Effective Bertini-type Theorem

• Effective Bertini-type Theorem is a quantitative framework that refines classical Bertini results by providing explicit control over invariants such as field of definition, degree, height, and genus.
• It employs iterative hyperplane sections and explicit candidate families, utilizing density formulas and controlled arithmetic to ensure smooth, well-behaved sections.
• The approach has significant applications in arithmetic geometry, enabling reductions of complex problems in abelian varieties to controlled settings on curves and Jacobians.

An effective Bertini-type theorem is a quantitative refinement of the classical Bertini principle that a general hyperplane section of a smooth variety is again well behaved. In effective forms, the hyperplane or hypersurface section is not only asserted to exist: the construction is accompanied by explicit control of invariants, a finite or open parameter space of admissible sections, a density formula, or a bound on the exceptional set. In "Bertini and Northcott" the effective content is arithmetic: one cuts a smooth projective variety by hyperplanes with coefficients in a prescribed infinite set of algebraic numbers while controlling the resulting curve’s field of definition, degree, height, and genus (Pazuki et al., 2020). Other works realize effectiveness through finite explicit families of candidate hyperplanes, density statements over finite fields, or open parameter loci in mixed characteristic (Rodak et al., 2021).

1. From generic Bertini to effective Bertini

The classical Bertini philosophy, as presented in several of the cited works, says that a “general” hyperplane section of a smooth variety has the expected regularity or smoothness property. Effective Bertini-type theorems replace this generic-existential formulation by one of several stronger outputs: an explicit family of candidate sections, an explicit open subset of parameter space, a density formula, or quantitative control of invariants of the section itself (Rodak et al., 2021).

In the arithmetic-geometric formulation of "Bertini and Northcott", the decisive strengthening is not only that a smooth geometrically irreducible curve can be obtained by repeated hyperplane section, but that its field of definition, degree, projective height, and genus are all bounded explicitly in terms of the ambient variety and the Northcott complexity of the coefficient set (Pazuki et al., 2020). This use of Northcott data is distinctive: the hyperplanes are required to have coefficients in a prescribed infinite set $S\subset \overline{\mathbf Q}$ with finite Northcott number $m(S)$.

Across the broader literature, the meaning of effectiveness varies with the ambient problem. Over $\mathbf C$, one finds finite explicit families of candidate hyperplanes or linear subspaces such that at least one member satisfies the Bertini conclusion (Rodak et al., 2021). Over finite fields, effectiveness often takes the form of explicit asymptotic densities, bounded extension degrees, or bounds for the number of bad sections (Wutz, 2016). In mixed characteristic, effectiveness is frequently encoded by a Zariski dense open subset of a projective parameter space whose lifts yield normal or regular hyperplane sections (Ochiai et al., 2011).

2. The Northcott-controlled theorem of Bertini and Northcott

The central statement in "Bertini and Northcott" is Theorem 1.3. Let $K\subset \overline{\mathbf Q}$ be a subfield, let $X\subset \mathbf P^N_K$ be a smooth closed subvariety with $\dim X\ge 2$, and let $S\subset \overline{\mathbf Q}$ be an infinite set with finite Northcott number $m(S)$. Then there exists a finite subset $s\subset S$ and a non-singular geometrically irreducible curve

$C\subset X$

defined over the extension $m(S)$0, such that

$m(S)$1

$m(S)$2

and

$m(S)$3

This theorem is the paper’s main geometric input, and it is the bridge that reduces arithmetic questions on general abelian varieties to Jacobians (Pazuki et al., 2020).

The theorem is effective in several simultaneous senses. The curve is defined over a controlled finite extension generated by finitely many elements of $m(S)$4. Its degree does not increase. Its height increases only by an explicit linear error term. Its genus is bounded solely in terms of $m(S)$5. The structure of the conclusion is therefore stronger than a generic Bertini theorem over an algebraically closed field: the section is not merely “general,” but arithmetically and geometrically controlled.

The appearance of the Northcott number $m(S)$6 is essential. The height bound depends on $m(S)$7, while the field of definition is constrained by a finite subset $m(S)$8. This places the hyperplane coefficients under simultaneous arithmetic and geometric control, and it is precisely this feature that makes the theorem suitable for downstream comparisons of projective heights and Faltings heights (Pazuki et al., 2020).

3. Quantitative hyperplane cutting and proof architecture

The proof proceeds by iterating hyperplane sections. First one uses classical Bertini to choose a hyperplane section that remains smooth and geometrically irreducible. One then refines this by ensuring that the hyperplane coefficients lie in $m(S)$9. The key observation is that since $\mathbf C$0 is infinite, one can choose a hyperplane in the open Bertini locus with coefficients in $\mathbf C$1 (Pazuki et al., 2020).

The quantitative engine for height control is Proposition 3.3, a result of Rémond giving a height-and-degree estimate for intersections of a variety with hypersurfaces defined by polynomials of bounded degree and bounded height. In the hyperplane case $\mathbf C$2, this yields Proposition 3.4: $\mathbf C$3 with

$\mathbf C$4

provided $\mathbf C$5 is defined by coefficients in $\mathbf C$6 of controlled height. The paper uses the estimates

$\mathbf C$7

for the linear form defining the hyperplane (Pazuki et al., 2020).

By repeating the hyperplane-cutting procedure $\mathbf C$8 times, Corollary 3.5 produces the desired curve, and at each step the height increment is controlled. After $\mathbf C$9 steps, the total error becomes

$K\subset \overline{\mathbf Q}$0

The genus estimate is obtained via a separate geometric argument: by repeated hyperplane sections and Castelnuovo-type bounds, one gets

$K\subset \overline{\mathbf Q}$1

Thus the proof combines an open Bertini locus, arithmetic control on coefficients, and intersection-theoretic height estimates into a single hyperplane-section machine (Pazuki et al., 2020).

A different model of effectiveness appears in "Effective Bertini theorem and formulas for multiplicity and the local Łojasiewicz exponent". There the goal is to replace “a general hyperplane works” by a finite explicit family of hyperplanes or linear spaces such that at least one member intersects transversally. The paper gives explicit bounds such as

$K\subset \overline{\mathbf Q}$2

$K\subset \overline{\mathbf Q}$3

and

$K\subset \overline{\mathbf Q}$4

and uses these finite families to derive formulas for multiplicity and the local Łojasiewicz exponent (Rodak et al., 2021). The contrast is instructive: in one case effectiveness controls invariants of the section; in the other it controls the search space of sections.

4. Passage from abelian varieties to Jacobians

The arithmetic significance of the theorem in "Bertini and Northcott" lies in its application to abelian varieties. For principally polarized abelian varieties satisfying the auxiliary condition $K\subset \overline{\mathbf Q}$5, meaning semi-stable reduction plus a theta embedding of level $K\subset \overline{\mathbf Q}$6 coming from the 16th power of a symmetric theta divisor, Proposition 1.6 asserts that for a principally polarized abelian variety $K\subset \overline{\mathbf Q}$7 of dimension $K\subset \overline{\mathbf Q}$8, there exists a finite $K\subset \overline{\mathbf Q}$9 and a smooth geometrically irreducible curve $X\subset \mathbf P^N_K$0, defined over $X\subset \mathbf P^N_K$1, such that there is a closed immersion

$X\subset \mathbf P^N_K$2

over the field of definition of $X\subset \mathbf P^N_K$3, the genus satisfies

$X\subset \mathbf P^N_K$4

and

$X\subset \mathbf P^N_K$5

for an explicit computable function $X\subset \mathbf P^N_K$6 (Pazuki et al., 2020).

The argument embeds $X\subset \mathbf P^N_K$7 into projective space via the theta embedding

$X\subset \mathbf P^N_K$8

applies Theorem 1.3 to obtain the curve $X\subset \mathbf P^N_K$9, and then converts the projective height bound for $\dim X\ge 2$0 into a Faltings-height bound for $\dim X\ge 2$1. Two comparison results are used in this conversion: $\dim X\ge 2$2 and

$\dim X\ge 2$3

A further input is Lemma 4.1: $\dim X\ge 2$4 Together these inequalities transfer the geometric control on $\dim X\ge 2$5 to arithmetic control on $\dim X\ge 2$6 (Pazuki et al., 2020).

The structural step that turns this into a reduction principle is Lemma 5.1: if

$\dim X\ge 2$7

then

$\dim X\ge 2$8

for some abelian variety $\dim X\ge 2$9. This is the input for Theorem 1.5, the abstract reduction principle. If a family $S\subset \overline{\mathbf Q}$0 of invariants is $S\subset \overline{\mathbf Q}$1-admissible and one has lower bounds of the form

$S\subset \overline{\mathbf Q}$2

for the Jacobians produced by the theorem, then one deduces

$S\subset \overline{\mathbf Q}$3

with

$S\subset \overline{\mathbf Q}$4

The paper applies this strategy to bounds for bad reduction and to rank. In particular, taking $S\subset \overline{\mathbf Q}$5, one gets a reduction of rank bounds to Jacobians, leading to Corollary 1.9 and Corollary 1.10; in the Honda-conjecture direction, a Jacobian bound of the form

$S\subset \overline{\mathbf Q}$6

implies

$S\subset \overline{\mathbf Q}$7

for all extensions $S\subset \overline{\mathbf Q}$8 of a suitable field of definition (Pazuki et al., 2020).

5. Variants across arithmetic, finite-field, analytic, and mixed-characteristic settings

The effective Bertini paradigm appears in several mathematically distinct forms.

Setting	Effective output	Representative paper
Smooth projective $S\subset \overline{\mathbf Q}$9, coefficients in $m(S)$0	Explicit control of field of definition, degree, height, and genus of a curve $m(S)$1	(Pazuki et al., 2020)
$m(S)$2, irreducible smooth cones or projective varieties	Finite explicit family of linear sections or hyperplanes, with at least one transversal member	(Rodak et al., 2021)
Finite fields, hypersurfaces containing a fixed subscheme	Explicit density formula for smooth sections satisfying local conditions	(Wutz, 2016)
Finite-field pencils of hypersurfaces	Finite extension $m(S)$3, depending only on $m(S)$4 and $m(S)$5, after which a hyperplane is transverse to every member of the pencil	(Asgarli et al., 2020)
Hypersurfaces over $m(S)$6 with no linear factors	At most $m(S)$7 bad planes through a fixed line	(Beneish et al., 27 Aug 2025)
Mixed characteristic local or projective geometry	Zariski dense open subset of a parameter space whose lifts give normal or regular sections	(Ochiai et al., 2011)

Over finite fields, "Bertini theorems for hypersurface sections containing a subscheme over finite fields" proves that hypersurfaces containing a fixed closed subscheme $m(S)$8 and satisfying finitely many local conditions still meet a smooth quasi-projective scheme $m(S)$9 smoothly, provided the stratified dimension condition

$s\subset S$0

holds. The asymptotic density is given explicitly by

$s\subset S$1

This is a finite-field analogue of Altman–Kleiman and a generalization of Poonen’s theorem (Wutz, 2016).

Another finite-field form appears in "A Bertini type theorem for pencils over finite fields". Under condition $s\subset S$2 and the separability hypothesis

$s\subset S$3

there exists a finite extension $s\subset S$4 such that over every finite field $s\subset S$5 there is a $s\subset S$6-hyperplane simultaneously transverse to every member of the pencil. The extension depends only on $s\subset S$7 and $s\subset S$8, not on the particular pencil (Asgarli et al., 2020).

A sharper local counting form appears in "Effective Bertini theorems and zeros of $s\subset S$9-adic forms of degree 7". For a hypersurface $C\subset X$0 over $C\subset X$1 with no linear factors over $C\subset X$2, the paper proves that there are at most

$C\subset X$3

planes through a fixed line that are bad for the relevant linear-factor condition through a marked point. This quantitative Bertini theorem is then used in the degree-$C\subset X$4 Artin application (Beneish et al., 27 Aug 2025).

In mixed characteristic, "Bertini theorem for normality on local rings in mixed characteristic" proves that if $C\subset X$5 is a complete local domain of mixed characteristic with $C\subset X$6 normal, $C\subset X$7, and $C\subset X$8 infinite, then there exists a Zariski dense open subset $C\subset X$9 such that for every point $m(S)$00, the quotient

$m(S)$01

is a normal domain of mixed characteristic $m(S)$02 (Ochiai et al., 2011). A global projective analogue is proved in "Normal hyperplane sections of normal schemes in mixed characteristic", where one obtains a Zariski-dense open subset

$m(S)$03

such that $m(S)$04 is normal and flat over $m(S)$05 for every

$m(S)$06

(Horiuchi et al., 2017).

Analytic and relative multiplier-ideal versions show that effectiveness need not be algebraic in form. For a free linear system $m(S)$07 on a compact complex manifold and a quasi-plurisubharmonic function $m(S)$08, "Injectivity theorem for pseudo-effective line bundles and its applications" proves that the set

$m(S)$09

is dense in $m(S)$10 in the classical topology (Fujino et al., 2016). The relative version in "Relative Bertini type theorem for multiplier ideal sheaves" proves the existence of a dense subset of hyperplanes with the same restriction property, and states that the complement is analytically meagre (Fujino, 2017).

Other variants shift the preserved property away from smoothness. Over real closed fields, "Generalized connectedness and Bertini-type theorems over real closed fields" proves the existence of a nonempty open family of hypersurface sections preserving formal reality and integrality, under the paper’s curve-level conjectural hypothesis (Ouyang et al., 5 Nov 2025). "Bertini theorems admitting base changes" formulates Bertini theorems over excellent rings for singularity classes stable under suitable base change (Tanaka, 2022). "Bertini type results and their applications" proves Bertini criteria for pullbacks of general hyperplanes or linear spaces under a dominant morphism in terms of the rank strata

$m(S)$11

with conclusions such as preservation of $m(S)$12 under codimension conditions on those strata (Biswas et al., 2023).

6. Failures, boundary cases, and mathematical role

Effective Bertini statements do not simply strengthen a single classical theorem; they also expose the limits of Bertini phenomena. Over finite fields, the literature repeatedly emphasizes that the classical infinite-field formulation fails in its naive form. In the pencil setting, a base extension may be necessary, and the need for a finite extension $m(S)$13 is part of the theorem itself (Asgarli et al., 2020).

The failure of Bertini-type restriction can be sharp for singularity-theoretic invariants. "Counterexamples To Bertini Theorems for Test Ideals" proves that for test ideals in characteristic $m(S)$14, the expected analogues

$m(S)$15

and

$m(S)$16

are false in general once $m(S)$17, even for general hyperplanes (Bydlon, 2016). By contrast, "Bertini theorems for $m(S)$18-singularities" proves that strongly $m(S)$19-regular, sharply $m(S)$20-pure, and $m(S)$21-pure singularities do satisfy Bertini-type theorems under the stated separability hypotheses, while $m(S)$22-injective singularities fail even basic Bertini properties (Schwede et al., 2011). This contrast shows that effectiveness is inseparable from the choice of preserved structure.

Even in positive results, the good locus may have unexpected topology. In the multiplier-ideal setting, the dense set of good divisors need not be Zariski open and may fail to be an intersection of countably many nonempty Zariski open sets (Fujino et al., 2016). In the real closed field setting, the relevant property is no longer smoothness alone, but the preservation of formal reality and integrality, detected through sign-definiteness on generalized connected subsets (Ouyang et al., 5 Nov 2025).

Within arithmetic geometry, the role of effective Bertini is especially structural. In "Bertini and Northcott", the theorem is not only a curve-construction device; it is the mechanism that funnels height and rank questions for arbitrary abelian varieties into the Jacobian setting (Pazuki et al., 2020). In computational algebraic geometry, effective Bertini can instead be the passage from generic transversality to explicit finite search procedures (Rodak et al., 2021). Over finite fields, it can be the bridge from asymptotic genericity to actual counting statements or bounded field extensions (Wutz, 2016).

In this sense, the effective Bertini-type theorem is less a single theorem than a recurring pattern: one starts from a Bertini statement about generic sections and then replaces genericity by explicit arithmetic, geometric, topological, or algorithmic control. The theorem of "Bertini and Northcott" is a particularly consequential instance because it combines all four controls that are decisive for arithmetic applications—field of definition, degree, height, and genus—within one hyperplane-section construction (Pazuki et al., 2020).