The singular locus of hypersurface sections containing a closed subscheme over finite fields
aa r X i v : . [ m a t h . N T ] A p r THE SINGULAR LOCUS OF HYPERSURFACE SECTIONSCONTAINING A CLOSED SUBSCHEME OVER FINITE FIELDS
FRANZISKA WUTZ
Abstract.
We prove that there exist hypersurfaces that contain a givenclosed subscheme Z of the projective space over a finite field and intersecta given smooth scheme X off of Z smoothly, if the intersection V = Z ∩ X issmooth. Furthermore, we can give a bound on the dimension of the singularlocus of the hypersurface section and prescribe finitely many local conditionson the hypersurface. This is an analogue of a Bertini theorem of Bloch overfinite fields and is proved using Poonen’s closed point sieve. We also show asimilar theorem for the case where V is not smooth. Introduction
The classical Bertini theorem over infinite fields guarantee the existence of asmooth hypersurface section for a smooth subscheme X of the projective space.For a given closed subvariety Z ⊆ X , Bloch showed that the hypersurface can beassumed to contain Z , if V = Z ∩ X is smooth and p > dim X , where < p is thecodimension of Z in X ([Blo71]). If the condition on the dimension is not fulfilled,this does not hold anymore. But Bloch showed that there still exists a hypersurfacesection Y of X such that the singular locus of Y is smooth, contained in Z and ofdimension n − p . Over finite fields, Poonen proved an analogue for the case where V is smooth and V < dim X ([Poo08]); here Z is a closed subscheme of theprojective space. In this paper, we generalize this and show an analogue over finitefields of Bloch’s result for V ≥ dim X , where we also add the possibility toimpose finitely many local conditions on the hypersurface (Theorem 2.1).Furthermore, we show that if the intersection V is not smooth and V =dim X , there exists a hypersurface H containing Z and intersecting X off of Z smoothly, such that the singular locus of H ∩ X consists at most of finitely manypoints. We use Poonen’s closed point sieve (cf. [Poo04]) to prove this, and alsoprescribe finitely many local conditions on the hypersurface (Theorem 3.1).We use the following notation: let F q be a finite field of q = p a elements. Let S = F q [ x , . . . , x n ] and let S d ⊆ S the F q -subspace of homogeneous polynomialsof degree d . Let S homog = S d ≥ S d and let S ′ d be the set of all polynomials in F q [ x , . . . x n ] of degree ≤ d .For a scheme X of finite type over F q , we define the zeta function of X as ζ X ( s ) := Y P ∈ X closed (1 − q − s deg P ) − . This product converges for Re ( s ) > dim X .Let Z be a fixed closed subscheme of P n = P n F q . For d ∈ Z ≥ let I d be the F q -subspace of polynomials f ∈ S d vanishing on Z , and I homog = S d ≥ I d . For apolynomial f ∈ I d let H f = Proj( S/ ( f )) be the hypersurface defined by f .As in Poonen’s paper ([Poo08]), we want to measure the density of a set ofpolynomials within the space of polynomials vanishing on Z , and define the density elative to the closed subscheme Z of a subset P ⊆ I homog to be µ Z ( P ) := lim d →∞ P ∩ I d ) I d , if the limit exists. Acknowledgments.
The author would like to thank Uwe Jannsen heartily forhis support and help. Furthermore, the author gratefully acknowledges supportfrom the Deutsche Forschungsgemeinschaft through the SFB 1085
Higher Invari-ants . 2.
Results for V smooth
Following [Poo04], we want to prescribe finitely many local conditions on thehypersurface. For this, let Y be a finite subscheme of P n . For a polynomial f ∈ I d we define f (cid:12)(cid:12) Y ∈ H ( Y, I Z · O Y ) as follows: on each connected component Y i of Y let f (cid:12)(cid:12) Y be equal to the restriction of x − dj f to Y i , where j = j ( i ) is the smallest j ∈ { , , . . . , n } such that the coordinate x j is invertible on Y i . Theorem 2.1.
Let X be a quasi-projective subscheme of P n and Y a finite sub-scheme of P n such that U = X − ( X ∩ Y ) is smooth of dimension m ≥ over F q . Let Z be a closed subscheme of P n such that Z ∩ Y = ∅ . Let T ⊆ H ( Y, I Z · O Y ) . As-sume that the intersection V := Z ∩ U is smooth of dimension l such that l ≤ m + k where k ∈ Z ≥ . Define P = { f ∈ I homog : H f ∩ U is smooth of dimension m − at all points P ∈ U − V,f (cid:12)(cid:12) Y ∈ T and dim( H f ∩ U ) sing ≤ k } . Then µ Z ( P ) = T H ( Y, I Z · O Y ) ζ U − V ( m + 1) − . In particular, there exists a hypersurface H containing Z , defined by a poly-nomial f such that f (cid:12)(cid:12) Y ∈ T and such that the singular locus of the hypersurfacesection is contained in Z and at most of dimension k . The existence of such ahypersurface has been shown for infinite fields as well ([Blo71], Proposition 1.2). Remark . The proof is a closed point sieve as introduced in [Poo04] and isorganized as follows: First we look at closed points of low degree, defining therelevant set of polynomials to be P r = { f ∈ I homog : H f ∩ U is smooth of dimension m − at all points P ∈ ( U − V )
Suppose m ⊆ O U is the ideal sheaf of a closed point P ∈ U . Let C ⊆ U be the closed subscheme whose ideal sheaf is m ⊆ O U . Then for any d ∈ Z ≥ we have H ( X, I Z · O C ( d )) = (cid:26) q ( m − l ) deg P , if P ∈ V,q ( m +1) deg P , else . Proof.
This is Lemma 2.2 of [Poo08]; the condition on the dimension is not neededhere. (cid:3)
Lemma 2.4 (Singularities of low degree) . For P r defined as in Remark 2.2, µ Z ( P r ) = T H ( Y, I Z · O Y ) Y P ∈ ( U − V ) The proof is parallel to the one of Lemma 2.4 in [Wut16], just use Lemma 2.3above and ( U − V ) This is Lemma 4.2. of [Poo08]. The assumption m > l is not used in theproof. (cid:3) Lemma 2.8 (Singularities of high degree on V) . For Q V defined as in Remark 2.2, µ Z ( Q V ) = 0 . roof. The proof of this claim uses the induction argument used by Poonen in theproof of Lemma 2.6 in [Poo04], and up to the definition of the polynomials g i , it issimilar to the one of Lemma 4.3 of [Poo08].We may assume U is contained in A n = { x = 0 } ⊆ P n . Dehomogenize by setting x = 1 , and identify S d with the space of polynomials S ′ d ⊆ F q [ x , . . . , x n ] = A ofdegree ≤ d and I d with a subspace I ′ d ⊆ S ′ d .Let P be a closed point of U . Choose a system of local parameters t , . . . , t n ∈ A at P on A n such that t m +1 = . . . = t n = 0 defines U locally at P , and t = . . . = t m − l = t m +1 = . . . = t n = 0 defines V locally at P . We may assume in additionthat t , . . . , t m − l vanish on Z (cf. [Poo08], Lemma 4.3). By definition, dt , . . . , dt n are a O A n ,P -basis for the stalk Ω A n | F q ,P . Let ∂ , . . . , ∂ n be the dual basis of thestalk of the tangent sheaf T A n | F q at P . Choose s ∈ A satisfying s ( P ) = 0 to cleardenominators such that D i = s∂ i defines a global derivation A → A for all i . Thenthere exists a neighbourhood N P of P in A n such that N P ∩{ t m +1 = . . . = t n = 0 } = N P ∩ U , Ω N P | F q = ⊕ ni =1 O N P dt i and s ∈ O ( N P ) ∗ . We can cover U with finitelymany N P , so we may assume that U is contained in N P for some P . For f ∈ I ′ d ∼ = I d ,the hypersurface section H f ∩ U fails to be smooth of dimension m − at somepoint P ∈ V if and only if ( D f )( P ) = . . . = ( D m f )( P ) = 0 .Let τ = max i (deg t i ) , γ = ⌊ ( d − τ ) /p ⌋ , and ν = ⌊ d/p ⌋ . If we choose f ∈ I ′ d , g ∈ S ′ γ , . . . , g l − k ∈ S ′ γ uniformly at random, then f = f + g p t + . . . + g pl − k t l − k is a random element of I ′ d , because of f and since by assumption l − k ≤ m − l .Note that we cannot define more polynomials g l − k +1 , . . . , g m as in [Poo08], sinceour condition on the dimension would not yield a polynomial in I ′ d .For i = 0 , . . . , l − k we define W i = V ∩ { D f = . . . = D i f = 0 } . By definition of the D i , this subscheme depends only on f , g , . . . , g i .As in Lemma 2.6 of [Poo04], one can show that for ≤ i ≤ l − k − , conditioned ona choice of f , g , . . . , g i for which dim( W i ) ≤ l − i , the probability that dim( W i +1 ) ≤ l − i − is − o (1) as d → ∞ .It follows that for i = 0 , . . . , l − k we have Prob (dim W i ≤ l − i ) = 1 − o (1) as d → ∞ and thus W l − k is at most of dimension k with probability − o (1) , which iswhat we claimed, since W l − k contains the points where H f ∩ U is not smooth. (cid:3) Proof of Theorem 2.1. We have the inclusions P ⊆ P r ⊆ P ∪ Q medium r ∪ Q high U − V ∩ Q V : The first inclusion is clear. For the second, let f ∈ P r . If f is not in P , then bydefinition, either H f ∩ U is not smooth at some point P ∈ U − V , or dim( H f ∩ U ) sing ≥ k + 1 . In the first case, this point P must be of some degree ≥ r , since f ∈ P r , and f ∈ Q medium r ∪ Q high U − V . For the second case, if H f ∩ U is smooth at allpoints in U − V , then the singular locus of the hypersurface section is completelycontained in V and f ∈ Q V . If H f ∩ U is not smooth at some point in U − V , thenwe are in the first case again.By Lemma 2.4, 2.6, 2.7 and 2.8, µ Z ( P ) = lim r →∞ µ Z ( P r ) = T H ( Y, I Z · O Y ) ζ U − V ( m + 1) − . (cid:3) . Results for V not smooth In the condition on the dimension we will need for the analogue of Theorem 2.1in the case where V is not smooth, the embedding dimension of a scheme X at apoint P will occur naturally. It is defined as e ( P ) = dim κ ( P ) (Ω X | F q ( P )) . Let X e = X (Ω X | F q , e ) be the subscheme such that a scheme morphism f : T → X factors through X e if and only if f ∗ Ω X | F q is locally free of rank e . Then X e is the locally closedsubscheme of X where the embedding dimension of X is e .The condition on the dimension for V non smooth in the Bertini smoothnesstheorem (cf. [Wut16], [Gun15]) for hypersurface sections containing a closed sub-scheme is max { e + dim V e } < m instead of < dim X for the case V smooth([Poo08]). Hence, one would expect an analogue of Theorem 2.1 in the case V nonsmooth to hold for max { e + dim V e } ≤ m + k . But using our methods, we cannotprove this exact analogue of Theorem 2.1 where V is not smooth, since we can onlybound the dimension of the bad points of high degree in each V e and not in V . Theonly case that still works is k = 0 : Theorem 3.1. Let X be a quasi-projective subscheme of P n and Y a finite sub-scheme of P n such that U = X − ( X ∩ Y ) is smooth of dimension m ≥ over F q .Let Z be a closed subscheme of P n such that Z ∩ Y = ∅ . Let T ⊆ H ( Y, I Z · O Y ) .Assume that for V = Z ∩ U we have max { e + dim V e } ≤ m . Define P = { f ∈ I homog : f (cid:12)(cid:12) Y ∈ T, ( H f ∩ U ) sing ⊆ Z and such that dim( H f ∩ U ) sing ≤ } . Then µ Z ( P ) = T H ( Y, O Y ) ζ U − V ( m + 1) − . The proof again uses the closed point sieve. We define P r , Q medium r and Q high U − V asin in the previous section. The proofs for 2.4, 2.6 and 2.7 do not need the conditionson the smoothness or dimension of V , and thus we only have to show the followinglemma on singularities of high degree: Lemma 3.2 (Singularities of high degree for V not smooth) . Define Q = { f ∈ I homog : dim( H f ∩ U ) sing ≥ } . Then µ Z ( Q ) = 0 .Proof. We may assume U is contained in A n = { x = 0 } ⊆ P n . As there are onlyfinitely many V e , it is enough to bound the probability that a polynomial gives asingular locus of dimension ≥ if we intersect this singular locus with V e .Let P be a closed point of V e . Since U is smooth, we can choose a system of localparameters t , . . . , t n ∈ A on A n such that t m +1 = . . . = t n = 0 defines U locally at P . Then dt , . . . , dt n are a basis for the stalk of Ω A n | F q at P and dt , . . . , dt m are abasis for the stalk of Ω U | F q at P . Using the exact sequence ([Har93], Section II.8) I V / I V → Ω X ⊗ O V φ → Ω V → , we show that dt , . . . , dt m − e form a basis of thekernel of φ at P and dt m − e +1 , . . . , dt m a basis of Ω V | F q ,P ⊗ O V e ,P . In particular, t , . . . , t m − e all vanish on V, since Ω V | F q ⊗ κ ( P ) ∼ = m V,P / m V,P . Again, we mayalso assume in addition that t , . . . , t m − e vanish on Z .As in Lemma 2.8, we define ∂ i , s and D i such that for f ∈ I ′ d ∼ = I d , the hyper-surface section H f ∩ U fails to be smooth of dimension m − at some point P ∈ V e if and only if ( D f )( P ) = . . . = ( D m f )( P ) = 0 . et τ = max ≤ i ≤ l e − (deg t i ) and γ = ⌊ ( d − τ ) /p ⌋ where l e = dim V e . We select f ∈ I ′ d and g ∈ S ′ γ , . . . , g l e − ∈ S ′ γ uniformly and independently at random. Thenthe distribution of f = f + g p t + . . . + g pl e − t l e is uniform over I ′ d , since by assumption, l e ≤ m − e .Now everything works as in the proof of Lemma 2.8: for i = 0 , . . . , l e we define W i = V e ∩ { D f = . . . = D i f = 0 } , depending only on f , g , . . . , g i . Using the induction as in Lemma 2.6 of [Poo04],we show that W l e is at most of dimension 0 with probability − o (1) , which is whatwe claimed. (cid:3) Proof of Theorem 3.1. As in Section 2, we have the inclusions P ⊆ P r ⊆ P ∪ Q medium r ∪ Q high U − V ∩ Q . By 2.6, 2.7 and 3.2, the error in our approximation is negligible, and Lemma 2.4yields Theorem 3.1. (cid:3) References [Blo71] S. Bloch, 1971, Ph.D. thesis, Columbia University.[Gun15] J. Gunther, Random hypersurfaces and embedding curves in surfaces over finitefields , arXiv:1510.04733v1.[Har93] R. Hartshorne, Algebraic Geometry , Graduate Texts in Mathematics ,Springer, New York, 1993.[Poo04] B. Poonen, Bertini theorems over finite fields , Annals of Mathematics (2004), 1099-1127.[Poo08] B. Poonen, Smooth hypersurface sections containing a given subscheme over afinite field , Math. Research Letters , no. 2, 265-271.[Wut16] F. Wutz, Bertini theorems for smooth hypersurface sections containing a sub-scheme over finite fields , arXiv:1611.09092., arXiv:1611.09092.