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**Bertini theorems over finite fields.**
*(English)*
Zbl 1084.14026

The paper under review contains several arithmetic counterparts of classical Bertini theorems: the goal is to understand whether a certain property of a variety embedded into a projective space remains true for its sufficiently general hyperplane section.

The first case is that of a finite ground field. This case became an object of special attention after the paper of N. Katz [Math. Res. Lett. 6, 613–624 (1999); Corrections 8, 689–691 (2001; Zbl 1016.11022)], where, among other things, there appeared an explicit example of a smooth hypersurface over a finite field without smooth hyperplane sections. The paper under review presents results of two opposite kinds: Theorem 3.1 (“Anti-Bertini Theorem”) strengthens the above mentioned counter-example, saying that given a finite field \(F\) and integers \(n\geq 2\), \(d\geq 1\), there exists a smooth projective geometrically integral hypersurface \(X\) in \({\mathbb P}^n\) over \(F\) such that no hyperplane section of \(X\) by a hypersurface of degree \(d\) is smooth of dimension \(n-2\). Surprisingly enough, this result is a corollary of a “positive” one (Theorem 1.1), saying that a smooth hypersurface section of an \(m\)-dimensional subvariety \(X\subset {\mathbb P}^n\) does exist if one is allowed to use hypersurfaces of any degree \(d\); moreover, this can be done with probability \(1/\zeta_X(m+1)\). As another consequence, the author gets positive answers to some questions posed in the above cited paper of N. Katz (this was independently done in O. Gabber [Geom. Funct. Anal. 11, 1192–1200 (2001; Zbl 1072.14513)]). Here is one of such results concerning “space-filling curves” (Corollary 3.5): if \(X\) is a smooth, projective, geometrically integral variety of dimension \(m\geq 1\) over a finite field \(F\) and \(E\) is a finite extension of \(F\), then there exists a smooth, projective, geometrically integral curve \(Y\) on \(X\) such that \(Y(E)=X(E)\). As yet another curious application, one can mention explicit formulas giving the probability for a projective plane curve over a finite field to be nonsingular, or singular with at worst nodes as singularities.

The second case treated in the paper under review is that of regular quasiprojective schemes \(X\) over \(\mathbb Z\). As before, the author looks for conditions under which one can guarantee the existence of a regular hyperplane section. His main result here (Theorem 5.1), which is a precise analogue of the above mentioned Theorem 1.1, is proved under two additional assumptions: the \(abc\) conjecture and another one (which is true, for example, provided the closure of the generic fibre of \(X\) in \({\mathbb P}^n_{\mathbb Q}\) has at most isolated singularities). Note that one has to be careful trying to replace regularity with smoothness: Theorem 5.13 shows that if \(X/{\mathbb Z}\) is smooth of relative dimension \(m\), then the set of homogeneous polynomials yielding smooth hypersurface sections of \(X\) of relative dimension \(m-1\) has zero density. The author also discusses a counter-example of N. Fakhruddin of a smooth \(\mathbb Z\)-scheme without smooth hypersurface sections of relative dimension \(m-1\), as well as positive results of P. Autissier [Ann. Inst. Fourier 51, 1507–1523 (2001); corrigendum ibid. 52, No. 1, 303–304 (2002; Zbl 1020.11044)] (of slightly different nature) where one is allowed to replace \(\mathbb Z\) with \(O_K\), the ring of integers of a finite extension \(K/\mathbb Q\).

The first case is that of a finite ground field. This case became an object of special attention after the paper of N. Katz [Math. Res. Lett. 6, 613–624 (1999); Corrections 8, 689–691 (2001; Zbl 1016.11022)], where, among other things, there appeared an explicit example of a smooth hypersurface over a finite field without smooth hyperplane sections. The paper under review presents results of two opposite kinds: Theorem 3.1 (“Anti-Bertini Theorem”) strengthens the above mentioned counter-example, saying that given a finite field \(F\) and integers \(n\geq 2\), \(d\geq 1\), there exists a smooth projective geometrically integral hypersurface \(X\) in \({\mathbb P}^n\) over \(F\) such that no hyperplane section of \(X\) by a hypersurface of degree \(d\) is smooth of dimension \(n-2\). Surprisingly enough, this result is a corollary of a “positive” one (Theorem 1.1), saying that a smooth hypersurface section of an \(m\)-dimensional subvariety \(X\subset {\mathbb P}^n\) does exist if one is allowed to use hypersurfaces of any degree \(d\); moreover, this can be done with probability \(1/\zeta_X(m+1)\). As another consequence, the author gets positive answers to some questions posed in the above cited paper of N. Katz (this was independently done in O. Gabber [Geom. Funct. Anal. 11, 1192–1200 (2001; Zbl 1072.14513)]). Here is one of such results concerning “space-filling curves” (Corollary 3.5): if \(X\) is a smooth, projective, geometrically integral variety of dimension \(m\geq 1\) over a finite field \(F\) and \(E\) is a finite extension of \(F\), then there exists a smooth, projective, geometrically integral curve \(Y\) on \(X\) such that \(Y(E)=X(E)\). As yet another curious application, one can mention explicit formulas giving the probability for a projective plane curve over a finite field to be nonsingular, or singular with at worst nodes as singularities.

The second case treated in the paper under review is that of regular quasiprojective schemes \(X\) over \(\mathbb Z\). As before, the author looks for conditions under which one can guarantee the existence of a regular hyperplane section. His main result here (Theorem 5.1), which is a precise analogue of the above mentioned Theorem 1.1, is proved under two additional assumptions: the \(abc\) conjecture and another one (which is true, for example, provided the closure of the generic fibre of \(X\) in \({\mathbb P}^n_{\mathbb Q}\) has at most isolated singularities). Note that one has to be careful trying to replace regularity with smoothness: Theorem 5.13 shows that if \(X/{\mathbb Z}\) is smooth of relative dimension \(m\), then the set of homogeneous polynomials yielding smooth hypersurface sections of \(X\) of relative dimension \(m-1\) has zero density. The author also discusses a counter-example of N. Fakhruddin of a smooth \(\mathbb Z\)-scheme without smooth hypersurface sections of relative dimension \(m-1\), as well as positive results of P. Autissier [Ann. Inst. Fourier 51, 1507–1523 (2001); corrigendum ibid. 52, No. 1, 303–304 (2002; Zbl 1020.11044)] (of slightly different nature) where one is allowed to replace \(\mathbb Z\) with \(O_K\), the ring of integers of a finite extension \(K/\mathbb Q\).

Reviewer: Boris Kunyavskii (Ramat Gan)