Bertini Theorem Research Articles

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 that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties.

Applications of Swan's Bertini theorem to unimodular rows

Let R be an affine algebra of dimension d≥4 over a perfect field k of char ≠2 and I be an ideal of R. Then(1)MSd+1(R) is uniquely divisible prime to char k if R is reduced and k is infinite with c.d.(k)≤1.(2)Umd+1(R,I)/Ed+1(R,I) has nice group structure if c.d.2(k)≤2.(3)Umd(R,I)/Ed(R,I) has nice group structure if k is algebraically closed of char k≠2,3 and either (i) k=F‾p or (ii) R is normal.

Bertini theorems revisited

AbstractWe prove several new Bertini theorems over arbitrary fields and discrete valuation rings.

Weak normality and seminormality in the mixed characteristic case

We study certain properties of Noetherian rings with weak normality and seminormality in mixed characteristic. It is known that the two concepts can differ in the equal prime characteristic case, while they coincide by definition in the equal characteristic zero case. We exhibit some examples in the mixed characteristic case. We also establish the local Bertini theorem for weak normality in mixed characteristic under a certain condition.

Analytic Bertini theorem

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

Hilbert–Kunz multiplicity of fibers and Bertini theorems

Let k be an algebraically closed field of characteristic p>0. We show that if X⊆Pkn is an equidimensional subscheme with Hilbert–Kunz multiplicity less than λ at all points x∈X, then for a general hyperplane H⊆Pkn, the Hilbert–Kunz multiplicity of X∩H is less than λ at all points x∈X∩H. This answers a conjecture and generalizes a result of Carvajal-Rojas, Schwede and Tucker, whose conclusion is the same as ours when X⊆Pkn is normal. In the process, we substantially generalize certain uniform estimates on Hilbert–Kunz multiplicities of fibers of maps obtained by the aforementioned authors that should be of independent interest.

Higher connectivity of tropicalizations

We show that the tropicalization of an irreducible d-dimensional variety over a field of characteristic 0 is (d-ell )-connected through codimension one, where ell is the dimension of the lineality space of the tropicalization. From this we obtain a higher connectivity result for skeleta of rational polytopes. We also prove a tropical analogue of the Bertini Theorem: the intersection of the tropicalization of an irreducible variety with a generic hyperplane is again the tropicalization of an irreducible variety.

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: Let X be a nonsingular closed subvariety ofPkn, where k is an algebraically closed field. Then there exists a hyperplaneH⊂Pknnot containing X and such that the schemeH∩Xis 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 show that one can effectively indicate a finite family of hyperplanes H such that at least one of them satisfies the assertion of the Bertini theorem, provided the characteristic of the field k is equal to zero. As an application of the method used in the proof we give effective formulas for the multiplicity and the Łojasiewicz exponent of polynomial mappings.

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 \(\lambda \) (respectively the Hilbert–Kunz multiplicity is less than \(\lambda \)) 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 \(\lambda \) (respectively less than \(\lambda \)) at all points \(x \in X \cap H\).

On the proportion of transverse-free plane curves

We study the asymptotic proportion of smooth plane curves over a finite field Fq which are tangent to every line defined over Fq. 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 Fq admits a transverse Fq-line with very high probability.

Arithmetic ampleness and an arithmetic Bertini theorem

Arithmetic ampleness and an arithmetic Bertini theorem

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.

Effective bounds on the dimensions of Jacobians covering abelian varieties

We show that any abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective and explicit version of Poonen’s Bertini theorem over finite fields, which allows us to show the existence of smooth curves arising as hypersurface sections of bounded degree and genus. Additionally, for simple abelian varieties we prove a better bound. As an application, we show that for any elliptic curve E E over a finite field and any n ∈ N n\in \mathbb {N} , there exist smooth curves of bounded genus whose Jacobians have a factor isogenous to E n E^n .

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.

Sharp Bertini Theorem for Plane Curves over Finite Fields

AbstractWe prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_{q}$ with $d\leqslant q+1$ , 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 $2p+1$ when $q=p^{r}$ .

Factorization of bivariate sparse polynomials

We prove a function field analogue of a conjecture of Schinzel on the factorization of univariate polynomials over the rationals. We derive from it a finiteness theorem for the irreducible factorizations of the bivariate Laurent polynomials in families with fixed set of complex coefficients and varying exponents. Roughly speaking, this result shows that the truly bivariate irreducible factors of these sparse Laurent polynomials, are also sparse. The proofs are based on a generalization of a toric Bertini's theorem due to Fuchs, Mantova and Zannier.

Sharp degree bounds for sum-of-squares certificates on projective curves

Given a real projective curve with homogeneous coordinate ring R and a nonnegative homogeneous element f∈R, we bound the degree of a nonzero homogeneous sum of squares g∈R such that the product fg is again a sum of squares. Better yet, our degree bounds only depend on geometric invariants of the curve and we show that there exist smooth curves and nonnegative elements for which our bounds are sharp. We deduce the existence of a multiplier g from a new Bertini Theorem in convex algebraic geometry and prove sharpness by deforming rational Harnack curves on toric surfaces. Our techniques also yield similar bounds for multipliers on surfaces of minimal degree, generalizing Hilbert's work on ternary forms.

Normal hyperplane sections of normal schemes in mixed characteristic

The aim of this article is to prove that, under certain conditions, an affine flat normal scheme that is of finite type over a local Dedekind scheme in mixed characteristic admits infinitely many normal effective Cartier divisors. For the proof of this result, we prove the Bertini theorem for normal schemes of some type. We apply the main result to prove a result on the restriction map of divisor class groups of Grothendieck–Lefschetz type in mixed characteristic.

Counterexamples to Bertini theorems for test ideals

In algebraic geometry, Bertini theorems are an extremely important tool. A generalization of the classical theorem to multiplier ideals show that multiplier ideals restrict to a general hyperplane section. The test ideal can be seen to be the characteristic p>0 analog of the multiplier ideal. However, in this paper it is shown that the same type of Bertini type theorem does not hold for test ideals.

F-THRESHOLDS OF GRADED RINGS

The$a$-invariant, the$F$-pure threshold, and the diagonal$F$-threshold are three important invariants of a graded$K$-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly$F$-regular rings. In this article, we prove that these relations hold only assuming that the algebra is$F$-pure. In addition, we present an interpretation of the$a$-invariant for$F$-pure Gorenstein graded$K$-algebras in terms of regular sequences that preserve$F$-purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition$S_{k}$. We also present analogous results and questions in characteristic zero.