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Bertini Theorems for Hypersurface Sections Containing a Subscheme Over Nite Elds

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Bertini Theorems for Hypersurface Sections Containing a Subscheme Over Nite Elds 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 Promotionsgesuch eingereicht am 04. November 2014. 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 Contents Introduction1 1 Scheme-theoretic intersections and embedding dimension5 2 Smooth hypersurface sections containing a closed subscheme over a nite eld 11 2.1 Singular points of low degree......................... 16 2.2 Singular points of medium degree...................... 24 2.3 Singular points of high degree........................ 26 3 Bertini with Taylor conditions 37 References 42 Introduction n Bertini theorems say that if a scheme X ⊆ P 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 be an innite eld and n be a k X ⊆ Pk quasi-projective smooth scheme. Then there exists a hyperplane H such that the inter- section H \ X is smooth. This can be shown in the following way. We have a parameterization of the hyperplanes in n by the dual projective space n _: a point corresponds to the Pk (Pk ) a = (a0 : ::: : an) hyperplane given by the equation a0x0+:::+anxn = 0, where xi denote the homogeneous coordinates of the projective space n. Then for any eld , there is a dense Zariski Pk k open set n _ parameterizing the hyperplanes that intersect smoothly. If is UX ⊆ (Pk ) X k innite as in Theorem 0.1, the set of -rational points is non-empty, since n UX (k) k Pk (k) is a Zariski dense set in n. Hence we get the hyperplane we wanted. Pk 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 UX 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. n Theorem 0.2 ([Poo04] Theorem 1.1). Let X be a quasi-projective subscheme of P 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 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. The goal of this project was to show that there exists such an analogue. 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 = X(Ω1 ; e) of the attening stratication of a scheme X for the rank of the e Xj Fq dierential sheaf Ω1 , i.e. the locus in X where Ω1 has rank e. Xj Fq Xj Fq The second section contains the main result of this thesis, the requested analogue of Theorem 0.2: n Theorem 0.3. Let X be a quasi-projective subscheme of P that is smooth of dimension over a nite eld . Let be a closed subscheme of n, and let . m ≥ 0 Fq Z P V = Z \ X Assume max fe + dim Veg < m and Vm = ;, where Ve are the subschemes of the 0≤e≤m−1 attening stratication of V for the rank of the dierential sheaf Ω1 . Then, for V j Fq 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 quasi- projective k-scheme, let Z be a subscheme of X and U a subscheme of Z. Assume the estimate 1 max dim(U(ΩZ jU; e)) + e < min(dimx(X)): e x2U Then there is a hypersurface section of X containing Z which is smooth along U and o the closure of Z in X. 2 If we take Z = U closed in X, this gives the analogue of Theorem 0.3 for innite elds, since the conditions max fe + dim Veg < m and Vm = ; 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 µZ (P) of a subset P of all homo- geneous polynomials in . Then we look at the set of all homogeneous Fq [X0;:::;Xn] P polynomials such that the hypersurface given by contains f 2 Fq [X0;:::;Xn] Hf f Z and intersects X smoothly. If the density µZ (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 1 µ (P) = : Z m−1 Q ζX−V (m + 1) ζVe (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 2 X, sieve out those polynomials f for which the intersection Hf \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 Pr 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 µZ (Pr) for r ! 1 gives the correct density. 3 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 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 −1 −1 X \ Z := X ×Y Z = i (Z) = j (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 IX and IZ , respectively. The intersection of X and Z is again a closed subscheme of Y and the ideal sheaf corresponding to it is given by IX\Z = IX + IZ ⊆ OY , where the sum of IX and IZ is the sheaf associated to the presheaf U 7! IX (U) + IZ (U) ⊆ OY (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 OY;P of a point P 2 Y given by the prime ideal p we have IX\Z;P = (a + b)p = ap + bp = (IX )P + (IZ )P = (IX + IZ )P : n In particular, if X is quasi-projective and Z a closed subscheme of P , 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 OX;P =IZ;P , where IZ is the sheaf of ideals of Z: To see this, we put .
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