M ath. Res. Lett. 15 2008), no. 2, 265 271 c International Press 2008 SMOOTH HYPERSURFACE SECTIONS CONTAINING A GIEN SUBSCHEME OER A FINITE FIELD Bjorn Poonen 1. Introduction Let F q be a finite field of q = p a elements. Let X be a smooth quasi-projective subscheme of P n of dimension m 0 over F q. N. Katz asked for a finite field analogue of the Bertini smoothness theorem, and in particular asked whether one could always find a hypersurface H in P n such that H X is smooth of dimension m 1. A positive answer was proved in [Gab01] and [Poo04] independently. The latter paper proved also that in a precise sense, a positive fraction of hypersurfaces have the required property. The classical Bertini theorem was extended in [Blo70, KA79] to show that the hypersurface can be chosen so as to contain a prescribed closed smooth subscheme Z, provided that the condition dim X > 2 dim Z is satisfied. The condition arises naturally from a dimension-counting argument.) The goal of the current paper is to prove an analogous result over finite fields. In fact, our result is stronger than that of [KA79] in that we do not require Z X, but weaker in that we assume that Z X be smooth. With a little more work and complexity, we could prove a version for a non-smooth intersection as well, but we restrict to the smooth case for simplicity.) One reason for proving our result is that it is used by [SS07]. Let S = F q [x 0,..., x n ] be the homogeneous coordinate ring of P n. Let S d S be the F q -subspace of homogeneous polynomials of degree d. For each f S d, let H f be the subscheme ProjS/f)) P n. For the rest of this paper, we fix a closed subscheme Z P n. For d Z 0, let I d be the F q -subspace of f S d that vanish on Z. Let I homog = d 0 I d. We want to measure the density of subsets of I homog, but under the definition in [Poo04], the set I homog itself has density 0 whenever dim Z > 0; therefore we use a new definition of density, relative to I homog. Namely, we define the density of a subset P I homog by #P I d ) µ Z P) := lim, d #I d For a scheme X of finite type over F q, define the zeta func- if the limit exists. tion [Wei49] ζ X s) = Z X q s ) := closed P X 1 q s deg P ) 1 = exp r=1 ) #XF q r) q rs ; r Received by the editors March 8, 2007. 1991 Mathematics Subject Classification. Primary 14J70; Secondary 11M38, 11M41, 14G40, 14N05. This research was supported by NSF grant DMS-0301280. 265
266 BJORN POONEN the product and sum converge when Res) > dim X. Theorem 1.1. Let X be a smooth quasi-projective subscheme of P n of dimension m 0 over F q. Let Z be a closed subscheme of P n. Assume that the scheme-theoretic intersection := Z X is smooth of dimension l. If is empty, take l = 1.) Define P := { f I homog : H f X is smooth of dimension m 1 }. i) If m > 2l, then µ Z P) = ζ m + 1) ζ m l) ζ X m + 1) = 1 ζ m l) ζ X m + 1). In this case, in particular, for d 1, there exists a degree-d hypersurface H containing Z such that H X is smooth of dimension m 1. ii) If m 2l, then µ Z P) = 0. The proof will use the closed point sieve introduced in [Poo04]. In fact, the proof is parallel to the one in that paper, but changes are required in almost every line. 2. Singular points of low degree Let I Z O P n be the ideal sheaf of Z, so I d = H 0 P n, I Z d)). Tensoring the surjection O n+1) O f 0,..., f n ) x 0 f 0 + + x n f n with I Z, twisting by Od), and taking global sections shows that S 1 I d = I d+1 for d 1. Fix c such that S 1 I d = I d+1 for all d c. Before proving the main result of this section Lemma 2.3), we need two lemmas. Lemma 2.1. Let Y be a finite subscheme of P n. Let φ d : I d = H 0 P n, I Z d)) H 0 Y, I Z O Y d)) be the map induced by the map of sheaves I Z I Z O Y on P n. Then φ d is surjective for d c + dim H 0 Y, O Y ), Proof. The map of sheaves O P n O Y on P n is surjective so I Z I Z O Y is surjective too. Thus φ d is surjective for d 1. Enlarging F q if necessary, we can perform a linear change of variable to assume Y A n := {x 0 0}. Dehomogenization setting x 0 = 1) identifies S d with the space S d of polynomials in F q[x 1,..., x n ] of total degree d. and identifies φ d with a map I d B := H 0 P n, I Z O Y ). By definition of c, we have S 1I d = I d+1 for d c. For d b, let B d be the image of I d in B, so S 1B d = B d+1 for d c. Since 1 S 1, we have I d I d+1, so B c B c+1. But b := dim B <, so B j = B j+1 for some j [c, c + b]. Then B j+2 = S 1B j+1 = S 1B j = B j+1. Similarly B j = B j+1 = B j+2 =..., and these eventually equal B by the previous paragraph. Hence φ d is surjective for d j, and in particular for d c + b.
SMOOTH HYPERSURFACE SECTIONS 267 Lemma 2.2. Suppose m O X is the ideal sheaf of a closed point P X. Let Y X be the closed subscheme whose ideal sheaf is m 2 O X. Then for any d Z 0. { #H 0 q m l) deg P, if P, Y, I Z O Y d)) = q ) deg P, if P /. Proof. Since Y is finite, we may now ignore the twisting by Od). The space H 0 Y, O Y ) has a two-step filtration whose quotients have dimensions 1 and m over the residue field κ of P. Thus #H 0 Y, O Y ) = #κ) = q ) deg P. If P or equivalently P Z), then H 0 Y, O Z Y ) has a filtration whose quotients have dimensions 1 and l over κ; if P /, then H 0 Y, O Z Y ) = 0. Taking cohomology of on the 0-dimensional scheme Y yields 0 I Z O Y O Y O Z Y 0 #H 0 Y, I Z O Y ) = #H0 Y, O Y ) #H 0 Y, O Z Y ) { q ) deg P /q l+1) deg P, if P, = q ) deg P, if P /. If U is a scheme of finite type over F q, let U <r be the set of closed points of U of degree < r. Similarly define U >r. Lemma 2.3 Singularities of low degree). Let notation and hypotheses be as in Theorem 1.1, and define P r := { f I homog : H f X is smooth of dimension m 1 at all P X <r }. Then µ Z P r ) = 1 q m l) deg P ) 1 q ) deg P ). P <r P X ) <r Proof. Let X <r = {P 1,..., P s }. Let m i be the ideal sheaf of P i on X. let Y i be the closed subscheme of X with ideal sheaf m 2 i O X, and let Y = Y i. Then H f X is singular at P i more precisely, not smooth of dimension m 1 at P i ) if and only if the restriction of f to a section of O Yi d) is zero. By Lemma 2.1, µ Z P) equals the fraction of elements in H 0 I Z O Y d)) whose restriction to a section of O Yi d) is nonzero for every i. Thus s #H 0 Y i, I Z O Yi ) 1 µ Z P r ) = i=1 = #H 0 Y i, I Z O Yi ) ) 1 q m l) deg P 1 q ) deg P ), P <r P X ) <r by Lemma 2.2.
268 BJORN POONEN Corollary 2.4. If m > 2l, then lim µ ζ m + 1) ZP r ) = r ζ X m + 1) ζ m l). Proof. The products in Lemma 2.3 are the partial products in the definition of the zeta functions. For convergence, we need m l > dim = l, which is equivalent to m > 2l. Proof of Theorem 1.1ii). We have P P r. By Lemma 2.3, µ Z P r ) ) 1 q m l) deg P, P <r which tends to 0 as r if m 2l. Thus µ Z P) = 0 in this case. From now on, we assume m > 2l. 3. Singular points of medium degree Lemma 3.1. Let P X is a closed point of degree e, where e d c. Then the fraction of f I d such that H f X is not smooth of dimension m 1 at P equals { q m l)e, if P, q )e, if P /. Proof. This follows by applying Lemma 2.1 to the Y in Lemma 2.2, and then applying Lemma 2.2. Define the upper and lower densities µ Z P), µ Z P) of a subset P I homog as µ Z P) was defined, but using lim sup and lim inf in place of lim. Lemma 3.2 Singularities of medium degree). Define Q medium r := { f I d : there exists P X with r deg P d b m + 1 d 0 such that H f X is not smooth of dimension m 1 at P }. Then lim r µ Z Q medium r ) = 0. Proof. By Lemma 3.1, we have #Q medium r I d ) #I d P Z r deg P d b q m l) deg P + P Z r q m l) deg P + P X Z r deg P d b P X Z) r q ) deg P. ) deg P q Using the trivial bound that an m-dimensional variety has at most Oq em ) closed points of degree e, as in the proof of [Poo04, Lemma 2.4], we show that each of the two sums converges to a value that is Oq r ) as r, under our assumption m > 2l.
SMOOTH HYPERSURFACE SECTIONS 269 4. Singular points of high degree Lemma 4.1. Let P be a closed point of degree e in P n Z. For d c, the fraction of f I d that vanish at P is at most q mind c,e). Proof. Equivalently, we must show that the image of φ d in Lemma 2.1 for Y = P has F q -dimension at least mind c, e). The proof of Lemma 2.1 shows that as d runs through the integers c, c + 1,..., this dimension increases by at least 1 until it reaches its maximum, which is e. Lemma 4.2 Singularities of high degree off ). Define Q high X := d 0{ f I d : P X ) > d c Then µ Z Q high X ) = 0. such that H f X is not smooth of dimension m 1 at P } Proof. It suffices to prove the lemma with X replaced by each of the sets in an open covering of X, so we may assume X is contained in A n = {x 0 0} P n, and that =. Dehomogenize by setting x 0 = 1, to identify I d S d with subspaces of I d S d A := F q[x 1,..., x n ]. Given a closed point x X, choose a system of local parameters t 1,..., t n A at x on A n such that t = t m+2 = = t n = 0 defines X locally at x. Multiplying all the t i by an element of A vanishing on Z but nonvanishing at x, we may assume in addition that all the t i vanish on Z. Now dt 1,..., dt n are a O An,x-basis for the stalk Ω 1 A n /F. Let q,x 1,..., n be the dual basis of the stalk T An /F q,x of the tangent sheaf. Choose s A with sx) 0 to clear denominators so that D i := s i gives a global derivation A A for i = 1,..., n. Then there is a neighborhood N x of x in A n such that N x {t = t m+2 = = t n = 0} = N x X, Ω 1 N x/f q = n i=1 O N x dt i, and s ON u ). We may cover X with finitely many N x, so we may reduce to the case where X N x for a single x. For f I d I d, H f X fails to be smooth of dimension m 1 at a point P U if and only if fp ) = D 1 f)p ) = = D m f)p ) = 0. Let τ = max i deg t i ), γ = d τ)/p, and η = d/p. If f 0 I d, g 1 S γ,..., g m S γ, and h I η are selected uniformly and independently at random, then the distribution of f := f 0 + g p 1 t 1 + + gmt p m + h p is uniform over I d, because of f 0. We will bound the probability that an f constructed in this way has a point P X > d c where fp ) = D 1 f)p ) = = D m f)p ) = 0. We have D i f = D i f 0 ) + g p i s for i = 1,..., m. We will select f 0, g 1,..., g m, h one at a time. For 0 i m, define W i := X {D 1 f = = D i f = 0}. Claim 1: For 0 i m 1, conditioned on a choice of f 0, g 1,..., g i for which dimw i ) m i, the probability that dimw i+1 ) m i 1 is 1 o1) as d. The function of d represented by the o1) depends on X and the D i.) Proof of Claim 1: This is completely analogous to the corresponding proof in [Poo04].
270 BJORN POONEN Claim 2: Conditioned on a choice of f 0, g 1,..., g m for which W m is finite, ProbH f W m X > d c = ) = 1 o1) as d. Proof of Claim 2: By Bézout s theorem as in [Ful84, p. 10], we have #W m = Od m ). For a given point P W m, the set H bad of h I η for which H f passes through P is either or a coset of kerev P : I η κp )), where κp ) is the residue field of P, and ev P is the evaluation-at-p map. If moreover deg P > d c, then Lemma 4.1 implies ) #H bad /#I η q ν where ν = min η, d c. Hence ProbH f W m X > d c ) #W m q ν = Od m q ν ) = o1) as d, since ν eventually grows linearly in d. This proves Claim 2. End of proof: Choose f I d uniformly at random. Claims 1 and 2 show that with probability m 1 i=0 1 o1)) 1 o1)) = 1 o1) as d, dim W i = m i for i = 0, 1,..., m and H f W m X > d c =. But H f W m is the subvariety of X cut out by the equations fp ) = D 1 f)p ) = = D m f)p ) = 0, so H f W m X > d c is exactly the set of points of H f X of degree > d c where H f X is not smooth of dimension m 1. Thus µ Z Q high X ) = 0. Lemma 4.3 Singularities of high degree on ). Define Q high := d 0{ f I d : P > d c Then µ Z Q high ) = 0. such that H f X is not smooth of dimension m 1 at P }. Proof. As before, we may assume X A n and we may dehomogenize. Given a closed point x X, choose a system of local parameters t 1,..., t n A at x on A n such that t = t m+2 = = t n = 0 defines X locally at x, and t 1 = t 2 = = t m l = t = t m+2 = = t n = 0 defines locally at x. If m w is the ideal sheaf of w on P n, then I Z mw m 2 w is surjective, so we may adjust t 1,..., t m l to assume that they vanish not only on but also on Z. Define i and D i as in the proof of Lemma 4.2. Then there is a neighborhood N x of x in A n such that N x {t = t m+2 = = t n = 0} = N x X, Ω 1 N x/f q = n i=1 O N x dt i, and s ON u ). Again we may assume X N x for a single x. For f I d I d, H f X fails to be smooth of dimension m 1 at a point P if and only if fp ) = D 1 f)p ) = = D m f)p ) = 0. Again let τ = max i deg t i ), γ = d τ)/p, and η = d/p. If f 0 I d, g 1 S γ,..., g l+1 S γ, are chosen uniformly at random, then f := f 0 + g p 1 t 1 + + g p l+1 t l+1 is a random element of I d, since l + 1 m l. For i = 0,..., l + 1, the subscheme W i := {D 1 f = = D i f = 0}
SMOOTH HYPERSURFACE SECTIONS 271 depends only on the choices of f 0, g 1,..., g i. The same argument as in the previous proof shows that for i = 0,..., l, we have Probdim W i l i) = 1 o1) as d. In particular, W l is finite with probability 1 o1). To prove that µ Z Q high ) = 0, it remains to prove that conditioned on choices of f 0, g 1,..., g l making dim W l finite, ProbW l+1 > d c = ) = 1 o1). By Bézout s theorem, #W l = Od l ). The set H bad of choices of g l+1 making D l+1 f vanish at a given point P W l is either empty or a coset of kerev P : S γ κp )). Lemma 2.5 of [Poo04] implies that the size of this kernel or its coset) as a fraction of #S γ is at most q ν where ν := min are done. Proof of Theorem 1.1i). We have γ, d c 5. Conclusion P P r P Q medium r ). Since #W l q ν = o1) as d, we Q high X Qhigh, so µ Z P) and µ Z P) each differ from µ Z P r ) by at most µ Z Q medium r )+µ Z Q high µ Z Q high ). Applying Corollary 2.4 and Lemmas 3.2, 4.2, and 4.3, we obtain µ Z P) = lim µ ζ m + 1) ZP r ) = r ζ m l) ζ X m + 1). Acknowledgements X )+ I thank Shuji Saito for asking the question answered by this paper, and for pointing out [KA79]. References [Blo70] S. Bloch, 1970. Ph.D. thesis, Columbia University. [Ful84] W. Fulton, Introduction to intersection theory in algebraic geometry, CBMS Regional Conference Series in Mathematics, vol. 54, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1984.MR735435 85j:14008) [Gab01] O. Gabber, On space filling curves and Albanese varieties, Geom. Funct. Anal. 11 2001), no. 6, 1192 1200.MR1878318 2003g:14034) [KA79] S. L. Kleiman and Allen B. Altman, Bertini theorems for hypersurface sections containing a subscheme, Comm. Algebra 7 1979), no. 8, 775 790.MR529493 81i:14007) [Poo04] B. Poonen, Bertini theorems over finite fields, Ann. of Math. 2) 160 2004), no. 3, 1099 1127. MR 2144974 2006a:14035) [SS07] S. Saito and Kanetomo Sato, Finiteness theorem on zero-cycles over p-adic fields April 11, 2007). arxiv:math.ag/0605165. [Wei49] A. Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 1949), 497 508.MR0029393 10,592e) Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA E-mail address: poonen@math.berkeley.edu URL: http://math.berkeley.edu/~poonen