A Bertini type theorem for pencils over finite fields

容许基变换的贝尔蒂尼定理

Bertini theorems admitting base changes

本文将经典贝尔蒂尼定理推广到优环情形，证明在扩大基域后线性系中合适成员的奇点性质保持稳定。

Sharp Bertini theorem for plane curves over finite fields

We prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_q$ with $d\leq q+1$, then there is an $\mathbb{F}_q$-line $L$ that intersects $C$ transversely. We also prove the same result for non-reflexive curves of degree $p+1$ and $2p+1$ where $q=p^{r}$.

An application of Bertini Theorem

Given an irreducible variety $X$ over a finite field, the density of hypersurfaces of varying degree $d$ intersecting $X$ in an irreducible subvariety is $1$, by a result of Charles and Poonen. In this note, we analyse the situation fixing $d=1$ and extend the base field instead of the degree $d$. We compute the probability that a random linear subspace of the right dimension intersects $X$ in a given number of points.

Connectedness Bertini Theorem via numerical equivalence

Let $X$ be an irreducible projective variety and $f$ a morphism $X \rightarrow \mathbb{P}^n$. We give a new proof of the fact that the preimage of any linear variety of dimension $k\ge n+1-\dim f(X)$ is connected. We prove that the statement is a consequence of the Generalized Hodge Index Theorem using easy numerical arguments that hold in any characteristic. We also prove the connectedness Theorem of Fulton and Hansen as application of our main theorem.

实闭域上的广义连通性与Bertini型定理

Generalized connectedness and Bertini-type theorems over real closed fields

本文建立了Bertini定理在实闭域上的类比，证明了可逆层截面的零轨迹若具形式实泛点则不变号，并证明维数≥2的拟射影簇存在保持形式实性与整性的超曲面截面非空开子集。

Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field

Let P^n denote the n-dimensional projective space defined over the algebraic closure of a finite field F_q, let V contained P^n be a complete intersection defined over F_q of dimension r and singular locus of dimension at most s, and let π:V-->P^{s+1} be a "generic" linear mapping. We obtain an effective version of the Bertini smoothness theorem concerning π, namely an explicit upper bound of the degree of a proper Zariski closed subset of P^{s+1} which contains all the points defining singular fibers of π. For this purpose we make essential use of the concept of polar variety associated to th…

Bertini and his two fundamental theorems

After reviewing Bertini's life story, a fascinating drama, we make a critical examination of the old statements and proofs of Bertini's two fundamental theorems, the theorem on variable singular points and the theorem on reducible linear systems. We explain the content of the statements in a way that is accessible to a nonspecialist, and we develop versions of the old proofs that are complete and rigorous by current standards. In particular, we prove a new extension of Bertini's first theorem, which treats variable $r$-fold points for any $r$.

A non-commutative Bertini theorem

We prove a version of the classical 'generic smoothness' theorem with smooth varieties replaced by non-commutative resolutions of singular varieties. This in particular implies a non-commutative version of the Bertini theorem.

Analytic Bertini theorem

We prove an analytic Bertini theorem, generalizing a previous result of Fujino and Matsumura.

The Bertini irreducibility theorem for higher codimensional slices

Poonen and Slavov recently developed a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing. In this paper, we extend their work by proving an analogous bound for the dimension of the exceptional locus in the setting of linear subspaces of higher codimensions.

Bertini 型结果及其应用

Bertini type results and their applications

本文证明 Bertini 型定理，并将其应用于 Nori 基本群的 Lefschetz 定理及光滑拟射影簇上不可约 ℓ-adic 层的曲线限制保持性。

Bertini theorems for F-singularities

We prove that strongly F-regular and F-pure singularities satisfy Bertini-type theorems (including in the context of pairs) by building upon a framework of Cumino, Greco and Manaresi (compare with the work of Jouanolou and Spreafico). We also prove that F-injective singularities fail to satisfy even the most basic Bertini-type results.

解析贝蒂尼定理II——局部情形

Analytic Bertini theorem II --- The local case

研究证明局部解析贝蒂尼定理，证实布克森猜想

Toric and tropical Bertini theorems in positive characteristic

We generalize the toric Bertini theorem of Fuchs, Mantova, and Zannier to positive characteristic. A key part of the proof is a new algebraically closed field containing the field \kk(t_1,\dots,t_d) of rational functions over an algebraically closed field \kk of prime characteristic. As a corollary, we extend the tropical Bertini theorem of Maclagan and Yu to arbitrary characteristic, which removes the characteristic dependence from the d-connectivity result for tropical varieties from that paper.

Le théorème de Bertini en famille

We give upper bounds for the dimension of the set of hypersurfaces of $\mathbb{P}^N$ whose intersection with a fixed integral projective variety is not integral. Our upper bounds are optimal. As an application, we construct, when possible, hypersurfaces whose intersections with all the varieties of a family of integral projective varieties are integral. The degree of the hypersurfaces we construct is explicit. ----- On majore la dimension de l'ensemble des hypersurfaces de $\mathbb{P}^N$ dont l'intersection avec une variété projective intègre fixée n'est pas intègre. Les majorations obtenues s…

Bertini theorems for singular schemes and nearby cycles in mixed characteristic

We first study hyperplane sections of some singular schemes over a field. We prove a Bertini theorem for the log smoothness of generic hyperplane sections of a large class of log smooth schemes over a log point. We also give an abstract generalization of the usual Bertini theorem for smoothness. We then apply this result to the study of hyperplane sections of regular schemes over a complete discrete valuation ring of characteristic zero with algebraically closed residue field. Lastly, give an exposition of a result of Faltings which states that one can compute the $p$-adic nearby cycles of smo…

Semiample Bertini theorems over finite fields

We prove a semiample generalization of Poonen's Bertini Theorem over a finite field that implies the existence of smooth sections for wide new classes of divisors. The probability of smoothness is computed as a product of local probabilities taken over the fibers of the morphism determined by the relevant divisor. We give several applications including a negative answer to a question of Baker and Poonen by constructing a variety (in fact one of each dimension) which provides a counterexample to Bertini over finite fields in arbitrarily large projective spaces. As another application, we determ…

The exceptional locus in the Bertini irreducibility theorem for a morphism

We introduce a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing over a finite field. Extending a result of Benoist, we prove that for a morphism $ϕ\colon X \to \mathbb{P}^n$ such that $X$ is geometrically irreducible and the nonempty fibers of $ϕ$ all have the same dimension, the locus of hyperplanes $H$ such that $ϕ^{-1} H$ is not geometrically irreducible has dimension at most $\operatorname{codim} ϕ(X)+1$. We give an application to monodromy groups above hyperplane sections.

A general homological Kleiman-Bertini theorem

Let G be a smooth algebraic group acting on a variety X. Let F and E be coherent sheaves on X. We show that if all the higher Tor sheaves of F against G-orbits vanish, then for generic g in G, the sheaf Tor^X_j(gF, E) vanishes for all j >0. This generalizes a result of Miller and Speyer for transitive group actions and a result of Speiser, itself generalizing the classical Kleiman-Bertini theorem, on generic transversality, under a general group action, of smooth subvarieties over an algebraically closed field of characteristic 0.

Effective Bertini theorem and formulas for multiplicity and the local Łojasiewicz exponent

The classical Bertini theorem on generic intersection of an algebraic set with hyperplanes states the following: \emph{Let X be a nonsingular closed subvariety of $\mathbb{P}^n_k$, where $k$ is an algebraically closed field. Then there exists a hyperplane $H\subset \mathbb{P}^n_k$ not containing $X$ and such that the scheme $H\cap X$ is regular at every point. Furthermore, the set of hyperplanes with this property forms an open dense subset of the complete linear system $|H|$ considered as a projective space. } We will show that one can effectively indicate a finite family of hyperplanes $H$ s…

Bertini Theorems for $F$-signature and Hilbert-Kunz multiplicity

We show that Bertini theorems hold for $F$-signature and Hilbert--Kunz multiplicity. In particular, if $X \subseteq \mathbb{P}^n$ is normal and quasi-projective with $F$-signature greater than $λ$ (respectively the Hilbert--Kunz multiplicity is less than $λ$) at all points $x \in X$, then for a general hyperplane $H \subseteq \mathbb{P}^n$ the $F$-signature (respectively Hilbert--Kunz multiplicity) of $X \cap H$ is greater than $λ$ (respectively less than $λ$) at all points $x \in X \cap H$.

On the Bertini regularity theorem for arithmetic varieties

Let $\mathcal{X}$ be a regular projective arithmetic variety equipped with an ample hermitian line bundle $\overline{\mathcal{L}}$. We prove that the proportion of global sections $σ$ with $\left\lVert σ\right\rVert_{\infty}<1$ of $\overline{\mathcal{L}}^{\otimes d}$ whose divisor does not have a singular point on the fiber $\mathcal{X}_p$ over any prime $p<e^{\varepsilon d}$ tends to $ζ_{\mathcal{X}}(1+\dim \mathcal{X})^{-1}$ as $d\rightarrow \infty$.

A short proof that all linear codes are weakly algebraic-geometric using Bertini theorems of B. Poonen

In this paper we give a simpler proof of a deep theorem proved by Pellikan, Shen and van Wee that all linear codes are weakly algebraic-geometric using a theorem of B.Poonen.

Bertini theorems for differential algebraic geometry

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely t…

Bertini Type Theorems

Let $X$ be a smooth irreducible projective variety of dimension at least 2 over an algebraically closed field of characteristic 0 in the projective space ${\mathbb{P}}^n$. Bertini's Theorem states that a general hyperplane $H$ intersects $X$ with an irreducible smooth subvariety of $X$. However, the precise location of the smooth hyperplane section is not known. We show that for any $q\leq n+1$ closed points in general position and any degree $a>1$, a general hypersurface $H$ of degree $a$ passing through these $q$ points intersects $X$ with an irreducible smooth codimension 1 subvariety on $X…

Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)

In this article, we prove a strong version of local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen-Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over Noetherian normal domains, which is fundamental in the study of Euler system theory over normal domains and Iwasawa main conjectures.

A Kleiman-Bertini Theorem for sheaf tensor products

Fix a variety X with a transitive (left) action by an algebraic group G. Let E and F be coherent sheaves on X. We prove that, for elements g in a dense open subset of G, the sheaf Tor_i^X(E, g F) vanishes for all i > 0. When E and F are structure sheaves of smooth subschemes of X in characteristic zero, this follows from the Kleiman-Bertini theorem; our result has no smoothness hypotheses on the supports of E or F, or hypotheses on the characteristic of the ground field.

Arithmetic ampleness and an arithmetic Bertini theorem

Let $\mathcal X$ be a projective arithmetic variety of dimension at least $2$. If $\overline{\mathcal L}$ is an ample hermitian line bundle on $\mathcal X$, we prove that the proportion of those effective sections of $\overline{\mathcal L}^{\otimes n}$ that define an irreducible divisor on $\mathcal X$ tends to $1$ as $n$ tends to $\infty$. We prove variants of this statement for schemes mapping to such an $\mathcal X$. On the way to these results, we discuss some general properties of arithmetic ampleness, including restriction theorems, and upper bounds for the number of effective sections o…

Bertini theorems over finite fields

Let X be a smooth quasiprojective subscheme of P^n of dimension m >= 0 over F_q. Then there exist homogeneous polynomials f over F_q for which the intersection of X and the hypersurface f=0 is smooth. In fact, the set of such f has a positive density, equal to zeta_X(m+1)^{-1}, where zeta_X(s)=Z_X(q^{-s}) is the zeta function of X. An analogue for regular quasiprojective schemes over Z is proved, assuming the abc conjecture and another conjecture.

有限域上Hilbert-Samuel重数的Bertini定理

Bertini theorems for Hilbert-Samuel multiplicity over finite fields

针对有限域上的约化等维拟射影概形，证明存在正密度超曲面集使得交点的重数保持。

Hilbert-Kunz Multiplicity of Fibers and Bertini Theorems

Let $k$ be an algebraically closed field of characteristic $p > 0$. We show that if $X\subseteq\mathbb{P}^n_k$ is an equidimensional subscheme with Hilbert--Kunz multiplicity less than $λ$ at all points $x\in X$, then for a general hyperplane $H\subseteq\mathbb{P}^n_k$, the Hilbert--Kunz multiplicity of $X\cap H$ is less than $λ$ at all points $x\in X\cap H$. This answers a conjecture and generalizes a result of Carvajal-Rojas, Schwede and Tucker, whose conclusion is the same as ours when $X\subseteq\mathbb{P}^n_k$ is normal. In the process, we substantially generalize certain uniform estimate…

Bertini irreducibility theorems over finite fields

Given a geometrically irreducible subscheme X in P^n over F_q of dimension at least 2, we prove that the fraction of degree d hypersurfaces H such that the intersection of H and X is geometrically irreducible tends to 1 as d tends to infinity. We also prove variants in which X is over an extension of F_q, and in which the immersion of X in P^n is replaced by a more general morphism.

On Resolving Singularities of Plane Curves via a Theorem attributed to Clebsch

This paper discusses a central theorem in birational geometry first proved by Eugenio Bertini in 1891. J.L. Coolidge described the main ideas behind Bertini's proof, but he attributed the theorem to Clebsch. He did so owing to a short note that Felix Klein appended to the republication of Bertini's article in 1894. The precise circumstances that led to Klein's intervention can be easily reconstructed from letters Klein exchanged with Max Noether, who was then completing work on the lengthy report he and Alexander Brill published on the history of algebraic functions [Brill/Noether 1894]. This …

Bertini定理对$F$-有理性的失效

Bertini's theorem for $F$-rationality is false

本文构造了九维$F$-有理仿射和射影簇的反例，证明Bertini定理对$F$-理性不成立，即超平面截口可能失去$F$-理性或$F$-内射性。

关于$F$-有理且$F$-纯奇点的Bertini定理

Bertini's theorem for $F$-rational $F$-pure singularities

本文证明特征$p>0$下$F$-有理且$F$-纯拟射影簇与一般超平面的交仍保持这些奇点性质，并给出孤立非$F$-正则的$\mathbb{Q}$-Gorenstein簇特征与指数的关系。

重访斋藤定理及其在超曲面自由束中的应用

Saito's theorem revisited and application to free pencils of hypersurfaces

本文通过多重特征概形重新解释斋藤自由性准则，构造出$\mathbb P^n$中大量新的自由约化超曲面族，解决了已知自由超曲面例子少的问题。

How many hypersurfaces does it take to cut out a Segre class?

We prove an identity of Segre classes for zero-schemes of compatible sections of two vector bundles. Applications include bounds on the number of equations needed to cut out a scheme with the same Segre class as a given subscheme of (for example) a projective variety, and a `Segre-Bertini' theorem controlling the behavior of Segre classes of singularity subschemes of hypersurfaces under general hyperplane sections. These results interpolate between an observation of Samuel concerning multiplicities along components of a subscheme and facts concerning the integral closure of corresponding ideal…

On rationality of nonsingular threefolds with a pencil of Del Pezzo surfaces of degree 4

We prove a criterion of nonsingularity of a complete intersection of two fiberwise quadrics in a scroll over $P^1$. As a corollary we derive the following addition to the Alexeev theorem on rationality of standard Del Pezzo fibrations of degree 4 over $P^1$: we prove that any fibration of this kind with the topological Euler characteristic -4 is rational.

更高杜布瓦和更高理性奇点

Higher Du Bois and Higher Rational Pairs

本文提出将更高杜布瓦和更高理性奇点的概念扩展到对，并通过广义注入定理证明相关性质。

Fibration by non-smooth projective curves of arithmetic genus two in characteristic two

Looking in positive characteristic for failures of the Bertini-Sard theorem, we determine, up to birational equivalence, the separable proper morphisms of smooth algebraic varieties in characteristic two, whose fibres are non-smooth curves of arithmetic genus two.

Gabber's presentation lemma for finite fields

We give a proof of Gabber's presentation lemma for finite fields. We use ideas from Poonen's proof of Bertini's theorem to prove this lemma in the special case of open subsets of the affine plane. We then reduce the case of general smooth varieties to this special case.

Motivic Euler products in motivic statistics

We formulate and prove an analog of Poonen's finite-field Bertini theorem with Taylor conditions that holds in the Grothendieck ring of varieties. This gives a broad generalization of the work of Vakil-Wood, who treated the case of smooth hypersurface sections. In fact, our techniques give analogs in motivic statistics of all known results in arithmetic statistics that have been proven using Poonen's sieve, including work of Bucur-Kedlaya on complete intersections and Erman-Wood on semi-ample Bertini theorems. A key ingredient is the use of motivic Euler products, as introduced by the first au…

Scheme-theoretic Whitney conditions and applications to tangency of projective varieties

We investigate a scheme-theoretic variant of Whitney condition a. If X is a projec-tive variety over the field of complex numbers and Y $\subset$ X a subvariety, then X satisfies generically the scheme-theoretic Whitney condition a along Y provided that the pro-jective dual of X is smooth. We give applications to tangency of projective varieties over C and to convex real algebraic geometry. In particular, we prove a Bertini-type theorem for osculating plane of smooth complex space curves and a generalization of a Theorem of Ranestad and Sturmfels describing the algebraic boundary of an affine …

Free pencils on divisors

Let X be a smooth projective variety defined over an algebraically closed field, and let Y in X be a reduced and irreducible ample divisor in X. We give a numerical sufficient condition for a base point free pencil on $Y$ to be the restriction of a base point free pencil on $X$. This result is then extended to families of pencils and to morphisms to arbitrary smooth curves. Serrano had already studied this problem in the case n=2 and 3, and Reider had then attacked it in the case $n=2$ using vector bundle methods based on Bogomolov's instability theorem on a surface (char(k)=0). The argument g…

Connectivity via nongeneric pencils

We use slicing by nongeneric pencils of hypersurfaces and prove a new theorem of Lefschetz type for singular non compact spaces, at the homotopy level. As applications, we derive results on the topology of the fibres of polynomial functions or the topology of complements of hypersurfaces in $\mathbb C^n$.

The Zariski-Lefschetz principle for higher homotopy groups of nongeneric pencils

We prove a general Zariski-van Kampen-Lefschetz type theorem for higher homotopy groups of generic and nongeneric pencils on singular open complex spaces.

射影超曲面的束、Griffiths 高度与几何不变量理论。II 具有半齐次奇点的超曲面

Pencils of projective hypersurfaces, Griffiths heights and geometric invariant theory. II Hypersurfaces with semihomogeneous singularities

本文确立了具有半齐次奇点的射影超曲面束的中间维上同调的稳定 Griffiths 高度公式，通过构造有限覆盖以获得半稳定光滑模型，成功规避了确定奇点纤维初等指数的难题。

Schubert induction

We describe a Schubert induction theorem, a tool for analyzing intersections on a Grassmannian over an arbitrary base ring. The key ingredient in the proof is the Geometric Littlewood-Richardson rule, described in a companion paper. Schubert problems are among the most classical problems in enumerative geometry of continuing interest. As an application of Schubert induction, we address several long-standing natural questions related to Schubert problems, including: the "reality" of solutions; effective numerical methods; solutions over algebraically closed fields of positive characteristic; so…

On the proportion of transverse-free plane curves

We study the asymptotic proportion of smooth plane curves over a finite field $\mathbb{F}_q$ which are tangent to every line defined over $\mathbb{F}_q$. This partially answers a question raised by Charles Favre. Our techniques include applications of Poonen's Bertini theorem and Schrijver's theorem on perfect matchings in regular bipartite graphs. Our main theorem implies that a random smooth plane curve over $\mathbb{F}_q$ admits a transverse $\mathbb{F}_q$-line with very high probability.

有限域上的泰勒条件

Taylor conditions over finite fields

论文扩展了Poonen的有限域上的Bertini定理，研究局部自由商的微分层上的泰勒条件，受Bilu和Howe在动机设定中的结果启发。

关于 $\mathbb{F}_{p}$ 上例外平面函数的分类

On the Classification of Exceptional Planar Functions over $\mathbb{F}_{p}$

本文旨在推进有限域上例外平面/PN单项式函数的分类，利用Weil界、Bezout定理和Bertini定理得出了多项强有力的部分结果。

Lefschetz Theorem Using Real Morse Theory

We prove the Lefschetz hyperplane section theorem using a simpler machinery by making the observation that we can compose the Lefschetz Pencil with a Real Morse function to get a map from the variety to $\mathbb{R}$ which is "close" to being a Real Morse function. The proof uses a new method unlike the conventional one which uses vanishing cycles, thimbles and monodromy. We prove the genus formula for plane curves using Morse theory, Lefschetz pencil and Bézout's theorem. And then we prove the Riemann Hurwitz formula for ramified maps between curves by employing techniques from deformation the…

The Hesse pencil and polarizations of type $(1,3)$ on Abelian surfaces

In this short note we prove two theorems, the first one is a sharpening of a result of Lange and Sernesi: the discriminant curve W of a general Abelian surface $A$ endowed with an irreducible polarization $D$ of type $(1,3)$ is an irreducible curve of degree $18$ whose singularities are exactly $36$ nodes and $72$ cusps. Moreover, we analyze the degeneration of the discriminant curve $W$ and its singularities as $A$ tends to the product of two equal elliptic curves. The second theorem, using the first one in order to prove a transversality assertion, shows that the general element of a family …

无理 pencils 与通过基本群的 profinite 完成来刻画与积同构的代数簇

Irrational pencils, and characterization of Varieties isogenous to a product, via the Profinite completion of the Fundamental group

本文通过基本群的 profinite 完成，刻画了与积同构的代数簇，并证明了无理 pencil 与紧 Kahler 流形的对应关系。