## 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 $$K/\mathbb Q$$.

### MathOverflow Questions:

Bertini’s theorem over non-algebraically closed field

### MSC:

 14G15 Finite ground fields in algebraic geometry 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14N05 Projective techniques in algebraic geometry 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties 11G25 Varieties over finite and local fields

### Keywords:

Bertini theorem; finite field; arithmetic scheme

### Citations:

Zbl 1016.11022; Zbl 1072.14513; Zbl 1020.11044
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