ) = 0 if i ≤ N − r. Now, the excision long exact sequence in compact cohomology [30, Rem. 1.30, p.94] gives:(14) · · · −→ H ic ( ¯ U , Q ℓ ) −→ H i ( P N , Q ℓ ) −→ H i ( ¯ X, Q ℓ ) −→ H i +1 c ( ¯ U , Q ℓ ) −→ · · · from which the result follows. (cid:3) We shall now study the cohomology in higher degrees of general complete inter-sections.
INGULAR VARIETIES OVER FINITE FIELDS 13
Proposition.
Let X be a complete intersection in P N of dimension n ≥ andmultidegree d with dim Sing X ≤ s . Then: (i) Relation (12) holds, and hence, b ′ i ( X ) = 0 , if n + s + 2 ≤ i ≤ n . (ii) b n + s +1 ,ℓ ( X ) ≤ b n − s − ( N − s − , d ) . (iii) Relation (12) holds, and hence, b ′ i ( X ) = 0 , if ≤ i ≤ n − .Proof. Thanks to Proposition 1.3, there are regular flags for X ; let ( E , . . . , E s +1 )be one of them, and let Y = X ∩ E s +1 . Then Y is a nonsingular variety by definitionof regular flags, and a complete intersection. Applying the Weak Lefschetz Theorem2.1, we deduce that the Gysin map ι ∗ : H i − s − ( ¯ Y , Q ℓ ( − s − −→ 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. (cid:3) Remarks. (i) The case s = 0 of this theorem is a result of I.E. Shparlinski˘ı andA.N. Skorobogatov [39, Thm. 2.3], but the proof of (a) and (b) in that Theoremis unclear to us: in any case their proof use results which are valid only if thecharacteristic 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 toprove Proposition 7.1 below.3.5. Remark.
For further reference, we note that if s ≥ i = n + s + 1 iseven, then H i ( X, Q ℓ ) contains a subspace isomorphic to Q ℓ ( − i/ ι ∗ : H n − s − ( ¯ Y , Q ℓ ( − s − −→ 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 The-orem [17, Satz 2.4, p. 136], [20, Cor. 7.5]; see [3, Cor 4.18] for an alternative proof.If ν = ( ν , . . . , ν r ) ∈ N r , and if d = ( d , . . . , d r ) ∈ ( N × ) r , we define d ν = d ν . . . d ν r r . If c ≥
1, let M ( c ) = { ( ν , . . . , ν r ) ∈ N r | ν + · · · + ν r = c and ν i ≥ ≤ i ≤ r } . Theorem.
The primitive n -th Betti number of any nonsingular complete in-tersection in P N of codimension r = N − n and of multidegree d is b ′ n ( N, d ) = ( − N − r +1 ( N − r + 1) + ( − N N X c = r ( − c (cid:18) N + 1 c + 1 (cid:19) X ν ∈ M ( c ) d ν . We now give estimates on the primitive n -th Betti number of a nonsingularcomplete intersection.4.2. Proposition.
Let d = ( d , . . . , d r ) ∈ ( N × ) r and b ′ n ( N, d ) be the n th primitiveBetti number of a nonsingular complete intersection in P N of dimension n = N − r and multidegree d . Let δ = max( d , . . . , d r ) and d = deg X = d · · · d r . Then, for r ≥ , we have b ′ n ( N, d ) ≤ ( − n +1 ( n + 1) + d (cid:18) N + 1 n (cid:19) ( δ + 1) n . INGULAR VARIETIES OVER FINITE FIELDS 14
In particular, b ′ n ( N, d ) ≤ (cid:18) N + 1 n (cid:19) ( δ + 1) N . Proof.
Observe that for any c ≥ r and ν ∈ M ( c ), we have d ν ≤ dδ c − r and | M ( c ) | = (cid:18) c − r − (cid:19) . Hence, by Theorem 4.1, b ′ n ( N, d ) − ( − n +1 ( n + 1) ≤ N X c = r (cid:18) N + 1 c + 1 (cid:19)(cid:18) c − r − (cid:19) dδ c − r . Now, for c ≥ r ≥
1, we have (cid:18) N + 1 c + 1 (cid:19) = N ( N + 1) c ( c + 1) (cid:18) N − c − (cid:19) ≤ N ( N + 1) r ( r + 1) (cid:18) N − c − (cid:19) , and moreover, N X c = r (cid:18) N − c − (cid:19)(cid:18) c − r − (cid:19) δ c − r = (cid:18) N − r − (cid:19) N X c = r (cid:18) nc − r (cid:19) δ c − r = (cid:18) N − r − (cid:19) ( δ + 1) n . It follows that b ′ n ( N, d ) − ( − n +1 ( n + 1) ≤ d N ( N + 1) r ( r + 1) (cid:18) N − r − (cid:19) ( δ + 1) n = d (cid:18) N + 1 r + 1 (cid:19) ( δ + 1) n . This proves the first assertion. The second assertion is trivially satisfied if n = 0,while if n ≥
1, then (cid:0) N +1 n (cid:1) ≥ N + 1 ≥ n + 1 , and since d = d · · · d r < ( δ + 1) r , we have d ( δ + 1) n ≤ ( δ + 1) N − − n +1 ( n + 1) + d (cid:0) N +1 n (cid:1) ( δ + 1) n ≤ ( n + 1) + ( δ + 1) N (cid:0) N +1 n (cid:1) − (cid:0) N +1 n (cid:1) ≤ (cid:0) N +1 n (cid:1) ( δ + 1) N , and the second assertion is thereby proved. (cid: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 ) = d − d (( d − N − ( − N ) ≤ ( d − N − ε N . In particular, if N = 2 (plane curves), then b ( d ) = ( d − d − N = 3(surfaces in 3-space), then b ′ (3 , d ) = ( d − d − d −
2) + 1) = d − d + 6 d − . (ii) The Betti number of a nonsingular curve which is a complete intersection of r = N − P N of multidegree d = ( d , . . . , d r ) is equal to b ( N, d ) = b ′ ( N, d ) = ( d · · · d r )( d + · · · + d r − N −
1) + 2 . Now observe that if r and d , . . . , d r are any positive integers, then d + · · · + d r ≤ ( d · · · d r ) + r − , and the equality holds if and only if either r = 1 or r > r − 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 d + 1 − ( d + d ) = ( d − d − . INGULAR VARIETIES OVER FINITE FIELDS 15
The inductive step follows readily using the assertion for r = 2.For the nonsingular curve above, if we let d = d · · · d r , then d is its degree and b ( N, d ) = d ( d + · · · + d r − r −
2) + 2 ≤ d ( d −
3) + 2 = ( d − d − . Since for a nonsingular plane curve of degree d the first Betti number is, as notedin (i), always equal to ( d − d − b ( N, d ) ≤ ( d − d − r = N − P N of multidegree d = ( d , . . . , d r ) is equal to b ( N, d ) = b ′ ( N, d ) + 1 = d (cid:18) r + 32 (cid:19) − ( r + 3) X ≤ i ≤ r d i + X ≤ i,j ≤ r d i d j − , where d = d d · · · d r is the degree of the surface.(iv) The primitive n -th Betti number of a complete intersection defined by r = 2forms of the same degree d is equal to b ′ n ( N, ( d, d )) = ( N − d − N + 2 d − d (( d − N − + ( − N ) . Zeta Functions and the Trace Formula
In this section k = F q is the finite field with q elements. We denote by k r thesubfield of k which is of degree r over k . Let X be a separated scheme of finite typedefined over the field k . As before, we denote by ¯ X = X ⊗ k k the scheme deducedfrom X by base field extension from k to k ; note that ¯ X remains unchanged if wereplace k by one of its extensions. The zeta function of X is(15) Z ( X, T ) = exp ∞ X r =1 T r r | X ( k r ) | , where T is an indeterminate. From the work of Dwork and Grothendieck (see,for example, the Grothendieck-Lefschetz Trace Formula below), we know that thefunction Z ( X, T ) is a rational function of T . This means that there are two familiesof complex numbers ( α i ) i ∈ I and ( β j ) j ∈ J , where I and J are finite sets, such that(16) Z ( X, T ) = Y j ∈ J (1 − β j T ) Y i ∈ I (1 − α i T ) . We assume that the fraction in the right-hand side of the above equality is irre-ducible. 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 toits multiplicity. We call the members of the families ( α i ) i ∈ I and ( β j ) j ∈ J the char-acteristic roots of Z ( X, T ). The degree deg Z ( X, T ) of Z ( X, T ) is the degree of itsnumerator 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 theusual way, from the above expression of Z ( X, T ), we deduce | X ( k r ) | = X i ∈ I α ri − X j ∈ J β rj . In order to simplify the notation, we write now H ic ( ¯ X ) = H ic ( ¯ X, Q ℓ ) , INGULAR VARIETIES OVER FINITE FIELDS 16 where ℓ is a prime number other than p ; in the case of proper subschemes, we mayuse H i ( ¯ X ) instead. If we denote by ϕ the element of g given by ϕ ( x ) = x q , thenthe geometric Frobenius element F of g is defined to be the inverse of ϕ . In thetwisted 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 ic ( ¯ X ) an endomor-phism denoted by F | H ic ( ¯ X ). A number α ∈ Q ℓ is pure of weight r if α is analgebraic integer and if | ι ( α ) | = q r/ for any embedding ι of 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 . Thenthe Grothendieck-Lefschetz Trace Formula holds: (17) | X ( k r ) | = n X i =0 ( − i Tr( F r | H ic ( ¯ X )) , and Deligne’s Main Theorem holds: (18)
The eigenvalues of F | H ic ( ¯ X ) are pure of weight ≤ i. See, for instance, [30, Thm. 13.4, p. 292] for the Grothendieck-Lefschetz TraceFormula, and [10, Thm. 1, p. 314] for Deligne’s Main Theorem.The Trace Formula (17) is equivalent to the equality(19) Z ( X, T ) = P ,ℓ ( X, T ) · · · P n − ,ℓ ( X, T ) P ,ℓ ( X, T ) · · · P n,ℓ ( X, T ) , where P i,ℓ ( X, T ) = det(1 − T F | H ic ( ¯ X )). We can write(20) P i,ℓ ( X, T ) = b i,ℓ Y j =1 (1 − ω ij,ℓ T ) , where ω ij,ℓ ∈ Q ℓ , and b i,ℓ = b i,ℓ ( ¯ X ) = deg P i,ℓ ( X, T ). The numbers ω ij,ℓ are calledthe reciprocal roots of P i,ℓ ( X, T ). The Trace Formula (17) may be written as | X ( k ) | = n X i =0 ( − i b i,ℓ X j =1 ω ij,ℓ , with the convention that if b i,ℓ = 0, then the value of the corresponding sum iszero. The compact ´etale ℓ -adic Euler-Poincar´e characteristic of X is χ ℓ ( X ) = n X i =0 ( − i b i,ℓ ( X ) . On the other hand, let us define σ ℓ ( X ) = n X i =0 b i,ℓ ( X ) . The equality between the right-hand sides of (16) and (19) implydeg Z ( X, T ) = χ ℓ ( X ) . Hence, the compact ´etale Euler-Poincar´e characteristic of X is independent of ℓ ,and the number deg Z ( X, T ) depends only on ¯ X . In the same way we findtot . deg Z ( X, T ) ≤ σ ℓ ( X ) . Here, the two sides may be different because of a possibility of cancellations occur-ring in the right-hand side of (19).
INGULAR VARIETIES OVER FINITE FIELDS 17 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 Nk ′ which can be defined, scheme-theoretically, by the vanishing of a system of m nonzero forms with coefficients in k ′ , of multidegree d = ( d , . . . , d m ).We now state a result of N. Katz [24, Cor. of Th. 3], whose proof is based ona result of Adolphson and Sperber [2, Th. 5.27]. If one checks the majorations inthe proof of [24, loc. cit. ], the number 9 appearing therein can be replaced by thenumber 8 in the inequality below. Theorem (Katz’s Inequality) . Let X be a closed subscheme in P N defined overan algebraically closed field, and of type ( m, N, d ) . If d = ( d , . . . , d m ) , then let δ = max( d , . . . , d m ) . We have (21) σ ℓ ( X ) ≤ × m × ( mδ + 3) N +1 . We would like to compare the number of points of X and that of the projectivespace of dimension equal to that of X . Hence, if dim X = n , we introduce therational function(22) Z ( X, T ) Z ( P n , T ) = exp ∞ X r =1 T r r ( | X ( k r ) | − | P n ( k r ) | ) . where Z ( P n , T ) = 1(1 − T ) . . . (1 − q n T ) . Proposition.
Given any projective scheme X of dimension n defined over k ,let τ ( X ) = tot . deg Z ( X, T ) Z ( P n , T ) . Also, given any nonnegative integers m , N , and d = ( d , . . . , 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 , . . . , d m ) , then τ k ( m, N, d ) ≤ × m × ( mδ + 3) N +1 , and, in particular, τ k ( m, N, d ) is bounded by a constant independent of the field k .Proof. Follows from Katz’s Inequality (21), since τ ( X ) ≤ σ ℓ ( X ) + n. (cid:3) Number of Points of Complete Intersections
In this section k = F q . We now state our main Theorem on the number of pointsof 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 Nk , defined by r = N − n equations, with multidegree d = ( d , . . . , d r ) , and choosean integer s such that dim Sing X ≤ s ≤ n − . Then (cid:12)(cid:12)(cid:12) | X ( k ) | − π n (cid:12)(cid:12)(cid:12) ≤ b ′ n − s − ( N − s − , d ) q ( n + s +1) / + C s ( X ) q ( n + s ) / , where C s ( X ) is a constant independent of k . If X is nonsingular, then C − ( X ) = 0 .If s ≥ , then C s ( X ) = n + s X i = n b i,ℓ ( ¯ X ) + ε i and upon letting δ = max( d , . . . , d r ) , we have C s ( X ) ≤ τ ( X ) ≤ τ ( r, N, d ) ≤ × r × ( rδ + 3) N +1 . INGULAR VARIETIES OVER FINITE FIELDS 18
Proof.
Equality (19) implies Z ( X, T ) Z ( P n , T ) = P ,ℓ ( X, T ) · · · P n − ,ℓ ( X, T )(1 − T ) . . . (1 − q n T ) P ,ℓ ( X, T ) · · · P n,ℓ ( X, T ) , and therefore, in view of (20) and (22), we get(23) | X ( k ) | − π n = n X i =0 ( − ε i q i/ + ( − i b i X j =1 ω ij,ℓ )where it may be recalled that ε i = 1 if i is even and 0 otherwise. Moreover, whenall the cancellations have been performed, the number of terms of the right-handside of (23) is at most equal to the total degree τ ( X ) of the rational fraction above.Now from Proposition 3.3(i) and (iii) we have P i,ℓ ( X, T ) = 1 − ε i q i/ T if i / ∈ [ n, n + s + 1] . Hence, these polynomials cancel if Z ( X, T ) /Z ( P n , T ) is in irreducible form. Ac-cordingly, from (23) we deduce | X ( k ) | − π n = n + s +1 X i = n ( − ε i q i/ + ( − i b i X j =1 ω ij,ℓ ) , and this gives (cid:12)(cid:12)(cid:12) | X ( k ) | − π n (cid:12)(cid:12)(cid:12) ≤ A + B, where A = 0 if s = n − A = (cid:12)(cid:12)(cid:12) − ε n + s +1 q ( n + s +1) / + ( − n + s +1 b n + s +1 X j =1 ω ( n + s +1) j,ℓ (cid:12)(cid:12)(cid:12) if s < n − , while B = (cid:12)(cid:12)(cid:12) n + s X i = n ( − ε i q i/ + ( − i b i X j =1 ω ij,ℓ ) (cid:12)(cid:12)(cid:12) if s ≤ n − . Now by Deligne’s Main Theorem (18), the numbers ω ij,ℓ are pure of weight ≤ i/ B ≤ C s ( X ) q ( n + s ) / where C s ( X ) = n + s X i = n b i,ℓ ( ¯ X ) + ε i . Clearly, C s ( X ) is independent of k and is at most equal to the number of terms inthe right-hand side of (23). Hence by Proposition 5.1, C s ( X ) ≤ τ ( X ) ≤ τ ( r, N, d ) ≤ × 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 − A = (cid:12)(cid:12)(cid:12) b n + s +1 X j =1 ω ( n + s +1) j,ℓ (cid:12)(cid:12)(cid:12) ≤ b n − s − ( N − s − , d ) q ( n + s +1) / . On the other hand, if 0 ≤ s ≤ n − n + s + 1 is even, then A = (cid:12)(cid:12)(cid:12) b n + s +1 X j =1 ω ( n + s +1) j,ℓ − q ( n + s +1) / (cid:12)(cid:12)(cid:12) . INGULAR VARIETIES OVER FINITE FIELDS 19
But in view of Remark 3.5, H n + s +1 ( X, Q ℓ ) contains a subspace which is isomorphicto Q ℓ ( − ( n + s +1) / q ( n + s +1) / is an eigenvalue of the highest possible weightof F | H ic ( ¯ X ), and hence it is a reciprocal root of P n + s +1 ( X, T ). It follows that A ≤ ( b n − s − ( N − s − , d ) − q ( n + s +1) / . Thus in any case, A ≤ b n − s − ( N − s − , d ) q ( n + s +1) / . The case when X is non-singular follows similarly using Proposition 3.1. (cid:3) The case where X is nonsingular is Deligne’s Theorem [9, Thm. 8.1]. In theopposite, since we always have s ≤ n −
1, Theorem 6.1 implies the following weakversion of the Lang-Weil inequality for complete intersections: | X ( k ) | − π n = O (cid:16) q n − (1 / (cid:17) . We shall obtain a much better result in Theorem 11.1. Nevertheless, the followingCorollary shows that we can obtain the Lang-Weil inequality (in fact, a strongerresult) 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 inthe beginning of Section 3, X is integral, and so the hypothesis that X is irreducibleis 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 Nk withmultidegree d , then (cid:12)(cid:12)(cid:12) | X ( k ) | − π n (cid:12)(cid:12)(cid:12) ≤ b ′ ( N − n + 1 , d ) q n − (1 / + C n − ( X ) q n − , where C n − ( X ) is as in Theorem 6.1. Moreover, if d = deg X , then b ′ ( N − n + 1 , d ) ≤ ( d − d − , with equality holding if and only if X is a hypersurface.Proof. Follows from Theorem 6.1 with s = n − (cid:3) Remark.
It is also worthwhile to write down explicitly the particular case of acomplete intersection X of dimension n in P Nk regular in codimension 2. Namely,if dim Sing X ≤ n −
3, then (cid:12)(cid:12)(cid:12) | X ( k ) | − π n (cid:12)(cid:12)(cid:12) ≤ b ′ ( N − n − , d ) q n − + C n − ( X ) q n − (3 / . Complete Intersections with isolated singularities
We now prove an inequality for the central Betti number of complete intersectionswith only isolated singularities. Unfortunately, the proof of this result depends onthe following condition:( R n , p ) The resolution of singularities holds in characteristic p for excellent localrings of dimension at most n .This condition means the following. Let A be an excellent local ring of equalcharacteristic p of dimension ≤ n and X = Spec A . Let U be a regular opensubscheme of X , and S = X \ U . Then, there is a commutative diagram e U −−−−→ e X ←−−−− e S y y π y U −−−−→ X ←−−−− S INGULAR VARIETIES OVER FINITE FIELDS 20 where e X is a regular scheme and where π is a proper morphism which is a birationalisomorphism, and an isomorphism when restricted to e U . Moreover e S = e X \ e U is a divisor with normal crossings , that is, a family of regular schemes of purecodimension 1 such that any subfamily intersects properly.Recall that the resolution of singularities holds in characteristic 0, and that( R , p ) holds for any p . For details, see the book [1] by Abhyankar and the survey[29] by Lipman.7.1. Proposition.
Assume that k = ¯ k and that ( R n , p ) holds. Let X be a completeintersection in P N of dimension n ≥ 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 , . . . , f r ).Since k is algebraically closed, there exists a nonsingular complete intersection Y of P N of codimension r of multidegree d . Let ( g , . . . , g r ) be a system defining Y .The one-parameter family of polynomials T g j ( X , . . . , X N ) + (1 − T ) f j ( X , . . . , X N ) , (1 ≤ j ≤ r )define a scheme Z coming with a morphism π : Z −→ A k . The morphism π is proper and flat. Let S = Spec k { T } where k { T } is the stricthenselization of k [ T ] at the ideal generated by T [30, p. 38]. Then S is the spectrumof 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 calleda trait strictement local in French). Let Z = Z × k S the scheme obtained by basechange : Z −−−−→ Z π y y π S −−−−→ A k Then π is a proper and flat morphism, and the closed fiber Z s = π − ( s ) is iso-morphic to X . If ¯ η is a geometric point of S mapping to η , the geometric fiber Z ¯ η = Z × κ (¯ η ) is a nonsingular complete intersection of codimension r . We thusget a diagram Z ¯ η −−−−→ Z ←−−−− X y π y y ¯ η −−−−→ S ←−−−− s By [8, Eq. 2.6.2, p. 9], there is a long exact sequence of cohomology . . . −→ φ i − gl −→ H i ( Z s , Q ℓ ) sp i −→ H i ( Z ¯ η , Q ℓ ) −→ φ igl −→ . . . where φ igl is the space of global vanishing cycles , and sp i is the specialization mor-phism . The Theorem on Sheaves of Vanishing Cycles for cohomology of low degree[8, Thm. 4.5 and Var. 4.8] states that φ igl = 0 for i ≤ n −
1. Hence sp i is anisomorphism for i ≤ n − i = n which proves the requiredresult, with the following warning: according to [8, 4.4, p. 14], the proof of theTheorem on Sheaves of Vanishing Cycles relies on ( R n , p ). (cid:3) From now on let k = F q . The following corollary is essentially a result of I.E.Shparlinski˘ı and A.N. Skorobogatov [39]. However, as noted in Remark 3.4, oneneeds to assume ( R n , p ) for proving such a result. INGULAR VARIETIES OVER FINITE FIELDS 21
Corollary.
Assume that ( R n , p ) holds. If X is a complete intersection of di-mension n in P Nk with multidegree d with only isolated singularities, then (cid:12)(cid:12)(cid:12) | X ( k ) | − π n (cid:12)(cid:12)(cid:12) ≤ b ′ n − ( N − , d ) q ( n +1) / + ( b n ( N, d ) + ε n ) q n/ . Proof.
Use Theorem 6.1 with s = 0 and apply Proposition 7.1 to ¯ X . (cid:3) 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 n +1 k of degree d with only isolated singularities, then (cid:12)(cid:12)(cid:12) | X ( k ) | − π n (cid:12)(cid:12)(cid:12) ≤ ( d − n +1 q ( n +1) / . Proof.
Follows from Corollary 7.2 and the inequality in Example 4.3 (i). (cid:3)
The following result may be thought of as an analogue of the Weil inequality forcertain singular curves and it is a more precise version of a result by Aubry andPerret [4, Cor. 2.5].7.4.
Corollary. If X is an irreducible curve in P Nk , then (cid:12)(cid:12)(cid:12) | X ( k ) | − ( q + 1) (cid:12)(cid:12)(cid:12) ≤ b ( ¯ X ) √ q. Moreover, if the curve X is a complete intersection with multidegree d , and if d denotes the degree of X , then b ( ¯ X ) ≤ b ( N, d ) ≤ ( d − d − , and the last inequality is an equality if and only if X is isomorphic to a nonsingularplane curve.Proof. The first statement follows from the Trace Formula (17) and Deligne’s MainTheorem (18). In the second statement, the first inequality follows from Proposition7.1 since ( R n , p ) is true for n = 1. The last assertion is a consequence of theobservations in Example 4.3 (ii). (cid:3) Example.
We work out here a simple case which will be used later on in Example9.2. Let C be a nonsingular plane curve of genus g and degree d , defined over k , andlet X be the projective cone in P k over C [16, Ex. 2.10, p. 13]. Thus X is a surfacewhich is a complete intersection, and with exactly one singular point, namely, thevertex of the cone. Then | X ( k m ) | = q m | C ( k m ) | + 1 . If α , . . . , α g are the roots of F in H ( C ), then(24) | X ( k m ) | = q m ( q m + 1 − g X j =1 α mj ) + 1 = q m − g X j =1 ( qα j ) m + q m + 1 , and therefore (cid:12)(cid:12)(cid:12) | X ( k ) | − π (cid:12)(cid:12)(cid:12) ≤ ( d − d − q / . Observe that the equality can occur if C has the maximum number of points allowedby Weil’s inequality. From (24) and the definition (15) of the zeta function, we get Z ( X, T ) = P ( C, qT )(1 − q T )(1 − qT )(1 − T ) . We compare this with the expression (19) of the zeta function. We note that X isirreducible, that the eigenvalues of the Frobenius in H ( X ) are pure of weight 3,and that those of H ( X ) are pure of weight ≤
1. Hence P ( X, T ) = 1 , P ( X, T ) = (1 − qT ) , P ( X, T ) = P ( C, qT ) . INGULAR VARIETIES OVER FINITE FIELDS 22
By looking at the degrees of these polynomials, we find b ( X ) = 0 , b ( X ) = 1 , b ( X ) = 2 g ( C ) . The Penultimate Betti Number
Let X be a separated scheme of finite type over k = F q of dimension n . Denote by H n − ( ¯ X ) the subspace of H n − c ( ¯ X ) generated by the (generalized) eigenvectors ofthe Frobenius endomorphism whose eigenvalues are pure of weight exactly equal to2 n −
1. This subspace is the component of maximal weight 2 n − H n − c ( ¯ X ) induced by the weight. Define P +2 n − ( X, T ) = det(1 − T F | H n − ( ¯ X )) ∈ Z ℓ [ T ] . The 2 n − -th virtual Betti number of Serre [22, p. 28] is b +2 n − ( X ) = deg P +2 n − ( X, T ) = dim H n − ( ¯ X ) . Recall that two schemes are birationally equivalent if they have isomorphic denseopen sets.8.1.
Proposition.
Let X be a separated scheme of dimension n defined over k . (i) The polynomial P +2 n − ( X, t ) has coefficients independent of ℓ . (ii) The space H n − ( ¯ X ) is a birational invariant. More precisely, if U is adense open set in X , then the open immersion j : U −→ X induces anisomorphism j ∗ : H n − ( ¯ U ) −→ H n − ( ¯ X ) . (iii) The polynomial P +2 n − ( X, T ) and the virtual Betti number b +2 n − ( X ) arebirational invariants. Assertion (iii) is reminiscent of [27, Cor. 6].
Proof.
Assertion (i) can be easily checked by the following formula, a particularcase of
Jensen’s formula . In the complex plane, let γ be the oriented boundary ofan annulus r ′ ≤ | w | ≤ r, with q − n < r ′ < q − n +(1 / < r < q − n +1 . If t is a complex number with | t | > q − n +1 , then P +2 n − ( X, t ) = exp 12 iπ Z γ log(1 − w − t ) Z ′ ( X, w ) Z ( X, w ) dw. Let us prove (ii). Set Z = X \ U and consider the long exact sequence of cohomologywith compact support [30, Rem. 1.30, p. 94]: . . . −→ H n − c ( ¯ Z ) i −→ H n − c ( ¯ U ) j ∗ −→ H n − c ( ¯ X ) −→ H n − c ( ¯ Z ) −→ . . . and recall that the homomorphisms of this exact sequence are g -equivariant. Now H n − c ( ¯ Z ) = 0 since dim Z ≤ n −
1. Hence we get an exact sequence0 −→ H n − c ( ¯ Z ) / ker i ˜ i −→ H n − c ( ¯ U ) j ∗ −→ H n − c ( ¯ X ) −→ H n − c ( ¯ Z ) is equal to the number of irreducible components of ¯ Z of dimension n −
1, and all the eigenvalues of the Frobenius automorphism in thisspace are pure of weight 2 n −
2. Hence the eigenvalues of the Frobenius in Im˜ i arealso pure of weight 2 n −
2. Thus H n − ( ¯ U ) ∩ Im˜ i = 0 and the restriction of j ∗ to H n − ( ¯ U ) is an isomorphism. Assertion (iii) is a direct consequence of (ii). (cid:3) The following elementary result will be needed below.
INGULAR VARIETIES OVER FINITE FIELDS 23
Lemma.
Let A and B be two finite sets of complex numbers included in thecircle | z | = M . If, for some λ with < λ < M and for every integer s sufficientlylarge, X β ∈ B β s − X α ∈ A α s = O ( λ s ) , then A = B .Proof. Suppose A = B . By interchanging A and B if necessary, we can supposethat there is an element β ∈ B \ A . Then the rational function R ( z ) = X α ∈ A − αz − X β ∈ B − βz is holomorphic for | z | < M − and admits the pole β − on the circle | z | = M − .But the Taylor series of R ( z ) at the origin is R ( z ) = ∞ X s =0 X α ∈ A α s − X β ∈ B β s z s , and the radius of convergence of this series is ≥ λ − > M − . (cid:3) Lemma.
Let X be an irreducible scheme of dimension n defined over k . Thefollowing are equivalent : (i) There is a constant C such that (cid:12)(cid:12)(cid:12) | X ( F q s ) | − q ns (cid:12)(cid:12)(cid:12) ≤ Cq s ( n − for any s ≥ . (ii) We have H n − ( ¯ X ) = 0 .Proof. Let A be the set of eigenvalues of the Frobenius automorphism in H n − ( ¯ X ).By the Trace Formula (17) and Deligne’s Main Theorem (18),(25) (cid:12)(cid:12)(cid:12) | X ( F q s ) | − cq ns − X α ∈ A α s (cid:12)(cid:12)(cid:12) ≤ C ′ ( X ) q s ( n − for any s ≥ , where c = dim H nc ( ¯ X ) is the number of irreducible components of X of dimension n and where C ′ ( X ) = n − X i =0 b i,l ( ¯ X )is independent of q . Here c = 1 since X is irreducible. Now assume that (i) holds.From Formula (25) above, we deduce (cid:12)(cid:12)(cid:12) X α ∈ A α s (cid:12)(cid:12)(cid:12) ≤ ( C ′ ( X ) + C ) q s ( n − for any s ≥ . So Lemma 8.2 implies A = ∅ , and hence, H n − ( ¯ X ) = 0. The converse implicationis an immediate consequence of (25). (cid:3) Lemma.
Let K be an algebraically closed field, and X an irreducible projectivecurve in P NK , with arithmetic genus p a ( X ) . Let e X be a nonsingular projectivecurve birationally equivalent to X , with geometric genus g ( e X ) . Then we have thefollowing. (i) If d denotes the degree of X , then g ( e X ) ≤ b ( X ) ≤ p a ( X ) ≤ ( d − d − . (ii) If K = ¯ k , where k is a finite field, and if X is defined over k , then b +1 ( X ) = 2 g ( e X ) , INGULAR VARIETIES OVER FINITE FIELDS 24
During the proof of the Lemma, we shall make use of the following standardconstruction, when X is a curve. This leads to an inequality between Hilbertpolynomials.8.5. Remark (Comparison of Hilbert polynomials) . Let K be an algebraically closedfield, X a closed subvariety in P NK distinct from the whole space, and r an inte-ger such that dim X + 1 ≤ r ≤ N . Let C r ( X ) be the subvariety of G N − r,N oflinear varieties of codimension r meeting X . From the properties of the incidencecorrespondence Σ defined byΣ = (cid:8) ( x, E ) ∈ P N × G N − r,N | x ∈ E (cid:9) , it is easy to see that C r ( X ) = π ( π − ( X )) is irreducible and that the codimensionof C r ( X ) in G N − r,N is equal to r − dim X . Hence, the set of linear subvarieties ofcodimension r in P NK disjoint from X is a nonempty open subset D r ( X ) of G N − r,N .If E belongs to D n +2 ( X ), where n = dim X , the projection π with center E gives rise to a diagram X i −−−−→ P NK − E y π X y π X ′ i ′ −−−−→ P n +1 K such that X ′ is an irreducible hypersurface with deg X ′ = deg X , and where therestriction π X is a finite birational morphism: denoting by S ( X ) the homogeneouscoordinate ring of X , we have an inclusion S ( X ′ ) ⊂ S ( X ), and S ( X ) is a finitelygenerated module over S ( X ′ ). Hence, if P X ( T ) ∈ Q [ T ] is the Hilbert polynomial of X [16, p. 52], we have P X ′ ( t ) ≤ P X ( t ) if t ∈ N and t → ∞ . Proof of Lemma 8.4.
Let U be a regular open subscheme of X . Then, there is acommutative diagram e U −−−−→ e X ←−−−− e S y y π y U −−−−→ X ←−−−− S where e X is a nonsingular curve, where π is a proper morphism which is a birationalisomorphism, and an isomorphism when restricted to e U , andSing X ⊂ S = X \ U, e S = e X \ e U .
The excision long exact sequence in compact cohomology [30, Rem. 1.30, p. 94]gives:0 −−−−→ H c ( X ) −−−−→ H c ( S ) −−−−→ H c ( U ) −−−−→ H c ( X ) −−−−→ X, U, S by e X, e U , e S . This implies b ( U ) = b ( X ) − | S | , b ( e U ) = b ( e X ) − | ˜ S | , and since U and e U are isomorphic, we obtain b ( X ) = b ( e X ) + d ( X ) = 2 g ( e X ) + d ( X ) , where d ( X ) = | ˜ S | − | S | , since, as is well-known, b ( e X ) = 2 g ( e X ). Let δ ( X ) = p a ( X ) − g ( e X ) . Then 0 ≤ d ( X ) ≤ δ ( X ) [34, Prop. 1, p. 68]. Hence b ( X ) = 2 g ( e X ) + d ( X ) ≤ g ( e X ) + 2 δ ( X ) = 2 p a ( X ) . INGULAR VARIETIES OVER FINITE FIELDS 25
This proves the first and second inequalities of (i). The Hilbert polynomial of X is([16, p. 54]): P X ( T ) = dT + 1 − p a ( X ) , Apply now the construction of Remark 8.5 to X , and obtain a morphism X −→ X ′ ,where X ′ is a plane curve of degree d . From the inequality P X ′ ( t ) ≤ P X ( t ) for t large, we get p a ( X ) ≤ p a ( X ′ ). Now by Example 4.3(ii), p a ( X ′ ) = ( d − d − / , since X ′ is a plane curve of degree d , and so p a ( X ) ≤ ( d − d − / , and this proves the third inequality of (i). Now, under the hypotheses of (ii), wehave by [4, Thm. 2.1]: P ( X, T ) = P ( e X, T ) d ( X ) Y j =1 (1 − ω j T ) , where the numbers ω j are roots of unity, and this implies the inequality in (ii). (cid:3) From now on, suppose k = F q . If X is a separated scheme of finite type over k , we say that the space H ic ( ¯ X ) is pure of weight i if all the eigenvalues of theFrobenius 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 Nk whichis regular in codimension one. Then: (i) The space H n − ( ¯ X ) is pure of weight n − . (ii) If Y is a typical curve on X over k , then P n − ( X, T ) divides P ( Y, q n − T ) . Proof.
By passing to finite extension k ′ of k , we can find a typical curve Y definedover k ′ . In that case the Gysin map ι ∗ : H ( ¯ Y , Q ℓ (1 − n )) −→ H n − ( ¯ X, Q ℓ )is a surjection by Corollary 2.1, and all the eigenvalues are pure of the sameweight, since ι ∗ is g -equivariant. This proves (i), and also that P n − ( X, T ) di-vides P ( Y, q n − T ) if k = k ′ , which proves (ii). (cid:3) In view of (i), a natural question is then to ask under which conditions the space H i ( ¯ X ) is pure of weight i . The following proposition summarizes the results onthis topic that we can state.8.7. Proposition.
Let X be a projective variety of dimension n defined over k andassume that dim Sing X ≤ s . (i) The space H i ( ¯ X ) 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 onlyisolated singularities. Then: (ii) The space H n ( ¯ X ) is pure of weight n .Proof. By Corollary 1.4, we can find a nonsingular proper linear section Y of X ofcodimension s + 1 defined over a finite extension k ′ /k . Since the Gysin maps areequivariant with respect to Gal(¯ k/k ′ ), Corollary 2.1 implies that the eigenvaluesof the Frobenius of k ′ in H n − ( ¯ X, Q ℓ ( n )) are pure, and the same holds for theeigenvalues of the Frobenius of k , since they are roots of the former. This proves(i). Finally, as in the proof of Proposition 7.1, we deduce from the Theorem onSheaves of Vanishing Cycles a g -equivariant exact sequence0 = φ n − s − gl −→ H n − s ( ¯ X ) −→ H n − s ( Z ¯ η ) −→ . . . where Z ¯ η is a nonsingular complete intersection. This implies (ii). (cid:3) INGULAR VARIETIES OVER FINITE FIELDS 26
Remark. If X is a complete intersection with only isolated singularities, thenthe spaces H n ( ¯ X ) and H n +1 ( ¯ X ) are the only ones for which the non-primitive partis nonzero. Hence, provided ( R n , p ) holds, Proposition 8.7 shows that(iii) H i ( ¯ X ) is pure of weight i for 0 ≤ i ≤ n .Thus for this kind of singular varieties, the situation is the same as for nonsingularvarieties. It is worthwhile recalling that if X is locally the quotient of a nonsingularvariety 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 abelianvarieties associated to a variety, namely the Albanese and Picard varieties. Laterwe shall discuss their relation with some ´etale cohomology spaces. For the generaltheory of abelian varieties, we refer to [28] and [31].Let X be a variety defined over a perfect field k and assume for simplicity that X has a k -rational nonsingular point x (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 isdefined at x and if g ( x ) = 0. An Albanese-Weil variety (resp. an
Albanese-Serrevariety ) of X is an abelian variety A defined over k equipped with an admissiblerational map (resp. an admissible morphism) f from X to A satisfying the followinguniversal 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 : @@R g X f y A ϕ −−−−→ B Assume that A exists. If U is an open subset of X containing x where f is defined,then the smallest abelian subvariety containing f ( U ) is equal to A . The abelianvariety 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 [35, Thm. 5].Let X be a variety, and let ι : e X −→ X be any birational morphism to X froma nonsingular variety e X (take for instance e X = Reg X ). Since any rational mapof 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 amorphism of e X into B , and factors through the Albanese-Serre variety Alb s e X : (cid:0)(cid:0)(cid:9) f w e X ι −−−−→ X ˜ f s y y g Alb s e X ϕ −−−−→ B This implies that we can take Alb w X = Alb s e X , if we define the canonical map as f w = e f s ◦ ι − . Hence, the Albanese-Weil variety Alb w X , together with a canonicalmap f w : X −→ Alb w X INGULAR VARIETIES OVER FINITE FIELDS 27 exists for any variety X , and two birationally equivalent varieties have the sameAlbanese-Weil variety. These two results have been proved by Weil [28, Thm. 11,p. 41], and [28, p. 152]. If X is a curve, then Jac X = Alb w X by definition,and the dimension of Jac X is equal to the genus of a nonsingular projective curvebirationally equivalent to X .We recall now the following result [35, 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 : HHHHj f s X f w y Alb w X ν −−−−→ Alb s X (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. (cid:3) Example.
Notice that Alb s X is not a birational invariant and moreover thatthe inequality dim Alb s X < dim Alb w X can occur. For instance, let C be a non-singular plane curve of genus g , defined over k , and let X be the normal projectivecone in P k over C , as in Example 7.5. Since X is an hypersurface, Alb s X is triv-ial by Remark 9.3 below. On the other hand, X is birationally equivalent to thenonsingular projective surface e X = C × P . Since any rational map from A to anabelian variety is constant, the abelian variety Alb w X = Alb w e X is equal to theJacobian 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],[38], 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
Pic
X/k of X [12], [6, Ch. 8], which is a separated commutative group scheme locally of finitetype over k . Its identity component Pic X/k is an abelian scheme defined over k ,and Pic s X = (Pic X/k ) red [12, Thm. 3.3(iii), p. 237].9.3. Remark (complete intersections) . The Zariski tangent space at the origin ofPic
X/k is the coherent cohomology group H ( X, O X ) [12, p. 236], and hence,dim Pic X/k ≤ dim H ( X, O X ) . For instance, if X is a projective normal complete intersection of dimension ≥
2, atheorem of Serre [16, Ex. 5.5, p. 231] asserts that H ( 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, thenthere 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 ˜ g w y ˜ f w y Alb w Y ϕ ∗ −−−−→ Alb w X We use this construction in the following situation. Recall that the set U r ( X ) hasbeen defined in section 1. INGULAR VARIETIES OVER FINITE FIELDS 28
Proposition.
Let X be a projective variety of dimension n embedded in P N .If ≤ r ≤ n − , and if E ∈ U r ( X ) , let Y = X ∩ E be the corresponding linearsection of dimension n − r , and let ι : Y −→ X be the canonical closed immersion. (i) If n − r ≥ , the set of E ∈ U r ( X ) such that ι ∗ is a purely inseparableisogeny contains a nonempty open set of G N − r,N . (ii) The set of E ∈ U n − ( X ) such that ι ∗ is surjective contains a nonemptyopen set of G N − n − ,N . If E belongs to this set, then Y = X ∩ E is a curvewith dim Alb w X ≤ dim Jac Y. (iii) If deg X = d , if Y = X ∩ E is a curve as in (ii), and if e Y is a nonsingularprojective curve birationally equivalent to Y , then dim Alb w X ≤ g ( e Y ) ≤ ( d − d − . Proof.
The results (i) and (ii) are classical. For instance, assertion (i) follows frominduction 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 alsoTheorem 11 and its proof in [35, p. 159] for very simple arguments to show that ι ∗ is surjective. The first inequality in (iii) is an immediate consequences of (ii) andthe second is Lemma 8.4(i). (cid:3) Remarks. (i) Let X be a projective variety regular in codimension one, and Y a typical curve on X . A theorem of Weil [43, Cor. 1 to Thm. 7] states that thehomomorphism 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 asurface. 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 ( k ) such that ma = 0. Let ℓ be aprime 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 bymultiplication by ℓ ; this is a free Z ℓ -module of rank 2 g , and the group g operateson T ℓ ( A ). The tensor product V ℓ ( A ) = T ℓ ( A ) ⊗ Z ℓ Q ℓ is a vector space of dimension 2 g over Q ℓ .We recall the following result [25, Lem. 5] and [30, Cor. 4.19, p. 131], whichgives a description in purely algebraic terms of H ( ¯ X, Q ℓ ) when X is a normalprojective variety.9.6. Proposition.
Let X be a normal projective variety defined over k . Then, thereis a g -equivariant isomorphism h X : V ℓ (Pic s X )( − ∼ −→ H ( ¯ X, Q ℓ ) . In particular, b ,ℓ ( ¯ X ) = 2 dim Pic s X is independent of ℓ . Remarks. (i) If X is a normal complete intersection, then Proposition 9.6 andExample 9.3 imply H ( ¯ X, Q ℓ ) = 0, in accordance with Proposition 3.3(iii).(ii) If X is a normal projective variety, we get from Proposition 9.6 a g -equivariantisomorphism H (Alb s ¯ X, Q ℓ ) ∼ −→ H ( ¯ X, Q ℓ ) . INGULAR VARIETIES OVER FINITE FIELDS 29
Proposition.
Let X be a normal projective variety of dimension n ≥ definedover k which is regular in codimension . Then there is a g -equivariant isomorphism j X : V ℓ (Alb w X ) ∼ −→ H n − ( ¯ X, Q ℓ ( n )) . If ( R n , p ) holds, the same conclusion is true if one only assumes that X is regularin codimension .Proof. Step
1. Assume that X is a subvariety in P Nk . Since X is regular in codimen-sion 2, we deduce from Proposition 1.3 and Corollary 1.4 that U n − ( X ) contains anonempty Zariski open set U in the Grassmannian G N − n +2 ,N . On the other hand,any open set defined over ¯ k is defined over a finite extension k ′ , and contains anopen set defined over k (take the intersection of the transforms by the Galois groupof k ′ /k ). Let U ⊂ U be an open set defined over k . If E ∈ U , then Y = X ∩ E isa typical surface on X over k , i. e. , a nonsingular proper linear section of dimension2 in X . For such a typical surface Y , the closed immersion ι : Y −→ X induces ahomomorphism ι ∗ : Alb w Y −→ Alb w X . By Proposition 9.4(i), the set U of linearvarieties E ∈ U such that ι ∗ is a purely inseparable isogeny contains as well anonempty open subset U ⊂ G N − n +2 ,N which is defined over k . Step
2. Assume that U ( k ) is nonempty. If E ∈ U ( k ), we get a g -equivariantisomorphism V ℓ ( ι ∗ ) : V ℓ (Alb w Y ) ∼ −→ V ℓ (Alb w X ) . Since Y is nonsingular, we get from Poincar´e Duality Theorem for nonsingularvarieties [30, Cor. 11.2, p. 276] a g -equivariant nondegenerate pairing H ( ¯ Y , Q ℓ ) × H ( ¯ Y , Q ℓ (2)) −→ Q ℓ , from which we deduce a g -equivariant isomorphism ψ : Hom( H ( ¯ Y , Q ℓ ) , Q ℓ ) −→ H ( ¯ Y , Q ℓ (2)) . Since (
X, Y ) is a semi-regular pair with Y nonsingular, from Corollary 2.1 we knowthat the Gysin map ι ∗ : H ( ¯ Y , Q ℓ (2 − n )) −→ H n − ( ¯ X, Q ℓ )is an isomorphism. Now a g -equivariant isomorphism of vector spaces over Q ℓ : j X : V ℓ (Alb w X ) ∼ −→ H n − ( ¯ X, Q ℓ ( n ))is defined as the isomorphism making the following diagram commutative:Hom( V ℓ (Pic s Y )( − , Q ℓ ) ̟ −−−−→ ∼ V ℓ (Alb w Y ) V ℓ ( ι ∗ ) −−−−→ ∼ V ℓ (Alb w X ) t h Y y ∼ j X y Hom( H ( ¯ Y , Q ℓ ) , Q ℓ ) ψ −−−−→ ∼ H ( ¯ Y , Q ℓ )(2) ι ∗ −−−−→ ∼ H n − ( ¯ X, Q ℓ ( n ))Here ̟ is defined by the Weil pairing, and t h Y is the transpose of the map h Y defined in Proposition 9.6. Hence, the conclusion holds if U ( k ) = ∅ . Step
3. Assume that k is an infinite field. One checks successively that if U isan open subset in an affine line, an affine space, or a Grassmannian, then U ( k ) = ∅ and the conclusion follows from Step 2. Step
4. Assume that k is a finite field. Then the following elementary resultholds (as a consequence of Proposition 12.1 below, for instance). Claim.
Let U be a nonempty Zariski open set in G r,N , defined over k , and k s = F q s the extension of degree s of k = F q . Then there is an integer s ( U ) such that U ( k s ) = ∅ for every s ≥ s ( U ) . INGULAR VARIETIES OVER FINITE FIELDS 30
Now take for U the open set in G N − n +2 ,N introduced in Step 1. Choose any s ≥ s ( U ), and let g s = Gal(¯ k/k s ). Since U ( k s ) = ∅ , upon replacing k by k s , wededuce from Step 2 a g s -equivariant isomorphism of Q ℓ -vector spaces: j X,s : V ℓ (Alb w X ) ∼ −→ H n − ( ¯ X, Q ℓ ( n )) . This implies in particular that if m = 2 dim Alb w X , thendim H n − ( ¯ X, Q ℓ ( n )) = dim V ℓ (Alb w X ) = m. In each of these spaces, there is an action of g = g . By choosing bases, we identifyboth of them with Q mℓ . Denote by g ∈ GL m ( Q ℓ ) the matrix of the endomorphism V ℓ ( ϕ ), where ϕ ∈ g is the geometric Frobenius, and by g ∈ GL m ( Q ℓ ) the matrixof the Frobenius operator in H n − ( ¯ X, Q ℓ ( n )). The existence of the g s -equivariantisomorphism j X,s implies that g s and g s are conjugate. In order to finish the proofwhen k is finite, we must show that g and g are conjugate. This follows from theConjugation Lemma below, since g is semi-simple by [31, p. 203]. Step
5. Assume now that ( R n , p ) holds and that X is regular in codimension 1.Take e X to be a nonsingular projective variety birationally equivalent to X over k .Then Alb w e X = Alb w X since the Albanese-Weil variety is a birational invariant,and H n − ( e X ⊗ ¯ k, Q ℓ ( n )) = H n − ( X ⊗ ¯ k, Q ℓ ( n )) , by Proposition 8.1(ii). Now it is well known that H n − ( e X ⊗ ¯ k, Q ℓ ) is pure, andthe same holds for X , by Prop. 8.7(i). Hence, H n − ( e X ⊗ ¯ k, Q ℓ ( n )) = H n − ( X ⊗ ¯ k, Q ℓ ( n )) . Since the conclusion is true for a nonsingular variety, we obtain a g -equivariantmap j e X : V ℓ (Alb w e X ) ∼ −→ H n − ( e X ⊗ ¯ k, Q ℓ ( n )) , and this gives the required g -equivariant isomorphism. (cid:3) Conjugation Lemma.
Let K be a field of characteristic zero, and let g and g betwo matrices in GL n ( K ) , with g semi-simple. If g s is conjugate to g s for infinitelymany prime numbers s , then g is conjugate to g .Proof. Let g = su be the multiplicative Jordan decomposition of g into its semi-simple and unipotent part. Take a and b prime with g a conjugate to g a . Then s a u a is conjugate to g a , and hence, u a = I , by the uniqueness of the Jordandecomposition. Similarly, we find u b = I . Hence u = I with the help of B´ezout’sequation, and g is semisimple.Take now two diagonal matrices d and d in GL n ( ¯ K ) such that g i is conjugate to d i in GL n ( ¯ K ). Two conjugate diagonal matrices are conjugate by an element of thegroup W of permutation matrices: if d s and d s are conjugate, then d s = ( w s d w − s ) s with w s ∈ W . Since W is finite, one of the sets T ( w ) = (cid:8) s ∈ N | d s = ( wd w − ) s (cid:9) contains infinitely many prime numbers. Take two prime numbers a and b in thatset, then d a = h a , d b = h b , h = wd w − , from which we deduce d = h by B´ezout’s equation. This implies that d and d are conjugate in GL n ( ¯ K ), and the same holds for g and g . But two elements ofGL n ( K ) which are conjugate in GL n ( ¯ K ) are conjugate in GL n ( K ). (cid:3) Remark.
Let X be a complete intersection of dimension ≥ w X is trivial, since Proposition 3.3(i) implies that H n − ( ¯ X, Q ℓ ) = 0. INGULAR VARIETIES OVER FINITE FIELDS 31
The following result is a weak form of Poincar´e Duality between the first andthe penultimate cohomology spaces of some singular varieties.9.10.
Corollary.
Let X be a normal projective variety of dimension n ≥ definedover k regular in codimension . Then there is a g -equivariant injective linear map H ( ¯ X, Q ℓ ) −→ Hom( H n − ( ¯ X, Q ℓ ( n )) , Q ℓ ) . If ( R n , p ) holds, the same conclusion is true if one only assumes that X is regularin codimension .Proof. Proposition 9.6 furnishes an isomorphism h − X : H ( ¯ X, Q ℓ ) ∼ −→ V ℓ (Pic s X )( − . From the surjective map ν : Alb w X −→ Alb s X defined in Proposition 9.1, we getby duality a homomorphism with finite kernel t ν : Pic s X −→ Pic w X generatingan 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 )( − −→ Hom( V ℓ (Alb w X ) , Q ℓ )and Proposition 9.8 gives an isomorphismHom( V ℓ (Alb w X ) , Q ℓ ) −→ Hom( H n − ( ¯ X, Q ℓ ( n )) , Q ℓ ) . The required linear map is the combination of all the preceding maps. (cid:3)
Recall that a projective variety, regular in codimension one, which is a localcomplete intersection, is normal. For such varieties, we obtain a sharper version ofProposition 9.1(ii) as follows.9.11.
Corollary.
Assume that ( R n , p ) holds. Let X be a projective variety, regularin codimension one, which is a local complete intersection. Then the canonical map ν : Alb w X −→ Alb s X is an isomorphism.Proof. Since ( R n , p ) holds and since X is a local complete intersection, we know,by Poincar´e Duality (Remark 2.7), thatdim H ( ¯ X, Q ℓ ) = dim H n − ( ¯ X, Q ℓ ( n )) , and the linear map of Corollary 9.10 is bijective. Hence, the homomorphism V ℓ ( t ν ) : V ℓ (Pic s X ) −→ V ℓ (Pic w X )is also bijective. By Tate’s Theorem [31, Appendix I], this implies that ν is anisogeny. Since the kernel of ν is connected by Prop. 9.1(ii), it is trivial. (cid:3) A Conjecture of Lang and Weil
We assume now that k = F q is a finite field. In order to state the results of thissection, 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 e X defined over k ,which is regular in codimension . As a special case of Abhyankar’s results [1], (
RS2 ) is valid in any characteristic p > X ≤ X = 3 and p = 2 , , description in purely algebraic terms of the birational in-variant H n − ( ¯ X, Q ℓ ). Recall that H n − ( ¯ X, Q ℓ ) is in fact pure as soon as X isregular in codimension 1. INGULAR VARIETIES OVER FINITE FIELDS 32
Theorem.
Let X be a variety of dimension n ≥ defined over k satisfying ( RS2 ) . Then there is a g -equivariant isomorphism j X : V ℓ (Alb w X ) ∼ −→ H n − ( ¯ X, Q ℓ ( n )) , and hence b +2 n − ( X ) = 2 dim Alb w X . In particular this number is even.Proof. Take e X birationally equivalent to X as in ( RS2 ). Then Alb w e X = Alb w X since the Albanese-Weil variety is a birational invariant, and H n − ( e X ⊗ ¯ k, Q ℓ ) = H n − ( X ⊗ ¯ k, Q ℓ )(equality as g -modules) by Propositions 8.1(ii) and 8.7(i). Now apply Proposition9.8 to e X . (cid:3) Remark.
The preceding result can be interpreted in the category of motives over k (cf. [36], [M]): this is what we mean by a “description in purely algebraicterms” 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 abelianvarieties up to isogenies with the category of pure motives of weight −
1. Denoteby L = h ( P ) be the Lefschetz motive , which is pure of weight 2. Proposition 9.6implies that if X is normal, then h ( X ) = Pic s ( X ) ⊗ L . Now, denote by h i + ( X ) the part of h i ( X ) which is pure of weight i . Theorem 10.1means that if X satisfies ( RS2 ), then h n − ( 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 conjec-tural statement of Grothendieck [14, p. 343] in the case i = n : “ In odd dimensions, the piece of maximal filtration of H i − ( X, Z ℓ ( i )) is also thegreatest “abelian piece”, and corresponds to the Tate module of the intermediateJacobian J i ( X ) (defined by the cycles algebraically equivalent to 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 andSperber state Theorem 10.1 without any assumption about resolution of singular-ities, but they give a proof only if dim X ≤
2, in which case (
RS2 ) holds in anycharacteristic, by Abhyankar’s results.For any separated scheme X of finite type over k , the group g operates on ¯ X andthe Frobenius morphism of ¯ X is the automorphism corresponding to the geometricelement ϕ ∈ g . If A is an abelian variety defined over k , then ϕ is an endomorphismof A . It induces an endomorphism T ℓ ( ϕ ) of T ℓ ( A ), anddeg( n. 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 2 g with coefficients in Z ,called the characteristic polynomial of A . Moreover, if dim A = g , f c ( A, T ) = g Y j =1 ( T − α j ) , INGULAR VARIETIES OVER FINITE FIELDS 33 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(26) deg( ϕ ) = det T ℓ ( ϕ ) = g Y j =1 α j = q g . The trace of ϕ is the unique rational integer Tr( ϕ ) such that f c ( A, T ) ≡ T g − Tr( ϕ ) T g − (mod T g − ) . In order to state the next results, we need to introduce some conventions. If X isa separated scheme of finite type over k , we call M the following set of conditionsabout H ic ( ¯ X, Q ℓ ):– The action of the Frobenius morphism F in H ic ( ¯ X, Q ℓ ) is diagonalizable. – The space H ic ( ¯ X, 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 polynomialas f ∨ ( T ) = T d f ( T − ).10.5. Corollary.
Let X be a normal projective variety defined over k . Then: (i) We have P ( X, T ) = f ∨ c (Pic s X, T ) . (ii) If ϕ is the Frobenius endomorphism of Pic s X , then Tr( F | H ( ¯ X, Q ℓ )) = Tr( ϕ ) . (iii) Conditions M hold for H ( ¯ X, Q ℓ ) .Proof. Let A be an abelian variety of dimension g defined over k with f c ( A, T ) = g Y j =1 ( T − α j ) . Since the arithmetic Frobenius F is the inverse of ϕ ,det( T − V ℓ ( F ) | V ℓ ( A )) = g Y j =1 ( T − α − j ) . Since the map α qα − is a permutation of the characteristic roots, we havedet( T − V ℓ ( F ) | V ℓ ( A )( − g Y j =1 ( T − qα − j ) = f c ( A, T ) , which implies det(1 − T V ℓ ( F ) | V ℓ ( A )( − f ∨ c ( A, T ) . If we apply the preceding equality to A = Pic s X , we get (i) with the help ofProposition 9.6. Now (ii) is an immediate consequence of (i) by looking at thecoefficient of T g − . As stated above, the polynomial f c ( A, T ) belongs to Z [ T ], andits 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. (cid:3) Corollary. If X is a normal projective surface defined over k , then the poly-nomials P i ( X, T ) are independent of ℓ for ≤ i ≤ . INGULAR VARIETIES OVER FINITE FIELDS 34
Proof.
The polynomial P ( X, T ) = P +3 ( X, T ) is independent of ℓ by Proposition8.1(i). Moreover P ( X, T ) is independent of ℓ by Corollary 10.5. Hence we haveproved 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 ) = P ( X, T ) P ( X, T ) P ( X, T ) P ( X, T ) P ( X, T ) , and the zeta function Z ( X, T ) is independent of ℓ . (cid:3) The following result gives an explicit description of the birational invariants P +2 n − ( X, T ) , b +2 n − ( X ) , Tr( F | H n − ( ¯ X, Q ℓ ))in purely algebraic terms.10.7. Theorem.
Let X be a variety of dimension n ≥ defined over k . (i) If g = dim Alb w X , then P +2 n − ( X, T ) = q − g f c (Alb w X, q n T ) . In particular, b +2 n − ( X ) = 2 g . (ii) If ϕ is the Frobenius endomorphism of Alb w X , then Tr( F | H n − ( ¯ X, Q ℓ )) = q n − Tr( ϕ ) . Remark.
Serge Lang and Andr´e Weil have conjectured [27, p. 826-827] thatthe equality P +2 n − ( 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 caseof Theorem 10.7(i).Example 9.2 shows that we cannot replace Pic w X by Pic s X in the statementof 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 10.7hold true. Moreover (iii) Conditions M hold for H n − ( ¯ X, Q ℓ ) .Proof. Let A be an abelian variety of dimension g defined over k . Then(27) f c ( A, T ) = g Y j =1 ( T − α j ) = q g g Y j =1 (1 − α − j T ) , the last equality coming from (26). Nowdet( T − V ℓ ( F ) | V ℓ ( A )( − n )) = g Y j =1 ( T − q n α − j ) . Hence(28) det(1 − T V ℓ ( F ) | V ℓ ( A )( − n )) = g Y j =1 (1 − q n α − j T ) , and we get from (27)det(1 − T V ℓ ( F ) | V ℓ ( A )( − n )) = q − g f c ( A, q n T ) . INGULAR VARIETIES OVER FINITE FIELDS 35
Hence, (i) follows from Proposition 10.1. From (28) we deduce that β is an eigen-value of F in H n − ( ¯ X, Q ℓ ) if and only if q n /β is among the characteristic roots of ϕ . Now since α j α j = q , we get:Tr( F | H n − ( ¯ X, Q ℓ )) = g X j =0 q n α j = q n − g X j =0 α j = q n − Tr( ϕ ) . Finally (iii) follows from Theorem 10.1 as in the proof of Corollary 10.5. (cid:3)
We end this section by stating a weak form of the functional equation relatingthe polynomials P ( X, T ) and P +2 n − ( X, T ) when X is nonsingular.10.10. Corollary.
Let X be a normal projective variety of dimension n ≥ , definedover k . If g = dim Alb w X , then q − g P ∨ ( X, q n T ) divides P +2 n − ( X, T ) . Proof.
By Corollary 10.5(i) and Theorem 10.7(i), we know that P ∨ ( X, T ) = f c (Pic s X, T ) , P +2 n − ( 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 ). (cid:3) With Corollary 9.11, we prove in the same way :10.11.
Corollary.
Assume that ( R n , p ) holds. Let X be a projective variety ofdimension n defined over k , regular in codimension one, which is a local completeintersection. If g = dim Alb w X , then q − g P ∨ ( X, q n T ) = P n − ( X, T ) . On the Lang-Weil Inequality
In this section, let k = F q . 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 Nk , of dimension n and of degree d , defined over k . Then (cid:12)(cid:12)(cid:12) | X ( k ) | − π n (cid:12)(cid:12)(cid:12) ≤ ( d − d − q n − (1 / + C + ( X ) q n − , where C + ( X ) depends only on ¯ X , and C + ( X ) ≤ × m × ( mδ + 3) N +1 , if X is of type ( m, N, d ) , with d = ( d , . . . , d m ) and δ = max( d , . . . , d m ) . Inparticular, 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 Nk of dimension n defined over k . Then (cid:12)(cid:12)(cid:12) | X ( k ) | − π n − Tr( F | H n − ( ¯ X, Q ℓ )) (cid:12)(cid:12)(cid:12) ≤ C + ( X ) q n − , where C + ( X ) is as in Theorem 11.1.Proof. Denote by H n − − ( ¯ X ) the subspace of H n − ( ¯ X ) corresponding to eigen-values of the Frobenius endomorphism of weight strictly smaller than 2 n − C + ( X ) = dim H n − − ( ¯ X ) + n − X i =0 b i,l ( ¯ X ) + ε i . INGULAR VARIETIES OVER FINITE FIELDS 36
Clearly, C + ( X ) depends only on ¯ X , and by Proposition 5.1, C + ( X ) ≤ τ ( X ) ≤ τ k ( m, N, d ) ≤ × m × ( mδ + 3) N +1 , with the notations introduced therein. Then the Trace Formula (17) and Deligne’sMain Theorem (18) imply the required result. (cid:3) Proof of Theorem 11.1.
By Theorem 10.7(i) and Proposition 9.4(iii), b +2 n − ( X ) = 2 dim Alb w X ≤ g ( e Y ) ≤ ( d − d − , if Y is a suitably chosen linear section of X of dimension 1, and if e Y is a nonsin-gular curve birationally equivalent to Y . Since all the eigenvalues of the Frobeniusautomorphism in H n − ( ¯ X, Q ℓ ) are pure of weight 2 n − (cid:12)(cid:12)(cid:12) | X ( k ) | − π n (cid:12)(cid:12)(cid:12) ≤ g ( e Y ) q n − (1 / + C + ( X ) q n − , which is better than the desired inequality. (cid:3) Remark.
In exactly the same way, applying [24, Th. 1], one establishes thatif X is a closed algebraic subvariety in A Nk , of dimension n , of degree d and of type( m, N, d ), defined over k , then (cid:12)(cid:12)(cid:12) | X ( k ) | − q n (cid:12)(cid:12)(cid:12) ≤ ( d − d − q n − (1 / + C + ( X ) q n − , where C + ( X ) depends only on ¯ X and is not greater than 6 × m × ( mδ + 3) N +1 .11.4. Remark.
As a consequence of Remark 11.3, we easily obtain the followingversion of a lower bound due to W. Schmidt [33] for the number of points of affinehypersurfaces. If f ∈ k [ T , . . . , T N ] is an absolutely irreducible polynomial of degree d , and if X is the hypersurface in A Nk with equation f ( T , . . . , T N ) = 0, then | X ( k ) | ≥ q N − − ( d − d − q N − (3 / − d + 3) N +1 q N − . It may be noted that with the help of Schmidt’s bound, one is able to replace12( d + 3) N +1 by a much better constant, namely 6 d , but his bound is only validfor large values of q .Proposition 11.2 gives the second term in the asymptotic expansion of | X ( F q s ) | when s is large. From Theorem 10.7, we deduce immediately the following preciseinequality, which involves only purely algebraic terms.11.5. Corollary.
Let X be a projective variety of dimension n ≥ defined over k ,and let ϕ be the Frobenius endomorphism of Alb w X . Then (cid:12)(cid:12)(cid:12) | X ( k ) | − π n + q n − Tr( ϕ ) (cid:12)(cid:12)(cid:12) ≤ C + ( X ) q n − , where C + ( X ) is as in Theorem 11.1. (cid:3) Corollary.
With hypotheses as above, the following are equivalent: (i)
There is a constant C such that, for every s ≥ , (cid:12)(cid:12)(cid:12) | X ( F q s ) | − q ns (cid:12)(cid:12)(cid:12) ≤ Cq s ( n − . (ii) The Albanese-Weil variety of X is trivial.Proof. By Lemma 8.3 we know that (i) holds if and only if H n − ( ¯ X ) = 0, and thislast condition is equivalent to (ii) by Theorem 10.7. (cid:3) INGULAR VARIETIES OVER FINITE FIELDS 37
For instance, rational varieties, and, by Remark 6.3 or 9.9, complete intersectionsregular in codimension 2 satisfy the conditions of the preceding Lemma.It remains to prove Theorem 10.7. For that purpose, we need the followingnon-effective estimate [5, p. 333].11.7.
Lemma (Bombieri and Sperber) . Let X be a projective variety of dimension n ≥ defined over k , and let ϕ as above. Then | X ( k ) | = π n − q n − Tr( ϕ ) + O ( q n − ) . Proof.
By induction on n = dim X . If dim X = 2, then ( RS2 ) is satisfied. Hence,the conclusions of Theorem 10.7 hold true by Proposition 10.9, and (cid:12)(cid:12)(cid:12) | X ( k ) | − π + q Tr( ϕ ) (cid:12)(cid:12)(cid:12) ≤ C + ( X ) q, by Proposition 11.2 and Theorem 10.7(ii). Suppose that n ≥ n −
1. We can assume that X isa projective algebraic subvariety in P Nk not contained in any hyperplane. In viewof Lemma 1.2, if q is sufficiently large, there is a linear subvariety E of P Nk , ofcodimension 2, defined over k , such that dim X ∩ E = n −
2. For any u ∈ P k , thereciprocal image of u by the projection π : P Nk − E −→ P k is a hyperplane H u containing E , and Y u = X ∩ H u is a projective algebraic set ofdimension ≤ n − ≤ deg X . Now we have X ( k ) = [ u ∈ P ( k ) Y u ( k ) , hence | X ( k ) | = X u ∈ P ( k ) | Y u ( k ) | + O ( q n − ) ;in fact the error term is bounded by ( q + 1) | X ( k ) ∩ E ( k ) | , and can be estimated byProposition 12.1 below. Let S be the set of u ∈ P k such that Y u is not a subvarietyof 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 X u ∈ S | Y u ( k ) | = O ( q n − ) , hence, | X ( k ) | = X u/ ∈ S | Y u ( k ) | + O ( q n − ) . Let ϕ u be the Frobenius endomorphism of Alb w Y u . By the induction hypothesis, | Y u ( k ) | = π n − − q n − Tr( ϕ u ) + O ( q n − ) . But if u / ∈ S then Tr( ϕ u ) = Tr( ϕ ) by hypothesis. The two preceding relations thenimply | X ( k ) | = ( q + 1 − | S | )( π n − − q n − Tr( ϕ )) + O ( q n − ) , and the lemma is proved. (cid:3) INGULAR VARIETIES OVER FINITE FIELDS 38
Proof of Theorem 10.7.
From Proposition 11.2 and from Lemma 11.7 we getTr( F | H n − ( ¯ X, Q ℓ )) − q n − Tr( ϕ ) = O ( q n − )But the eigenvalues of F in H n − ( ¯ X, Q ℓ ) and the numbers q n − α j , where ( α j )is the family of characteristic roots of Alb w X , are pure of weight 2 n −
1. FromLemma 8.2 we deduce that these two families are identical, andTr( F | H n − ( ¯ X, Q ℓ )) = q n − Tr( ϕ ) . This yields the desired result. (cid:3)
One can also improve the Lang-Weil inequality when the varieties are of smallcodimension:11.8.
Corollary.
Let X be a projective subvariety in P Nk , and assume dim Sing X ≤ s, codim X ≤ dim X − s − . If dim X = n , then (cid:12)(cid:12)(cid:12) | X ( k ) | − π n (cid:12)(cid:12)(cid:12) ≤ C N + s ( ¯ X ) q ( N + s ) / , where C N + s ( ¯ X ) is as in Theorem 6.1.Proof. From Barth’s Theorem [15, Thm. 6.1, p. 146] and Theorem 2.4, we deducethat (12) holds if i ≥ N + s + 1. Now apply the Trace Formula (17) and Deligne’sMain Theorem (18). (cid:3) Number of Points of Algebraic Sets
In most applications, it is useful to have at one’s disposal some bounds on generalalgebraic sets. If X ⊂ P N is a projective algebraic set defined over a field k wedefine the dimension (resp. the degree ) of X as the maximum (resp. the sum) ofthe dimensions (resp. of the degrees) of the k -irreducible components of X . Fromnow on, let k = F q . The following statement is a quantitative version of Lemma 1of 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 di-mension 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 generatedby X is equal to P N . If n = 0 then | X ( k ) | ≤ d . Assume now that n ≥
1. We firstprove the desired inequality when X is k -irreducible. Let E be the linear subvarietyin 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 ւ ց p X ( k ) ( P m ) ∗ ( k )If x ∈ X ( k ) then p − ( x ) is in bijection with the set of hyperplanes H ∈ ( P m ) ∗ ( k )with x ∈ H ( k ); hence | p − ( x ) | = π m − and(29) | T ( k ) | = π m − | X ( k ) | . INGULAR VARIETIES OVER FINITE FIELDS 39
On the other hand, if H ∈ ( P m ) ∗ ( k ), then p − ( H ) is in bijection with to X ( k ) ∩ H ( k ), hence(30) | T ( k ) | = X H | X ( k ) ∩ H ( k ) | , where H runs over the whole of ( P m ) ∗ ( k ). Since X is a nondegenerate k -irreduciblesubset 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 B´ezout’s Theorem. Thus by the induction hypothesis, | X ( k ) ∩ H ( k ) | ≤ dπ n − , and from (29) and (30) we deduce | X ( k ) | ≤ π m π m − dπ n − . Now it is easy to check that if m ≥ n , then q ≤ π m π m − ≤ π n π n − ≤ q + 1 , so the desired inequality is proved when X is irreducible of dimension n . In thegeneral case, let X = Y ∪ · · · ∪ Y s be the irredundant decomposition of X in k -irreducible components, in such a waythat dim Y i ≤ n, d + · · · + d s = d, (deg Y i = d i ) . Then | X ( k ) | ≤ | Y ( k ) | + · · · + | Y s ( k ) | ≤ ( d + · · · + d s ) π n = dπ n . (cid:3) We take this opportunity to report the following conjecture on the number ofpoints of complete intersections of small codimension.12.2.
Conjecture (Lachaud) . If X ⊂ P Nk is a projective algebraic set defined over k of dimension n ≥ N/ and of degree d ≤ q + 1 which is a complete intersection,then | X ( k ) | ≤ dπ n − ( d − π n − N = d ( π n − π n − N ) + π n − N . 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 [37]: if X is an hypersurface of dimension n andof degree d ≤ q + 1, then | X ( k ) | ≤ dq n + π n − . Now assume that X is the union of d linear varieties G , . . . , G d of dimension n ≥ N/
2. We prove (ii) by induction on d . Write G i ( k ) = G i (1 ≤ i ≤ d ) forbrevity. If d = 1 then | G | = π n = ( π n − π N − n ) + π N − n , and the assertion is true. Now if G and G are two linear varieties of dimension n , then dim G ∩ G ≥ n − N . Hence for d > | G d ∩ ( G ∪ · · · ∪ G d − ) | ≥ π n − N . Now note that | G ∪ · · · ∪ G d | = | G ∪ · · · ∪ G d − | + | G d | − | G d ∩ ( G ∪ · · · ∪ G d − ) | . INGULAR VARIETIES OVER FINITE FIELDS 40
If we apply the induction hypothesis we get | G ∪ · · · ∪ G d | ≤ ( d − π n − π n − N ) + π n − N + π n − π n − N = d ( π n − π n − N ) + π n − N , which proves the desired inequality. Acknowledgements
This research was partly supported by the Indo-French Mathematical ResearchProgram of the Centre National de la Recherche Scientifique (CNRS) of France andthe National Board for Higher Mathematics (NBHM) of India, and we thank theseorganisations for their support. We would also like to express our warm gratitudeto Jean-Pierre Serre for his interest, and to Alexe¨ı Skorobogatov for his valuablesuggestions. Thanks are also due to the referees for pointing out some correctionsin earlier versions of this paper and making suggestions for improvements.
References [1] Abhyankar, S.S.,
Resolution of Singularities of Embedded Algebraic Surfaces , 2nd enlargeded., Springer-Verlag, New York, 1998.[2] Adolphson, A., Sperber, S.,
On the degree of the L -function associated with an exponentialsum , Compositio Math. (1988), 125-159.[3] Adolphson, A., Sperber, S., On the degree of the zeta Function of a complete intersection ,Contemporary Math. (1999), 165-179.[4] Aubry, Y., Perret, M.,
A Weil Theorem for singular curves , in “Arithmetic, Geometry, andCoding Theory”, Marseille-Luminy, 1993, de Gruyter, Berlin, 1996, 1-7.[5] Bombieri, E., Sperber, S.,
On the estimation of certain exponential sums , Acta Arith. (1995), 329-358.[6] Bosch, S., Lutkebohmert, W., Raynaud, M., N´eron Models , Ergeb. Math. Grenzgeb. (3) Bd.21, Springer-Verlag, Berlin, 1990.[7] Bourbaki, N.,
Alg`ebre Commutative, chapitres 8 et 9 , Masson, Paris, 1983.[8] Deligne, P.,
R´esum´e des premiers expos´es de A. Grothendieck , Exp. I in [SGA 7.1], 1-24.[9] Deligne, P.,
La conjecture de Weil I , Publ. Math. I.H.E.S. (1974), 273-308.[10] Deligne, P., La conjecture de Weil II , Publ. Math. I.H.E.S. (1981), 313-428.[11] Ghorpade, S.R., Lachaud, G., Number of solutions of equations over finite fields and aconjecture of Lang and Weil , in “Number Theory and Discrete Mathematics” (Chandigarh,2000), Hindustan Book Agency, New Delhi, & Birkh¨auser, Basel, 2002, 269–291.[12] Grothendieck, A.,
Technique de descente et th´eor`emes d’existence en g´eom´etrie alg´ebrique,VI. Les sch´emas de Picard , S´em. Bourbaki, 1961/62, Soci´et´e Math´ematique de France, Paris,1995 : (a)
Th´eor`emes d’existence , n ◦ Propri´et´es g´en´erales , n ◦ Profondeur et th´eor`emes de Lefschetz en cohomologie ´etale ,Exp. XIV in [SGA 2], 203-284.[14] Grothendieck, A.,
Lettre `a J.-P. Serre,16 Aoˆut 1964 , annexe `a [36]; = CorrespondanceGrothendieck-Serre, Soc. Math. France, Paris, 2001, 172-175.[15] Hartshorne, R.,
Equivalence relations on algebraic cycles and subvarieties of small codimen-sion , in “Algebraic Geometry”, Arcata 1974, Proc. Symp. Pure Math. , Amer. Math. Soc.,Providence, 1975, 129-164.[16] Hartshorne, R., Algebraic Geometry , Grad. Texts in Math. vol. 52, Springer-Verlag, Berlin,1977.[17] Hirzebruch, F.,
Der Satz von Riemann-Roch in Faisceau-theoretischer Formulierung: einigeAnwendingen und offene Fragen , Proc. Int. Cong. Math., 1954, Amsterdam, North-Holland,Amsterdam, 1956, vol. III, 457-473; = Coll. papers, Springer-Verlag, Berlin, 1987, vol. I, n ◦
12, 128-144.[18] Hooley, C.,
On the number of points on a complete intersection over a finite field , J. NumberTheory (1991), 338-358.[19] Jouanolou, J.-P., Th´eor`emes de Bertini et applications , Progress in Math. vol. 42, Birkh¨auser,Boston, 1983.[20] Jouanolou, J.-P.,
Cohomologie de quelques sch´emas classiques et th´eorie cohomologique desclasses de Chern , Exp. VII in [SGA 5], 282-350.[21] Katz, N.M.,
Number of points of singular complete intersections , Appendix to [18], 355-358.[22] Katz, N.M.,
Review of ℓ -adic Cohomology , in [M], 21-30. INGULAR VARIETIES OVER FINITE FIELDS 41 [23] Katz, N.M.,
Estimates for “singular” exponential sums , International Mathematics ResearchNotices, 1999, No. 16, 875-899.[24] Katz, N.M.,
Sums of Betti numbers in arbitrary characteristic , Finite Fields and their App. (2001), 29-44.[25] Katz, N.M., Lang, S., Finiteness theorems in geometric classfield theory , Enseign. Math. (1981), 285-319; = Lang, S., Collected Papers, vol. III, [1981a], 101-135, Springer-Verlag,New York, 2000.[26] Lachaud, G., Number of points of plane sections and linear codes defined on algebraic vari-eties , in “Arithmetic, Geometry and Coding Theory”, Marseille-Luminy 1993, De Gruyter,Berlin 1996, 77-104.[27] Lang, S., Weil, A.,
Number of points of varieties over finite fields , Amer. Journal of Math. (1954), 819-827; = Weil, A., Œuvres Scientifiques, vol. II, [1954f], 165-173, Springer-Verlag,New York, 1979; = Lang, S., Collected Papers, vol. I, [1954b], 54-62, Springer-Verlag, NewYork, 2000.[28] Lang, S., Abelian Varieties , Interscience Publishers, New York, 1959; reed., Springer-Verlag,Berlin, 1983.[29] Lipman, S.,
Introduction to resolution of singularities , in “Algebraic Geometry”, Arcata1974, Proc. Symp. Pure Math. , Amer. Math. Soc., Providence, 1975, 187-230.[30] Milne, J.S., ´Etale Cohomology , Princeton Math. Series vol. 33, Princeton University Press,Princeton, 1980.[31] Mumford, D., Abelian Varieties , 2d ed., Oxford Univ. Press, Bombay, 1974.[32] Northcott, D.G.,
Lessons on Rings, Modules and Multiplicities , Cambridge University Press,Cambridge, 1968.[33] Schmidt, W.M.,
A lower bound for the number of solutions of equations over finite fields , J.Number Theory (1974), 448-480.[34] Serre, J.-P., Groupes alg´ebriques et corps de classes , Publications de l’Institut de Math´e-matique de l’Universit´e de Nancago, No. VII. Hermann, Paris 1959.[35] Serre, J.-P.,
Morphismes universels et vari´et´es d’Albanese , in “Vari´et´es de Picard”, S´eminaireC. Chevalley (1958/59), n ◦
10, 22 pp ; = Expos´es de s´eminaires 1950-1999, Soc. Math.France, Paris, 2001, 141-160.[36] Serre, J.-P.,
Motifs , in “Journ´ees Arithm´etiques de Luminy (1989)”, Ast´erisque (1991), 333-349; = Œuvres, Vol. IV, n ◦ Lettre `a M. Tsfasman , in “Journ´ees Arithm´etiques de Luminy (1989)”,Ast´erisque (1991), 351-353; = Œuvres, Vol. IV, n ◦ La vari´et´e de Picard d’une vari´et´e compl`ete , in “Vari´et´es de Picard”,S´eminaire C. Chevalley (1958/59), n ◦
8, 25 pp.[39] Shparlinski˘ı, I.E., Skorobogatov, A.N.,
Exponential sums and rational points on completeintersections , Mathematika (1990), 201-208.[40] Skorobogatov, A.N. Exponential sums, the geometry of hyperplane sections, and some dio-phantine problems , Israel J. Math (1992), 359-379.[41] Weil, A., (a) Sur les Courbes Alg´ebriques et les Vari´et´es qui s’en d´eduisent , Act. Sci. Ind.vol. 1041, Hermann, Paris, 1948; (b)
Vari´et´es Ab´eliennes et Courbes Alg´ebriques , Act. Sci.Ind. vol. 1064, ibid. [42] Weil, A.,
Number of solutions of equations in finite fields , Bull. Amer. Math. Soc. (1949),497-508 ; = Œuvres Scient., Springer-Verlag, [1949b], Vol. I, 399-410.[43] Weil, A., Sur les crit`eres d’´equivalence en g´eom´etrie alg´ebrique , Math. Ann., (1954),95-127 ; = Œuvres Scient., Springer-Verlag, [1954d], Vol. II, 127-159.
Abbreviations [EGA 4.2] Grothendieck, A., Dieudonn´e, J., ´El´ements de G´eom´etrie Alg´ebrique , Ch. IV, Sec-onde Partie, ´Etude locale des sch´emas et des morphismes de sch´emas , Publ. Math.I.H.E.S. (1965).[SGA 2] Cohomologie Locale des Faisceaux Coh´erents et Th´eor`emes de Lefschetz Locaux etGlobaux , par A. Grothendieck, North-Holland, Amsterdam, 1968.[SGA 5]
Cohomologie ℓ -adique et Fonctions L , par A. Grothendieck, Lect. Notes in Math. , Springer-Verlag, Berlin, 1977.[SGA 7.1] Groupes de Monodromie en G´eom´etrie Alg´ebrique , Vol. 1, dirig´e par A. Grothen-dieck, Lect. Notes in Math. , Springer-Verlag, Berlin, 1972.[M] Jannsen, U., Kleiman, S., Serre, J.-P.,
Motives , Seattle 1991, Proc. Symp. PureMath. , Amer. Math. Soc., Providence, 1994. INGULAR VARIETIES OVER FINITE FIELDS 42
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