Bertini theorems for hypersurface sections containing a subscheme over nite elds

Transcription

1 Bertini theorems for hypersurface sections containing a subscheme over nite elds Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) der Fakultät für Mathematik der Universität Regensburg vorgelegt von Franziska Wutz Regensburg, November 2014

2 Promotionsgesuch eingereicht am 04. November Die Arbeit wurde angeleitet von Prof. Dr. Uwe Jannsen. Prüfungsausschuss: Vorsitzender: Prof. Dr. Harald Garcke 1. Gutachter: Prof. Dr. Uwe Jannsen 2. Gutachter: Prof. Dr. Kiran Kedlaya, University of California weiterer Prüfer: Prof. Dr. Guido Kings Ersatzprüfer: Prof. Dr. Klaus Künnemann

3 Contents Introduction 1 1 Scheme-theoretic intersections and embedding dimension 5 2 Smooth hypersurface sections containing a closed subscheme over a nite eld Singular points of low degree Singular points of medium degree Singular points of high degree Bertini with Taylor conditions 37 References 42

4

5 Introduction Bertini theorems say that if a scheme X P n has a certain property, for example if it is smooth or geometrically irreducible, then there exists a hyperplane H such that the scheme-theoretic intersection H X has this property as well. For the projective space over an innite eld k, we have the following classical Bertini smoothness theorem: Theorem 0.1 ([Jou83] Théorème 6.3). Let k be an innite eld and X P n k be a quasi-projective smooth scheme. Then there exists a hyperplane H such that the intersection H X is smooth. This can be shown in the following way. We have a parameterization of the hyperplanes in P n k by the dual projective space (P n k) : a point a = (a 0 :... : a n ) corresponds to the hyperplane given by the equation a 0 x a n x n = 0, where x i denote the homogeneous coordinates of the projective space P n k. Then for any eld k, there is a dense Zariski open set U X (P n k) parameterizing the hyperplanes that intersect X smoothly. If k is innite as in Theorem 0.1, the set U X (k) of k-rational points is non-empty, since P n k(k) is a Zariski dense set in P n k. Hence we get the hyperplane we wanted. Of course, one would like to have an analogue of Theorem 0.1 over nite elds as well. Unfortunately, if k is a nite eld, it may happen that U X does not have any k-rational points, and therefore none of the nitely many hyperplanes over k intersect X smoothly. But B. Poonen showed in [Poo04], that in this case there always exists a smooth hypersurface section of X. Theorem 0.2 ([Poo04] Theorem 1.1). Let X be a quasi-projective subscheme of P n that is smooth of dimension m 0 over a nite eld k. Then there always exists a hypersurface H such that H X is smooth of dimension m 1. Independently, O. Gabber proved in Corollary 1.6 in [Gab01] the existence of good hypersurfaces of any suciently large degree that is divisible by the characteristic of the eld k. 1

6 Poonen also proved in [Poo08] that the hypersurface H can be chosen such that it contains a given closed subscheme Z, if Z X is smooth and dim X > 2 dim(z X). It is already mentioned there, that it should be possible to prove a version for Z X non-smooth as well. analogue. The goal of this project was to show that there exists such an In the rst section of this thesis, we will present some basic results about intersections of schemes. Furthermore, the embedding dimension will be introduced and calculated in situations that are relevant for us. In this context, we will also look at the schemes X e = X(Ω 1 X F q, e) of the attening stratication of a scheme X for the rank of the dierential sheaf Ω 1 X F q, i.e. the locus in X where Ω 1 X F q has rank e. The second section contains the main result of this thesis, the requested analogue of Theorem 0.2: Theorem 0.3. Let X be a quasi-projective subscheme of P n that is smooth of dimension m 0 over a nite eld F q. Let Z be a closed subscheme of P n, and let V = Z X. Assume max {e + dim V e} < m and V m =, where V e are the subschemes of the 0 e m 1 attening stratication of V for the rank of the dierential sheaf Ω 1 V F q. Then, for d 1, there exists a hypersurface H of degree d containing Z such that H X is smooth of dimension m 1. There is a similar result for innite elds by Altman and Kleiman in [AK79]; the theorem there states the following: Theorem 0.4. ([AK79] Theorem 7) Let k be an innite eld, let X be a smooth quasiprojective k-scheme, let Z be a subscheme of X and U a subscheme of Z. Assume the estimate { max dim(u(ω 1 Z U, e)) + e } < min (dim x(x)). e x U Then there is a hypersurface section of X containing Z which is smooth along U and o the closure of Z in X. 2

7 If we take Z = U closed in X, this gives the analogue of Theorem 0.3 for innite elds, since the conditions max {e + dim V e} < m and V m = of Theorem 0.3 coincide with 0 e m 1 the condition here. But in [AK79] the scheme Z must be contained in X, whereas in our case Z need not be contained in X; it does not even have to intersect X smoothly. To prove Theorem 0.3, we will dene the density µ Z (P) of a subset P of all homogeneous polynomials in F q [X 0,..., X n ]. Then we look at the set P of all homogeneous polynomials f F q [X 0,..., X n ] such that the hypersurface H f given by f contains Z and intersects X smoothly. If the density µ Z (P) is positive, the set is nonempty. Hence if we show that the density of P is larger than zero, we get the hypersurface section we want; more precisely, we will show that the density is equal to µ Z (P) = 1. ζ X V (m + 1) m 1 ζ Ve (m e) e=0 For this calculation we apply the so called closed point sieve, which has been used by Poonen in [Poo04] for the proof of Theorem 0.2. The idea is to start with all homogeneous polynomials f of degree d that vanish at Z and, for each closed point P X, sieve out those polynomials f for which the intersection H f X is singular at P. This works since smoothness can be tested locally, and since a scheme of nite type over a nite eld is smooth if and only if it is regular at all closed points. In a rst step we consider only points of degree bounded by some r > 0 and calculate the density of the set P r of the remaining polynomials, i.e. those polynomials that give a hypersurface containing Z and intersecting X smoothly at all closed points of degree bounded by r. Unfortunately, this does not generalize to all closed points: the fact that we only look at a nite set of points is crucial for the proof. But the points of degree r do not give a nite set. The main diculty of the proof lies in its second step, in which we show that the set of polynomials that are sieved out at the innitely many points of degree r is of density zero. Then the limit of µ Z (P r ) for r gives the correct density. 3

8 Finally, in the third section we prove a rened version of Theorem 0.3, in which we prescribe the rst terms of the Taylor expansion of the polynomial f that give the hypersurface at nitely many points that are not in Z. Using this theorem, we show for a scheme X that is smooth in all but nitely many closed points, that there exists a hypersurface H that contains Z but none of those nitely many points, and intersects X smoothly. Acknowledgments First, I would like to thank my advisor Prof. Dr. Uwe Jannsen cordially for giving me the chance to research in such an interesting eld. Thank you for always encouraging me and giving me important advices. Further, I want to thank Patrick Forré, Andreas Weber and the other colleagues at the University of Regensburg for many inspiring discussions and providing a very nice working atmosphere. I am grateful for many useful ideas you gave me. Finally, I want to thank Matthias Rother for always supporting me and enriching my life. 4

9 1 Scheme-theoretic intersections and embedding dimension Let X and Z be two subschemes of a scheme Y with morphisms i : X Y and j : Z Y. Then X Z := X Y Z = i 1 (Z) = j 1 (X) is the (scheme-theoretic) intersection of X and Z. Remark 1.1. Let X and Z be closed subschemes of Y with ideal sheaves I X and I Z, respectively. The intersection of X and Z is again a closed subscheme of Y and the ideal sheaf corresponding to it is given by I X Z = I X + I Z O Y, where the sum of I X and I Z is the sheaf associated to the presheaf U I X (U) + I Z (U) O Y (U). This can be proven in the following way: We can cover Y by ane open subsets and assume Y = Spec A to be ane. Let X = Spec(A/a) and Z = Spec(A/b). In this situation, the intersection of X and Z is given by X Z = X Y Z = Spec(A/a) Spec A Spec(A/b) = Spec(A/a A A/b) = Spec(A/(a + b)). Thus, in the local ring O Y,P of a point P Y given by the prime ideal p we have I X Z,P = (a + b) p = a p + b p = (I X ) P + (I Z ) P = (I X + I Z ) P. In particular, if X is quasi-projective and Z a closed subscheme of P n, as will be the case in the second section, then the local ring of the intersection V = X Z at a closed point P is given by O X,P /I Z,P, where I Z is the sheaf of ideals of Z: To see this, we put S = F q [x 0,..., x n ]. In some ane open neighbourhood of P let X be given by Spec A and let Z, as a closed subscheme of P n, be given by Spec(S/b). Then by denition, X Z = Spec A S Spec(S/b) = Spec(A S S/b) = Spec(A/b). Hence the local ring of V at P is equal to O V,P = O X,P /I Z,P with maximal ideal m V,P = m X,P /I Z,P, and we have an equality κ V (P ) = κ(p ). 5

10 Denition 1.2. Let X be a scheme and let F be an O X -module of nite type. We call the function rk(f) : X N 0 dened by rk(f)(x) = rk x (F) = dim κ(x) F(x) = dim κ(x) F x OX,x κ(x) the rank of F. Theorem 1.3. ([GW10] Theorem 11.17) Let F be a quasi-coherent O X -module of nite type and let r 0 be an integer. Then there exists a unique subscheme X(F, r) of X such that a morphism of schemes f : T X factors through X(F, r) if and only if f (F) is locally free of rank r. By this theorem, a point x X lies in X(F, r) if and only if i xf is locally free of rank r, where i x : Spec(κ(x)) X is the canonical morphism. Hence the underlying set of X(F, r) is {x X : rk x (F) = r}. Set-theoretically, X is therefore the union of the locally closed subsets X(F, r). The family X(F, r) for r 0 is called attening stratication. If F is a locally free O X -module, rk(f) is a locally constant function. Conversely, we have the following corollary of Theorem 1.3 above: Corollary 1.4. ([GW10] Corollary 11.18) Let X be a reduced scheme and let F be a quasi-coherent O X -module of nite type. Then F is locally free if and only if rk(f) is a locally constant function. Let k be a eld, let X be a scheme locally of nite type over k and let x X be a point. We dene the embedding dimension e(x) of X at x to be the integer e(x) = dim κ(x) (Ω 1 X k(x)), i.e. the rank of Ω 1 X k at x. Then we have a attening stratication of X given by the locally closed subschemes X e = X(Ω 1 X k, e). 6

11 By denition those are the subschemes such that for all points x X e the embedding dimension e(x) of X at x is equal to e. The situation in the next sections will be the following: let X be a quasi-projective subscheme of P n that is smooth of dimension m 0 over F q and let Z be a closed subscheme of P n. Let V = Z X be the scheme-theoretical intersection of Z and X. In order to calculate the fraction of homogeneous polynomials f F q [X 0,..., X n ] of degree d that give us a hypersurface containing Z and intersecting X smoothly, we will need to know the embedding dimension e V (P ) of V at a point P V. We will see that e V (P ) equals dim κ(p ) ( mx,p /(m 2 X,P, I Z,P ) ). This dimension will arise naturally from the calculation of the fraction of the polynomials named above. For the calculation we need some properties of the sheaf of dierentials. Lemma 1.5. ([Har93] Proposition II 8.4A and Proposition II 8.7) Let A be a ring, let B be an A-algebra, and I be an ideal of B. Dene C = B/I. Then there exists a canonical exact sequence of C-modules I/I 2 δ Ω 1 B A B C Ω 1 C A 0, where for any b I with image b in I/I 2 we have δ(b) = db 1. If B is a local ring which contains a eld k isomorphic to its residue eld B/ m, then the map δ : m / m 2 Ω 1 B k B k is an isomorphism. Lemma 1.6. ([Har93] Proposition II.8.12) Let f : X Y be a morphism of schemes and let Z be a closed subscheme of X with ideal sheaf I. Then there exists an exact sequence of sheaves on Z I/I 2 Ω 1 X Y O Z Ω 1 Z Y 0. Lemma 1.7. ([Har93] Theorem II 8.25A) Let A be a complete local ring containing a eld k. Assume that the residue eld κ(a) = A/ m is a separably generated extension of k. Then there exists a subeld K A, containing k, such that K A/ m is an isomorphism. 7

12 Lemma 1.8. (cf. [Har93] Exercise II 8.1) Let B be a local ring containing a eld k such that the residue eld κ(b) = B/ m of B is a separably generated extension of k. Then there exists an isomorphism m / m 2 Ω 1 B k B κ(b). Proof. Since B/ m 2 is a complete local ring, by Lemma 1.7 there exists a subeld K B/ m 2 and an isomorphism K = κ(b). Now Lemma 1.5 yields an isomorphism m / m 2 Ω 1 (B/ m 2 ) k (B/ m 2 ) κ(b). By ([Mat70], p. 187, Theorem 58 (ii)) we have an isomorphism Ω 1 B k B κ(b) = Ω 1 (B/ m 2 ) k (B/ m 2 ) κ(b); this shows the Proposition. Proposition 1.9. Let X be a scheme of nite type over a perfect eld k and let Z be a closed subscheme of P n. Let V = Z X be the intersection of Z and X. Then for a closed point P V, Ω 1 V k(p ) = m V,P / m 2 V,P = m X,P /(I Z,P, m 2 X,P ). Proof. Since V is of nite type over k, the local ring O V,P contains k and the residue eld κ(p ) of X at P is a nite separable eld extension of k. By Lemma 1.8, there are isomorphisms m V,P / m 2 V,P = Ω 1 O V,P k OV,P κ V (P ) = Ω 1 V k(p ), where κ V (P ) is the residue eld of V at P. We have seen in Remark 1.1 that the local ring of V at P is equal to O V,P = O X,P /I Z,P and we have an equality κ V (P ) = κ(p ). Now the Proposition follows from (m X,P /I Z,P )/(m X,P /I Z,P ) 2 = mx,p /(I Z,P, m 2 X,P ). Remark Let X be a quasi-projective subscheme of P n that is smooth of dimension m 0 over F q, let Z be a closed subscheme of P n and V = Z X be the intersection of Z and X. By Proposition 1.9, we can calculate the embedding dimension of V at P as e V (P ) = dim κ(p ) Ω 1 V F q (P ) = dim κ(p ) m X,P /(I Z,P, m 2 X,P ). The points in the subschemes V e of the attening stratication of V are exactly the points P V such that dim κ(p ) m X,P /(I Z,P, m 2 X,P ) = e. In particular, since X is smooth, it is 8

13 also regular and we get dim X dim O X,P = dim κ(p ) m X,P / m 2 X,P dim κ(p ) m X,P /(I Z,P, m 2 X,P ) = e V (P ), i.e. the dimension of X is a uniform bound for the embedding dimension e V (P ) for all closed points P V. The relation between smoothness of a scheme over a eld k at a point x and the stalk of the sheaf of dierentials Ω 1 X k at x that we will need is the following: Lemma ([Liu06] Proposition 6.2.2) Let X be a scheme of nite type over a eld k and x X. Then the following properties are equivalent: (i) X is smooth in a neighbourhood of x. (ii) X is smooth at x. (iii) Ω 1 X k,x is free of rank dim x X := inf {dim U U is an open neighbourhood of x}. Note that for a closed point x X, we have dim x X = dim O X,x. Theorem ([AK70] Theorem VII 5.7) Let S be a locally Noetherian scheme, X an S-scheme locally of nite type, Y a closed S-subscheme, and J its sheaf of ideals. Let x be a point of Y and g 1,..., g n local sections of O X. Suppose X is smooth over S at x. Then the following conditions are equivalent: (i) There exists an open neighbourhood X 1 of X such that g 1,..., g n dene an étale morphism g : X 1 A n S and g 1,..., g p generate J on X 1. (ii) (a) Y is smooth over S at x. (b) g 1,x,..., g p,x J x. (c) dg 1 (x),..., dg n (x) form a basis of Ω 1 X S (x). (d) dg p+1 (x),..., dg n (x) form a basis of Ω 1 Y S (x). 9

14 (iii) g 1,x,..., g p,x generate J x and dg 1 (x),..., dg n (x) form a basis of Ω 1 X S (x). (iv) Y is smooth over S at x, g 1,x,..., g p,x form a minimal set of generators of J x and dg p+1 (x),..., dg n (x) form a basis of Ω 1 Y S (x). Furthermore, if these conditions hold, then, at x, the sequence 0 J/J 2 Ω 1 X S OX O Y Ω 1 Y S 0 is exact and composed of free O Y -modules with bases that are induced by {g 1,..., g p }, {dg 1,..., dg n } and {dg p+1,..., dg n }. We also need the following property of coherent sheaves: Lemma ([Har93] Exercise II 5.7) Let X be a Noetherian scheme and F a coherent sheaf on X. If the stalk F x at a point x X is a free O X,x -module, then there exists a neighbourhood U of x such that F U is free. 10

15 2 Smooth hypersurface sections containing a closed subscheme over a nite eld In this section we want to prove the analogue of Theorem 1.1 of [Poo08] in the case where the intersection of X and Z is not smooth. Let F q be a nite eld of q = p a elements. Let S = F q [x 0,..., x n ] be the homogeneous coordinate ring of the projective space P n over F q and S d S the F q -subspace of homogeneous polynomials of degree d. Let S d be the set of all polynomials in F q[x 0,... x n ] of degree d and S homog = S d. Let X be a scheme of nite type over F q. The degree of a point P X is dened as deg P = [κ(p ) : F q ]. By [GW10] Proposition 3.33, a point P of a scheme locally of nite type over a eld is closed if and only if the degree of P is nite. Furthermore, the schemes that we look at are always of nite type over a nite eld F q, and therefore they are smooth over F q if and only if they are regular at all closed points. For a scheme X of nite type over F q we dene the zeta function ζ X (s) := P X closed (1 q s deg P ) 1. This product converges for Re(s) > dim X by [Ser65] Chapter 1.6. Let Z be a xed closed subscheme of P n. For d Z 0 let I d be the F q -subspace of polynomials f S d vanishing on Z, and I homog = d 0 I d. We can identify S d with S d by the dehomogenization x 0 = 1. We dene the density of a subset P I homog by µ Z (P) := lim d #(P I d ) #I d, if the limit exists (cf. [Poo08]). We cannot measure the density using the denition of [Poo04], since if the dimension of Z is positive, the density of I homog would always be zero (cf. Lemma 3.1 [CP13]), and hence we have to use this density relative to I homog. We further dene the upper and lower density µ Z (P) and µ Z (P) of a subset P I homog by µ Z (P) := lim sup d #(P I d ) #I d, d 0 11

16 and using lim inf in place of lim sup. A set of density zero does not need to be nonempty; but if the density of a set is positive, then the set contains innitely many polynomials. For a polynomial f I d let H f = Proj(S/(f)) be the hypersurface dened by f. Let X be a quasi-projective subscheme of P n that is smooth of dimension m 0 over F q. We will show that the density of the set of polynomials f I homog, such that the hypersurface H f contains Z and intersects X smoothly, is positive and therefore such a hypersurface always exists. Theorem 2.1. Let X be a quasi-projective subscheme of P n that is smooth of dimension m 0 over F q. Let Z be a closed subscheme of P n and let V := Z X be the intersection. We dene P = {f I homog : H f X is smooth of dimension m 1}. (i) If max 0 e m 1 {e + dim V e} < m and V m =, then µ Z (P) = ζ V (m + 1) ζ X (m + 1) m 1 ζ Ve (m e) e=0 = 1. ζ X V (m + 1) m 1 ζ Ve (m e) e=0 (ii) If In particular, there exists a hypersurface H of degree d 1 containing Z such that H X is smooth of dimension m 1. max 0 e m 1 {e + dim V e} m or V m, then µ Z (P) = 0. At the end of this section, we will give an example involving simple normal crossings, in which the conditions of Theorem 2.1 (i) are fullled. Before we start the proof, we want to make a few remarks regarding the density. The rst two remarks show that Theorem 2.1 implies both Theorem 1.1 of [Poo04] and Theorem 1.1 of [Poo08]. 12

17 Remark 2.2. If we choose Z to be empty, then the density of a set P I homog = S homog is just µ (P) = lim d #(P S d ) #S d. This is the same density as used in [Poo04]. Furthermore, the conditions of Theorem 2.1(i) are fullled, since V is also empty, and the density µ (P) = 1 ζ X V (m + 1) m 1 ζ Ve (m e) e=0 = ζ X (m + 1) 1 given by Theorem 2.1(i) is the same density as in Theorem 1.1 of [Poo04]. Remark 2.3. If the intersection V = Z X is smooth of dimension l 0 as required in [Poo08], then for a closed point P V, l = dim O V,P = dim O X,P /I Z,P = dim κ(p ) m X,P /(m 2 X,P, I Z,P ) = e V (P ). Hence in this case, the embedding dimension of V at all points is equal to the dimension l of the intersection V and V e = for all e l. It follows that dim V l = dim V and the requirement max {e + dim V e} < m of Theorem 2.1 implies l + dim V = 2l < m. 0 e m 1 Therefore, if this condition is fullled, Theorem 2.1 (i) yields the statement of Theorem 1.1 of [Poo08] µ Z (P) = ζ V (m + 1) ζ X (m + 1)ζ V (m l). Thus, Theorem 1.1 of [Poo08] is implied by Theorem 2.1. Remark 2.4. The density in Theorem 2.1 is independent of the embedding X P n. Remark 2.5. If X is a subscheme of X, then obviously µ Z (X ) µ Z (X). We can say even more about the density of X if X is the union of two disjoint open subschemes X 1 and X 2 of X. Since the embedding dimension is calculated locally and X 1 is open in X, we have the equality e X (P ) = e X1 (P ) for any point P X 1, and 13

18 similarly for X 2. Therefore, the set of points in (Z X) e is the union of the points in (Z X 1 ) e and (Z X 2 ) e, and for Re(s) > dim(z X) e we have ζ (Z X)e (s) = (1 q s deg P ) 1 P (Z X) e closed = (1 q s deg P ) 1 (1 q s deg P ) 1 P (Z X 1 ) e closed P (Z X 2 ) e closed = ζ (Z X1 ) e (s) ζ (Z X2 ) e (s). The zeta function for X and for Z X is also multiplicative in the same way; hence if the requirements of Theorem 2.1 (i) are fullled, µ Z (P X ) = = ζ Z X (m + 1) ζ X (m + 1) m 1 ζ (Z X)e (m e) e=0 ζ Z X1 (m + 1) ζ Z X2 (m + 1) ζ X1 (m + 1) ζ X2 (m + 1) m 1 ζ (Z X1 ) e (m e) m 1 ζ (Z X2 ) e (m e) e=0 = µ Z (P X1 ) µ Z (P X2 ), where P X = {f I homog : H f X is smooth of dimension m 1} and P X1 and P X2 are dened similarly. e=0 Remark 2.6. It is important that we x the scheme Z at the beginning. Densities calculated for two dierent closed subschemes cannot be compared easily, because in the denition of the density we use the ideal sheaf of the closed subscheme; the density µ Z is relative to I homog. So in general, we cannot combine the result of Theorem 2.1 for two arbitrary but dierent closed subschemes Z 1 and Z 2 to get a result for example for the union of those subschemes. But if Z 1 and Z 2 are disjoint closed subschemes such that the requirements of Theorem 2.1 are fullled for Z 1, Z 2 and the union Z 1 Z 2, then µ Z1 Z 2 (P) = µ Z1 (P) µ Z2 (P). The reason for this is again the multiplicativity of the zeta function as in the remark above. 14

19 If Z 1 and Z 2 are two distinct closed subschemes of P n with Z 1 X = Z = Z 2 X, such that the requirements of Theorem 2.1 (i) are fullled, then the density is in both cases given by µ Z1 (P) = µ Z2 (P) = ζ V (m + 1), ζ X (m + 1) m 1 ζ Ve (m e) where again V = Z X. Note that the density in Theorem 2.1 does not depend on the points of Z outside of X: if we consider two closed subschemes Z and Z := Z X of P n, then the density µ Z (P) must be equal to µ Z (P), since the right hand side of the equality in Theorem 2.1 (i) does not depend on the points in Z Z. This may seem suprising, since in general, for a xed degree, there will be more hypersurfaces that contain Z than hypersurfaces that contain Z; so one would expect the density calculated for Z to be larger than that for Z. But as stated above, the two densities cannot be compared. e=0 As mentioned in the introduction, the proof of Theorem 2.1 will use the closed point sieve introduced in [Poo04]. It will be parallel to the one in [Poo08]; but there the intersection of X and the closed subscheme Z is assumed to be smooth, which does not have to be the case here. Therefore we will have to make signicant changes in almost every line of the proof. For this closed point sieve we will consider closed points of X of low, medium and high degree in the next three sections. At rst, we will calculate in Lemma 2.12 the density of the set P r of polynomials f I homog that give a good hypersurface section in the points of low degree bounded by r. In the subsequent sections we will show in 2.15, 2.16 and 2.19, that this density does not change if we also consider points of medium and high degree. More precisely, we will see that µ Z (P) diers from the density µ Z (P r ) of the polynomials that give a good hypersurface section at points of degree bounded by r at most by the upper density of the polynomials that do not give a good hypersurface section at points of medium and high degree. Hence we need to prove for r, that this upper density is zero, and that the limit of µ Z (P r ) is the value that we claimed for 15

20 µ Z (P). The requirements max 0 e m 1 {e + dim V e} < m and V m = of Theorem 2.1 (i) will be used in each lemma mentioned above. 2.1 Singular points of low degree Let the notation be as in Theorem 2.1. The goal in this section is to calculate the density of the set of polynomials that give a smooth hypersurface section in the points of low degree. For this, we need to study the zeroth Zariski-cohomology group of a nite subscheme of P n. Lemma 2.7. Let Y be a nite subscheme of P n over the nite eld F q. Then H 0 (Y, O Y (d)) = H 0 (Y, O Y ), i.e. we may ignore the twist on nite schemes. Proof. First we can assume Y A n = {x 0 0}: if this is not true, we can enlarge F q if necessary and perform a linear change of variable to achieve that the nitely many points of Y are contained in D + (x 0 ). Hence the canonical morphism φ d : H 0 (P n, O P n(d)) H 0 (Y, O Y (d)) factors through H 0 (D + (x 0 ), O P n (d)). D+ (x 0 For the standard-open set ) D + (x 0 ) we have (S(d)) D+ (x 0 ) = (S(d) (x0 ) ) and S (x 0 ) = S(d) (x0 ) for all d Z. Thus, O P n(d) D+ (x 0 ) = (S(d)) D+ (x 0 ) = (S(d) (x0 )) = (S (x0 )) = S D+ (x 0 ) = O P n D+ (x 0 ). This shows H 0 (D + (x 0 ), O P n (d)) D+ (x 0 = H 0 (D ) + (x 0 ), O P n ) and, since φ D+ (x 0 ) d factors through H 0 (D + (x 0 ), O P n ), we get D+ (x 0 ) H0 (Y, O Y (d)) = H 0 (Y, O Y ). Let I Z O P n be the ideal sheaf of Z. We want to show I d = H 0 (P n, I Z (d)) (cf. [GW10] Remark 13.26). First of all, note that S is saturated as a graded S-module, i.e. α : S Γ ( S) = n Z Γ(P n, S(n)) is an isomorphism of graded S-modules. This is true because we have isomorphisms S d = Γ(Pn, O P n(d)) in every grade d. Therefore, by [GW10] Proposition 13.24, there 16

21 exists a unique saturated homogeneous ideal J S such that Z = Proj(S/J); in particular, J = IZ. As J is saturated, we have an isomorphism α J : J Γ ( J) of graded S-modules and hence an isomorphism J d = Γ(P n, J(d)) = Γ(P n, I Z (d)) for any d. By writing Z = Proj(S/J), we can interpret Z to be the intersection of hypersurfaces given by the polynomials that generate the ideal J S. In particular, J d is the set of homogeneous polynomials of degree d that vanish on Z, and thus J d equals I d. Next we consider the surjection O (n+1) P n O P n (1) (f 0,..., f n ) x 0 f x n f n. Tensoring it with I Z gives a surjection ϕ : I (n+1) Z I Z (1). By a vanishing theorem of Serre ([Har93] III.5.2), if F is a coherent sheaf on P n, there exists an integer d 0, depending on F, such that H i (X, F(d)) = 0 for each i > 0 and each d d 0. The ideal sheaf I Z and therefore also the nite direct sum I (n+1) Z is coherent, since P n is Noetherian and hence the category of coherent O P n-modules is an abelian category ([Har93] Proposition 5.9). Thus we can apply the above theorem to get H 1 (P n, I (n+1) Z (d)) = 0 for each d d 0. This yields a short exact sequence 0 H 0 (P n, ker φ(d)) H 0 (P n, I (n+1) Z (d)) H 0 (P n, I Z (d + 1)) 0, and therefore a surjection for d 1 I (n+1) d = H 0 (P n, I (n+1) Z (d)) H 0 (P n, I Z (d + 1)) = I d+1. Since x 0 f x n f n S 1 I d for f i I d, we get S 1 I d = I d+1 for d 1. We x an integer c such that S 1 I d = I d+1 for all d c. 17

22 Lemma 2.8. ([Poo08], Lemma 2.1.) Let Y be a nite subscheme of P n over F q. 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. Then φ d is surjective for d c + dim H 0 (Y, I Z O Y ). Proof. For reasons of completeness, we add the proof following the one of [Poo08]. The map of sheaves O P n O Y is surjective, so the induced map I Z I Z O Y is surjective as well. Taking cohomology and using the vanishing theorem of Serre ([Har93] III.5.2) as in the remark previous to this lemma, we can show that φ d is surjective for d 1. As seen in the proof of Lemma 2.7, we can assume Y A n = {x 0 0}. Dehomogenization by setting x 0 = 1 identies S d with the space S d of polynomials in F q [x 1,..., x n ] of degree d and I d with the image I d of I d under this dehomogenization. This identies φ d with a map I d B = H 0 (Y, I Z O Y ). The dimension b of B is nite as Y is a nite scheme and therefore the local ring at each of its nitely many points is a local nite F q -algebra. For d c, let B d be the image of I d in B. By denition of c, we have S 1I d = I d+1 and hence S 1B d = B d+1 for d c. Since 1 S 1, we get B c B c+1... B. There exists a j [c, c + b] such that B j = B j+1 : suppose this is not true. Then dim B j+1 dim B j + 1 for all j [c, c + b] and therefore dim B c+b+1 dim B c + b + 1 b + 1, which contradicts B c+b+1 B since b is the dimension of B. Using S 1B d = B d+1 for d c, we get B j+2 = S 1B j+1 = S 1B j = B j+1. Similarly B j+2 = B j+3 =..., and thus B j = B j+k for all k N. Now by the rst paragraph of this proof, φ d is surjective for some d 1. Thus there exists an l N such 18

23 that B j+l = B and hence B j = B. This shows that φ d is surjective for d j, and in particular for d c + b = c + dim H 0 (Y, I Z O Y ). Lemma 2.9. Let X be a quasi-projective subscheme of P n that is smooth of dimension m 0 over F q. Let P be a closed point of X and let f I homog. Then H f X is smooth of dimension m 1 at P if and only if f / m 2 X,P. Further let m O X be the ideal sheaf of P and let Y X be the closed subscheme of P n corresponding to the ideal sheaf m 2 O X. Then H f X is smooth of dimension m 1 at P if and only if the restriction of f to a section of I Z O Y (d) is not equal to zero. Proof. Let P H f X and thus f m X,P. Since F q is perfect, H f X is smooth of dimension m 1 at P if and only if H f X is regular of dimension m 1 at P, i.e. O X,P /f is regular, where f also denotes the image of f under the map S O X,P. By Krull's principal ideal theorem, dim(o X,P /f) = m 1. Since f m X,P {0}, Corollary 2.12 in [Liu06] yields O X,P /f is regular if and only if f / m 2 X,P. This shows the rst claim. For the second claim, we observe that Y is the support of the quotient sheaf given by O X / m 2. Hence Y = Spec(O X,P / m 2 X,P ). Because both O X,P /m X,P and m X,P /m 2 X,P are nitely generated F q -modules, O X,P /m 2 X,P is also a nitely generated F q-module and Y is a nite scheme. Thus Lemma 2.7 yields H 0 (Y, O Y (d)) = H 0 (Y, O Y ) = O X,P / m 2 X,P. By what we have shown above, H f X is smooth at P if and only if f I d is not an element of m 2 X,P, i.e. f is not zero in H0 (Y, I Z O Y (d)). Let P be a closed point of X. If we dene the scheme Y as above, we have seen in the proof of Lemma 2.9 that Y = Spec(O X,P /m 2 X,P ) is a nite scheme. Hence we can apply Lemma 2.8 to Y to get a surjective homomorphism φ d : I d H 0 (Y, I Z O Y (d)) and in particular an isomorphism I d / ker φ d = H 0 (Y, I Z O Y (d)). Then Lemma 2.9 shows that the polynomials f I d, which are not zero in I d / ker φ d and thus not in the kernel of φ d, are exactly the polynomials that give us a hypersurface H containing Z such that 19

24 H X is smooth of dimension m 1 at the point P. Therefore, if we want to calculate the fraction of those polynomials, we need to know the size of H 0 (Y, I Z O Y (d)) {0}. Lemma Let m O X be the ideal sheaf of a closed point P X. Let Y X be the closed subscheme of P n, which corresponds to the ideal sheaf m 2 O X. Then for all d Z 0 q (m+1) deg P, if P / V, #H 0 (Y, I Z O Y (d)) = q (m e V (P )) deg P, if P V. Proof. As seen in the proof of Lemma 2.9, Y = Spec(O X,P /m 2 X,P ) is a nite scheme. So we have H 0 (Y, I Z O Y (d)) = H 0 (Y, I Z O Y ). We have an exact sequence of sheaves 0 I Z O Y O Y O Z Y 0. By a vanishing theorem of Grothendieck ([Har93] Theorem III 2.7), H i (Y, F) = 0 for all i > dim Y = 0 and all sheaves of abelian groups F on Y. Thus, taking cohomology of this sequence on Y yields an exact sequence 0 H 0 (Y, I Z O Y ) H 0 (Y, O Y ) H 0 (Y, O Z Y ) 0. Now we calculate #H 0 (Y, O Y ) and #H 0 (Y, O Z Y ) to get #H 0 (Y, I Z O Y (d)). There is a ltration of H 0 (Y, O Y ) = O X,P / m 2 X,P given by 0 m X,P / m 2 X,P O X,P / m 2 X,P O X,P / m X,P 0, whose quotients are vector spaces of dimensions m and 1 respectively over the residue eld κ(p ) of P since X is smooth and hence regular at the point P. So by additivity of length of modules, #H 0 (Y, O Y ) = #κ(p ) m+1 = q (m+1) deg P. Next we determine #H 0 (Y, O Z Y ). Since Y = Spec(O X,P / m 2 X,P ), Remark 1.1 shows H 0 (Y, O Z Y ) = O X,P /(I Z,P, m 2 X,P ). If P X V, then I Z,P is not contained in m X,P and H 0 (Y, O Z Y ) = 0. If P V, then H 0 (Y, O Z Y ) has a ltration given by 0 m X,P /(I Z,P, m 2 X,P ) H 0 (Y, O Z Y ) O X,P / m X,P 0. 20

25 We have seen in Remark 1.10, that e V (P ) = dim κ(p ) m X,P /(I Z,P, m 2 X,P ). Hence, dim κ(p ) H 0 (Y, O Z Y ) = 1 + dim κ(p ) m X,P /(I Z,P, m 2 X,P ) = 1 + e V (P ). Thus, #H 0 (Y, I Z O Y ) = #H0 (Y, O Y ) #H 0 (Y, O Z Y ) q (m+1) deg P, if P / V, = q (m+1) deg P /q (e V (P )+1) deg P, if P V, which is what we wanted to show. For a scheme X of nite type over F q we dene X <r to be the set of closed points of X of degree < r. Let X >r be dened similarly. Remark X <r is a nite set: since X is of nite type over F q, there exists a nite covering of X by ane open subschemes X i, where X i = Spec(F q [x 1,..., x n ] /a i ) for an ideal a i F q [x 1,..., x n ] and n N. Then X i (F q r) = Hom Fq (F q [x 1,..., x n ] /a i, F q r) Hom Fq (F q [x 1,..., x n ], F q r) = F n q r. The number of closed points of X i with degree r is less than or equal to the number of elements in X i (F q r), because for every such point P there exists an F q -homomorphism Spec F q r X mapping the unique point of Spec F q r to P. Since X i (F q r) is nite, it follows that X <r is a nite set. Lemma 2.12 (Singularities of low degree). Let X be a quasi-projective subscheme of P n that is smooth of dimension m 0 over F q and let Z be a closed subscheme of P n. Let V := Z X be the intersection. Dene P r := {f I homog : H f X is smooth of dimension m 1 at all points P X <r }. Then µ Z (P r ) = P (X V ) <r (1 q (m+1) deg P ) m e=0 P (V e) <r (1 q (m e) deg P ). 21

26 Proof. Let X <r = {P 1,..., P s }. Let m i be the ideal sheaf of P i on X and let Y i be the closed subscheme of X whose ideal sheaf is m 2 i O X. Let Y = s Y i. By Lemma 2.9, the intersection H f X is not smooth of dimension m 1 at P i if and only if the restriction of f to a section of I Z O Yi (d) is zero. Hence P r I d is the inverse image of s (H 0 (Y i, I Z O Yi (d)) {0}) under the F q -linear map s φ d : I d = H 0 (P n, I Z (d)) H 0 (Y, I Z O Y (d)) = H 0 (Y i, I Z O Yi (d)). We can ignore the twist by Lemma 2.7, and we may further assume that the condition d c + dim H 0 (Y, I Z O Y ) of Lemma 2.8 is fullled, since in the density that we want to calculate we only look at the limit d. Hence Lemma 2.8 implies that φ d is s surjective and the inverse image of (H 0 (Y i, I Z O Yi (d)) {0}) is the disjoint union of # s (H 0 (Y i, I Z O Yi (d)) {0}) cosets of the kernel of φ d. Thus s ( #(P r I d ) = # H 0 (Y i, I Z O Yi (d)) {0} ) # ker φ d. Again the surjectivity of φ d yields s #I d = # ker φ d # H 0 (Y i, I Z O Yi (d)). Inserting this into the denition of density and applying Lemma 2.10, we get s #H 0 (Y i, I Z O Yi ) 1 µ Z (P r ) = #H 0 (Y i, I Z O Yi ) = (1 q (m+1) deg P ) (1 q (m ev (P )) deg(p ) ) P (X V ) <r P V <r = (1 q (m+1) deg P ) P (X V ) <r m e=0 P (V e) <r (1 q (m e) deg P ) Note, that this proof only works since there are only nitely many points in X <r and hence Y is a nite subscheme of P n. If we wanted to use the same argument for the set of polynomials P dened as in Theorem 2.1, and therefore considered points of X of 22

27 arbitrary degree, we would have to let r tend to innity before we calculate the density µ Z (P), i.e. before we let d tend to innity. But the proof of Lemma 2.12 does not work there anymore, as then we would have innitely many points to deal with. So as mentioned in the introduction, we see now, that rst we need to look only at points of some bounded degree r as above. Then we show that when d r 1, the number of polynomials f I d of degree d that do not give a smooth intersection at the innitely many points of degree at least r is insignicant, i.e. the upper density of this set of polynomials is zero. Corollary Let max 0 e m 1 {e + dim V e} < m and V m =, then lim µ Z(P r ) = r ζ V (m + 1). ζ X (m + 1) m 1 ζ Ve (m e) Proof. The rst product in Lemma 2.12 converges anyway, since m + 1 > dim(x V ). The factor for e = m in the second product in this lemma does not appear since V m is empty. For all 0 e m 1, the product (1 q (m e) deg P ) is just the P (V e) <r partial product used in the denition of the zeta function of V e. This converges for m e > dim V e, i.e. for dim V e + e < m. Since we want every product in Lemma 2.12 to converge, we need dim V e + e < m for all e 0. Proof of Theorem 2.1 (ii). If max {e + dim V e} m, then there exists a 0 e 0 < m 0 e m 1 such that e 0 + dim V e0 m, i.e. m e 0 dim V e0. Applying Lemma 2.12 gives µ Z (P r ) P (V e0 ) <r (1 q (m e0) deg P ) e=0 P (V e0 ) <r (1 q dim Ve0 deg P ). This is the inverse of the partial product used in the denition of the zeta function of V e0. This zeta function has a pole at dim V e0 zero for r. (cf. [Tat65] Ÿ4), thus the product tends to As a locally closed subscheme of the Noetherian scheme X, the scheme V m is again Noetherian. Therefore, if it is nonempty, it contains a closed point P and the factor 23

28 (1 q (m m) deg P ) in the density of P r in Lemma 2.12 is equal to zero; hence the density µ(p r ) is zero for V m. The inclusion P P r implies µ Z (P) µ Z (P r ). We have seen above that the density of P r tends to zero for r if max e 0 {e + dim V e} m or V m. Hence the result follows. From now on, we assume max {e + dim V e} < m and V m =. 0 e m Singular points of medium degree Lemma Let P X be a closed point of degree d c. Then the fraction of m+1 polynomials f I d such that H f X is not smooth of dimension m 1 at P is equal to q (m+1) deg P, if P / V, q (m e V (P )) deg P, if P V. Proof. Let Y be dened as in Lemma 2.9. Then H f X is not smooth of dimension m 1 at P if and only if the restriction of f to a section of I Z O Y (d) is equal to zero. Applying Lemma 2.10 and using deg P d c we obtain m+1 q (d c), if P / V, #H 0 (Y, I Z O Y (d)) q (m e V (P ))(d c)/(m+1), if P V. Now m e V (P ) m+1 1 and hence dim H 0 (Y, I Z O Y (d)) d c. Therefore we can apply Lemma 2.8 and get an isomorphism H 0 (P n, I Z (d))/ ker φ d = H 0 (Y, I Z O Y (d)), where φ d is dened as in Lemma 2.8. As the polynomials we consider are exactly those with image zero in H 0 (Y, I Z O Y (d)), the fraction we want to calculate equals # ker φ d #H 0 (P n, I Z (d)) = 1 #H 0 (Y, I Z O Y (d)). 24

29 But by Lemma 2.10 we get q (m+1) deg P, if P / V, #H 0 (Y, I Z O Y (d)) = q (m e V (P )) deg P, ifp V. This shows the lemma. Lemma 2.15 (Singularities of medium degree). Let Q medium r := {f I d : there exists a point P X with r deg P d c m + 1 such d 0 that H f X is not smooth of dimension m 1 at P }. Then lim r µ Z (Q medium r ) = 0. Proof. If we take the union over all sets of polynomials that we considered in Lemma 2.14 for all points P X with degree between r and d c, we get Qmedium m+1 r. Hence, µ Z (Q medium r ) is at most equal to the sum over all those points of all fractions that we calculated in Lemma Moreover, just like in this lemma, we can split this sum to get a sum for points in V and one for points in X V. For a subscheme U of P n F q the number of points P of degree g in U is less than or equal to #U(F q g) = # Hom(Spec F q g, U), thus #(Q medium r I d ) #I d m e=1 + m e=1 + (d c)/(m+1) g=r (d c)/(m+1) g=r (d c)/(m+1) g=r (d c)/(m+1) g=r (number of points of degree g in V e ) q (m e)g (number of points of degree g in X V ) q (m+1)g #V e (F q g) q (m e)g #(X V )(F q g) q (m+1)g. By [LW54] Lemma 1, there exist constants C e and C for V e and X V that depend only g dim Ve on V e and X V, respectively, such that #V e (F q g) C e q and #(X V )(F q g) 25

30 Cq g dim(x V ). Then, using the assumption V m =, we obtain The other assumption hence #(Q medium m 1 r I d ) #I d e=1 #(Q medium m 1 r I d ) #I d m 1 e=1 C e q g dim Ve q (m e)g + g=r C e g=r q g(dim Ve+e m) + C Cq gm q (m+1)g g=r q g. max {e + dim V e} < m yields dim V e + e m 1 for all e, 0 e m 1 e=1 m 1 e=1 C e g=r Cq r q g + 1 q 1 C e q r Cq r + 1 q 1 Since this tends to zero for r, we get lim µ Z(Q medium r r which is what we claimed. ) = lim r lim sup d m 1 1 q = 1 q r ( e=1 g=r C e 1 q + C ). 1 1 q 1 #(Q medium r I d ) #I d = 0, 2.3 Singular points of high degree In this section we will show that the upper density for polynomials f I homog that do not give a smooth intersection at a point P X of high degree is equal to zero. We will split this up in two problems: First we prove that this upper density is zero if we only consider points in X V. This is just a result of [Poo08]. Then we show the same for points in V. Lemma 2.16 (Singularities of high degree o V ). Dene Q high X V := { f I d : there exists a point P (X V ) > d c m+1 d 0 Then µ Z (Q high X V ) = 0. is not smooth of dimension m 1 at P }. such that H f X 26

31 Proof. This is the statement of Lemma 4.2. in [Poo08] for the case in which the intersection V of X and Z is smooth, whereas in our case, V does not have to be smooth. But the proof does not use the fact that V is smooth, since in this lemma only the points that are not in V are considered; hence it also shows Lemma The proof for the analogue of this lemma for points on V will use the following version of Bézout's theorem, and a lemma that counts the polynomials f S d that vanish at some closed point P A n. Lemma ([Ful98] Example ) Let V 1,... V r be equidimensional closed subschemes of P n over F q. Let W 1,..., W s be the irreducible components of V j. r Then j=1 r s r deg W i deg V j. j=1 Lemma ([Poo04] Lemma 2.5) Let P be a closed point in A n over F q. Then the fraction of f S d that vanish at P is at most q min(d+1,deg P ). The last lemma we need for the proof of Theorem 2.1 is the following: Lemma 2.19 (Singularities of high degree on V ). Dene Q high V := d 0 { f I d : there exists a point P V > d c m+1 such that H f X is not smooth of dimension m 1 at P }. Then µ Z (Q high V ) = 0. For this lemma we cannot use a result of [Poo08] as we did above: an analogue of this lemma does exist there as well, namely Lemma 4.3. But the fact that V is assumed to be smooth there is crucial for the proof. Therefore we need to prove this lemma in a dierent way, but we will use the same technique as in [Poo08], i.e. the induction argument introduced by Poonen in [Poo04], Lemma

32 Proof. If the lemma is proven for all subsets X i of a nite ane open cover of X, then it holds for X as well, because the sum of the corresponding upper densities for the X i is an upper bound for µ Z (Q high V ). Hence we can assume without loss of generality, that X A n F q = {x 0 0} P n F q is ane. Again we identify S d, i.e. the homogeneous polynomials in F q [x 0,..., x n ] of degree d, with the space of polynomials S d F q[x 1,..., x n ] = A of degree d by setting x 0 = 1. This dehomogenization also identies I d with a subspace I d S d. Let P be a closed point of X. Since X is smooth, we can choose a system of local parameters t 1,..., t n A on A n such that t m+1 =... = t n = 0 denes X locally at P. Then dt 1,..., dt n are a basis for the stalk of Ω 1 A n F q at P and by Theorem 1.12, dt 1,..., dt m are a basis for the stalk of Ω 1 X F q at P. By using those local parameters we now want to nd suitable derivations D 1,..., D m : A A such that for f I d, the intersection H f X is not smooth at a point P V if and only if (D 1 f)(p ) =... = (D m f)(p ) = 0. We will then show that the probability that H f X is not smooth at a point in V e tends to zero for d and any e. Since we only have nitely many subschemes V e for 0 e m in the attening stratication for Ω 1 X F q, the upper density µ Z (Q high ), that we actually want to calculate, is a nite sum of those probabilities, and hence zero. As we have seen in the rst section, V is the disjoint union of the locally closed subsets V e = V (Ω 1 V F q, e) for 0 e m. By Proposition 3.52 of [GW10], we can give V e the structure of a reduced subscheme of V. For all P V e, by denition of the embedding dimension and the V e, we have dim κ(p ) Ω 1 V F q OV O Ve (P ) = dim κ(p ) Ω 1 V F q OV κ(p ) = e V (P ) = e. Thus the rank of Ω 1 V F q O Ve on V e is a constant function. Because V e is reduced and Ω 1 V F q O Ve is a quasi-coherent O Ve -module of nite type, Corollary 1.4 shows that Ω 1 V F q O Ve is a locally free O Ve -module. Next we consider the map Ω 1 X F q O V Ω 1 V F q, which is surjective by Lemma

33 Tensoring it with O Ve gives a surjective map φ : Ω 1 X F q O Ve Ω 1 V F q O Ve where Ω 1 X F q O Ve is a locally free sheaf of rank m and Ω 1 V F q O Ve is a locally free sheaf of rank e. At P, the sequence 0 ker φ Ω 1 X F q O Ve Ω 1 V F q O Ve 0 is an exact sequence of free modules and splits, since Ω 1 V F q,p O V e,p is free and therefore projective. Hence, dt 1,..., dt m e form a basis of the kernel of φ at P and dt m e+1,..., dt m a basis of Ω 1 V F q,p O V e,p. In particular, t 1,..., t m e all vanish on V, since by Proposition 1.9 Ω 1 V F q κ(p ) = m V,P / m 2 V,P. We want to show, that t 1,..., t m e can be assumed to vanish even on Z. For this, let the closed scheme Z be given by Spec A/I Z. Because the quasi-projective scheme X can be assumed to be projective in a neighbourhood of P and the above calculation was done locally, we can assume without loss of generality X = Spec A/J, such that P corresponds to a maximal ideal of A/J. Since t 1,..., t m are local parameters of X at P, it follows that t m+1,..., t n generate J P, and t 1,..., t m generate the maximal ideal m X,P. In particular, t 1,..., t m are not in m 2 X,P, because X is regular at P and therefore the images of t 1,..., t m in m X,P / m 2 X,P are a basis of m X,P / m 2 X,P as a κ(p )-vector space. Thus, t 1,..., t m are not in J P. The intersection V of X and Z is given by Spec A/(J+I Z ). By what we have shown above, t 1,..., t m e vanish on V and thus are elements of the ideal J + I Z localized at P. Since t 1,..., t m e are not in J P, there exist a i J P b i I Z,P \ {0} such that t i = a i + b i for all 1 i m e. Then t i b i mod J P and therefore b 1,..., b m e, t m e+1,..., t m are again local parameters of X at P. Furthermore, db 1,..., db m e are still a basis of the kernel of φ at P and dt m e+1,..., dt m are a basis of Ω 1 V F q,p O V e,p. Since t i A, we can choose b 1,..., b m e I Z and therefore assume that they vanish at Z. Hence we may also assume that t 1,..., t m e already vanish at Z. Let 1,..., n T A n F q,p be the basis of the stalk of the tangent sheaf, dual to dt 1,..., dt n. We can nd an s A with s(p ) 0 such that D i = s i gives a global derivation A A for i = 1,..., n. Since Ω 1 A n F q is a locally free and coher- and 29

34 ent O A n-module, by Lemma 1.13 there exists a neighbourhood N P of P in A n such that N P X = N P {t m+1 =... = t n = 0} and Ω 1 NP A n F q = n O NP dt i. Furthermore, we can choose s A such that s O(N P ) holds, due to the fact that s(p ) is not equal to zero. Since X is quasi-compact, we can cover X with nitely many N P and assume X N P. Hence in particular, Ω 1 X F q = m O X dt i. Let P V e be a closed point. Lemma 2.9 shows that for a polynomial f I d, the hypersurface section H f X is not smooth at P if and only if f m 2 X,P. By denition of the derivations D i, this is equivalent to (D 1 f)(p ) =... = (D m f)(p ) = 0. Note that we do not have to demand f(p ) to be zero, since Z is contained in the hypersurface H f for f I d, and thus f vanishes at all points in V e Z anyway. Now we want to bound the f I d induction argument in Lemma 2.6 of [Poo04]. for which there exists such a point by using the Let τ = max (deg t i) and γ = (d τ)/p where l e = dim V e. We select f 0 I d and 1 i l e+1 g 1 S γ,..., g le+1 S γ uniformly and independently at random. Then the distribution of f = f 0 + g p 1t g p l e+1 t l e+1 is uniform over I d : rst of all, we have to show that the sum on the right hand side is again a polynomial in I d. By our assumption we have e + l e < m for all 0 e m and therefore l e + 1 m e. But t 1,..., t m e and consequently t 1,..., t le+1 all vanish on Z; hence the sum vanishes as well and denes an element in I d since the degree of f is d. To prove that the distribution is uniform, note that every set mentioned above is nite, and thus we only need to show that all f I d have the same number of representations of this kind. First, every polynomial f I d can be constructed in this way because we can choose f 0 = f and g 1 =... = g e = 0. Now let f and F be any two polynomials in I d and let f = f 0 + g p 1t g p l e+1 t l e+1. Then, F = (f 0 f 1 ) + g p 1t g p l e+1 t l e+1 where f 1 = f F I d. That way, we get for any two dierent representations of f also two dierent representations of F and similarly vice versa. Hence any two polynomials 30

35 have the same number of representations of this kind. Since the distribution of the polynomials f in this representation is uniform over I d, it is enough to bound the probability for an f constructed in this way to have a point P V e,> d c such that (D 1 f)(p ) =... = (D m f)(p ) = 0. We will see, that we can m+1 even consider only the rst l e + 1 derivations D i and still show the claim. Here we are using the construction above because by denition of D i = s i with s O(N P ) we have D i f = D i f 0 + g p i s. Therefore D if does only depend on f 0 and g i. We will select the polynomials f 0, g 1,..., g le+1 one at a time. For 0 i l e + 1, dene W i = V e {D 1 f =... = D i f = 0}. Then W le+1 V e,> d c is the set of points P V e of degree > d c where H m+1 m+1 f X may be singular. Because we want to show that for d the upper density of the set of polynomials f of degree d that have such a point is equal to zero, we will show by an induction argument that W le+1 V e,> d c is empty with probability 1 o(1). Again m+1 note that we do not have to intersect this with the hypersurface H f or demand f(p ) to be zero since V e,> d c, is already contained in H f. At rst, we will use the polynomials m+1 f 0, g 1,... g le to show that the dimension of W le is bounded. In a second claim we will show by using the polynomial g le+1 in the construction of f above that, if W le is nite, the next derivation D le+1 does not vanish for f at P with probability 1 o(1) as d. Claim If the polynomials f 0, g 1,..., g i for 0 i l e have been chosen such that dim(w i ) l e i holds, then the probability for dim(w i+1 ) l e i 1 is equal to 1 o(1) as d. The function of d represented by o(1) depends on V e and the derivations D i. Proof of Claim Let Y 1,..., Y s be the (l e i) - dimensional F q - irreducible components of (W i ) red. The degree of a hypersurface generated by a polynomial of degree d is equal to d; hence Bézout's theorem Lemma 2.17 yields s (deg V e )(deg D 1 f)... (deg D i f) = O(d i ) 31