Bertini Theorem Research Articles

On Bertini's theorem in characteristic ${p}$ for families of canonical curves in ${\Bbb P}^{(p-3)/2}$

Bertini's theorem on variable singular points may fail in characteristic p; there are algebraic fibrations that are pathological in the sense that all fibres are non-smooth though the total space admits only a finite number of singular points. We prove that if an algebraic fibration by plane projective quartic curves in characteristic 7 is pathological then, after an eventual cyclic base extension of degree 3, it is, up to birational equivalence, obtained by a base extension from the pencil of quartic curves cut out by the equations zx3 + xy3 + t yz3 = 0. More generally, we classify pathological fibrations by canonical curves of arithmetic genus g = p − 1 2 in the projective space of dimension p − 3 2 . We prove that the gap sequences of the smooth Weierstrass points of the fibres have the property that a positive integer ℓ is a gap if and only if 2g + 1 − ℓ is a non-gap. In analogy to the Kodaira–Néron classification of special fibres of minimal fibrations by elliptic curves, we construct minimal proper regular models of pathological fibrations, determine the structure of the bad fibres, and study the global geometry of the total spaces. 2000 Mathematics Subject Classification 14H05, 14H10, 14H20, 14H50.

An effective Bertini theorem over finite fields

Let p be a prime, q a power of p and K the algebraic closure of the finite field GFð pÞ. Let X H P N ðKÞ be an irreducible variety. Set n :¼ dimðX Þ and d :¼ degðX Þ. Here we prove that if qd dðd � 1Þ n there is a hyperplane H of P N ðKÞ defined over GFðqÞ and transversal to X. 2000 Mathematics Subject Classification. 14N05

Pencils of Curves on Smooth Surfaces

Although the theory of singularities of curves - resolution, classification, numerical invariants - goes through with comparatively little change in finite characteristic, pencils of curves are more difficult. Bertini's theorem only holds in a much weaker form, and it is convenient to restrict to pencils such that, when all base points are resolved, the general member of the pencil becomes non-singular. Even here, the usual rule for calculating the Euler characteristic of the resolved surface has to be modified by a term measuring wild ramification. We begin by describing this background, then proceed to discuss the exceptional members of a pencil. In characteristic 0 it was shown by Ha and Le and by Le and Weber, using topological reasoning, that exceptional members can be characterised by their Euler characteristics. We present a combinatorial argument giving a corresponding result in characteristic p. We first treat pencils with no base points, and then reduce the remaining case to this.

Covers of algebraic varieties IV. A Bertini theorem for scandinavian covers

Let Y be an integral scheme and fix locally free 𝒪Y – sheaves ℰ, 𝒜 and ℬ of ranks 5, 3 and 3 respectively. Consider the projective bundle and a morphism δ : π*𝒜 → π*ℬ(1) X ≔ D1(δ) be the locus of points x ∈ ℙ where rk(δx) ≤ 1. Then the map ϱ ≔ π|X : X → Y is a cover of degree d = 6 if dim(X ∩ π–1(y)) = 0 for each y ∈ Y. We call such a cover scandinavian. We prove a Bertini – type theorem and we give some examples of scandinavian and non scandinavian covers of degree 6.

Non-hyperbolic complex space with a hyperbolic normalization

We construct an example of a non-hyperbolic singular projective surface X whose normalization V is the square of a genus 3 C and hence, hyperbolic. Let C be a smooth irreducible projective of genus g > 2. Then the smooth projective surface V = C x C is Kobayashi hyperbolic, that is, the Kobayashi pseudodistance on it is a distance [2]. Let V ___ pN be a projective embedding. Consider a generic projection ir V __ p3. By Bertini's theorem, the singular locus S (i.e. the closure of the set of points) of the image surface X = ir(V) c is an irreducible curve, and ir: V -* X is a normalization map (see [4]). The question arises whether the surface X is also hyperbolic. The answer is positive [5], and hence by the stability of hyperbolicity theorem [6], any (smooth) surface X' in I3 close enough to X is hyperbolic, as well. In that way examples of degree 32 smooth hyperbolic surfaces in p3 were produced [5]. By Proposition 1.1 in [5], hyperbolicity of a (singular) surface X as above is equivalent to hyperbolicity of its S. Actually, in [5] it is shown that the geometric genus of the S is > 225, which provides that X is hyperbolic. On the other hand, by the Kobayashi-Kwack theorem [2, 3], a normalization of a hyperbolic complex space is also hyperbolic. In this note we give an example which shows that in general, the converse is not true. To describe this example, denote by C the Fermat quartic x4 + y4 + z4 0 O in P2. Then the Cartesian square V = C X C C Ip2 X Ip2 c-_ p8 (the Segre embedding) is a smooth surface of degree 32 in Ip8. We construct a singular projective surface X whose normalization is V, which has a fibration X -* C over C with general fibre isomorphic to C and with four degenerate fibres Ci, i = 1, . . ., 4, isomorphic to Pl1. The double curve S = Cl U ... U C4 C X of X is neither irreducible nor hyperbolic, in contrast with the situation studied in [5]. Thus, the assumption in [5] that the projection ir is generic, is likely to be essential to provide hyperbolicity of the image surface X = 7r(V). Actually, in our example the surface X does not appear as a projection of V = C x C; but it has a natural embedding into a four-dimensional Brauer-Severi variety Y (see [1]) which is a smooth projective fiber bundle over Ip2 with general fibre Ip2. Received by the editors July 30, 1999. 2000 Mathematics Subject Classification. Primary 32H15, 32H20. (?2000 American Mathematical Society

Application to global Bertini theorems

Application to global Bertini theorems

On Bertini's theorem

We study the singularities of a generic element of a linear system of divisors on a smooth projective variety inside the base locus of the linear system, and give a generalization of Bertini's theorem.

A topological version of Bertini's theorem

We give a topological version of a Bertini type theorem due to Abhyankar. A new definition of a branched covering is given. If the restriction πV : V → Y of the natural projection π : Y × Z → Y to a closed set V ⊂ Y × Z is a branched covering then, under certain assumptions, we can obtain generators of the fundamental group π1((Y × Z) \ V ). Introduction. In his book [1, pp. 349–356], Abhyankar proves an interesting theorem called by him a “Bertini theorem” or a “Lefschetz theorem”. The theorem expresses a topological fact in complex analytic geometry. The purpose of this paper is to restate this theorem and its proof in purely topological language. Our formulation reads as follows: Theorem 1. Let Z be a connected topological manifold (without boundary) modeled on a real normed space E of dimension at least 2 and let Y be a simply connected and locally simply connected topological space. Suppose that V is a closed subset of Y × Z and π : Y × Z → Y denotes the natural projection. Assume that πV = π|V : V → Y is a branched covering whose regular fibers are finite and whose singular set ∆ = ∆(πV ) does not disconnect Y at any of its points. Set X = (Y × Z) \ V and L = {p} × Z, where p ∈ Y \∆. If there exists a continuous mapping h : Y → Z whose graph is contained in X, then the inclusion i : L \ V ↪→ X induces an epimorphism i∗ : π1(L \ V )→ π1(X). We have adopted the following definition. For any topological spaces Y and Y ∗, a continuous, surjective mapping ψ : Y ∗ → Y is a (topological) branched covering if there exists a nowhere dense subset ∆ of Y such that ψ|Y ∗ \ ψ−1(∆) : Y ∗ \ ψ−1(∆)→ Y \∆ is a covering mapping. Notice that the singular set ∆ of the branched covering ψ is not unique, but there is a smallest ∆(ψ) among such sets. Clearly, ∆(ψ) is a closed subset of Y . Thus 1991 Mathematics Subject Classification: 57M12, 55Q52.

Bertini theorems for ideals linked to a given ideal

We prove a generalization of Flenner’s local Bertini theorem for complete intersections. More generally, we study properties of the ‘general’ ideal linked to a given ideal. Our results imply the following. LetR be a regular local Nagata ring containing an infinite perfect fieldk, andI⊂R is an equidimensional radical ideal of heightr, such thatR/I is Cohen-Macaulay and a local complete intersection in codimension 1. Then for the ‘general’ linked idealJα, R/Jα is normal and Cohen-Macaulay.

A local bertini theorem in mixed characteristic

Flenner's proof in [F] of the Bertini theorem for local rings of mixed characteristic is incomplete. In this note, we modify Flenner's argument to fix this gap.

Generically rational polynomials

Introduction. Let k be an algebraically closed field of characteristic zero. Let β[#,;y] be a polynomial ring of two variables and let Al= Spec(k[x,y]). Embed Al into the projective plane Pf as the complement of a line /«,. Let /^k[x,y] be an irreducible polynomial, let FΛ be the curve on Al defined by f=a for every a^k and let CΛ be the closure of FΛ in Pf. Then the set Λ(/): — {CΛ; a^kU (°°)} is a linear pencil on Pi defined by/, where Cco^dL, d being the degree of/. The set Λ0(/):— {FΛ; a^k} is called the linear pencil on Al defined by /. The polynomial / is called generically rational when the general members of Λ(/) (or Λ0(/)) are irreducible rational curves. Since the algebraic function field k(x, y) of one variable over the subfield k(f) then has genus 0, Tsen's theorem says that / is generically rational if and only if / is a field generator in the sense of Russell [9, 10], i.e., there is an element g^k(x,y) such that ^(tfjj)=&(/£). If / is a generically rational polynomial, we can associate with / a non-negative integer n, where n-\-1 is the number of places at infinity of a general member FΛ of Λ0(/), i.e., the number of places of FΛ whose centers lie outside Al. If d^ 1, the pencil Λ(/) has base points situated outside Al. Let φ: W-+PI be the shortest succession of quadratic transformations with centers at the base points (including infinitely near base points) of Λ(/) such that the proper transform Λ' of Λ(/) by φ has no base points. Then the linear pencil Λ' defines a morphism p: W—*Pk9 whose general fibers are the proper transforms of general members of Λ(/); thence they are nonsingular rational curves by virtue of Bertini's theorem. Moreover, W contains in a canonical way an open subset isomorphic to Al. A generically rational polynomial / is said to be of simple type if the morphism p has n+l cross-sections contained in the boundary set W-Al (cf. Definition 1.8, below). If n=0, a generically rational polynomial / is sent to one of the coordinates x, y of Al by a biregular automorphism of Al (cf. Abhyankar-Moh's theorem [1, 4]); hence/ is of simple type. If w=l, a generically rational polynomial is always of simple type (cf. Theorem 2.3, below). However, if n>l, a generi-

Bourbaki's theorem and prime ideals

If R is a normal domain and M a finitely generated torsion free module, then by Bourbaki's theorem [2, Sect. 9, Theorem 6] there is a free submodule F of M such that M/F is isomorphic to an ideal. We shall call such an ideal a Bourbaki ideal for M. Of particular interest is the case when this ideal may be chosen to be prime; if I 'MfF, then homological properties of M will be inherited by the domain Rfl, for instance via the isomorphisms Exti(M, R) ~ Exti(J, R) ' Exti+1(Rfl, R), for i ~ 2. In this situation, one can, by taking for M direct sums of suitable syzygies of given modules of finite length, produce a domain Rfl with a prescribed sequence of finite length local cohomology modules. A detailed statement and proof, as well as further applications, may be found in the paper of Evans and Griffith [7]. In this paper, we present sufficient conditions on the ring R that all finitely generated reflexive modules give rise to prime Bourbaki ideals. We will also present a partial converse, which demonstrates that, for the most part, these conditions are actually necessary. The proof, and especially the construction of the ideal in part I, follows that of [7], although it is somewhat more general, and avoids the use of Swan's theorem (concerning projectives over Laurent polynomial rings). The use of Bertini's theorem was first suggested in a preprint of [7]; the timely appearance of a particularly apropos version of this theorem by Flenner [8] enabled us to complete the proof along the lines suggested. This part of the proof actually demonstrates more; the prime ideals which are produced in fact are analytically irreducible, and even normal under certain conditions. We may moreover produce such ideals with arbitrarily many minimal generators. Moh has given an explicit sequence of primes in k[[ X, Y, Z]] having no bound on the number of minimal generators [I2], and Bresinsky has done the same for power series in four or more variables [3]. In each of

On the Fundamental Group of the Complement of a Node Curve

where dl, * dt are the degrees of the irreducible components of C, would follow if one knew that the intersection of all subgroups of finite index in r(P 2C) were trivial. Zariski's original goal of determining all finite coverings of P2 C follows from the theorem stated here: the profinite completions of the displayed groups are the same, and similarly for the tame fundamental group and the prime-to-p completion in characteristic p (cf. [11, [31). The theorem also implies the same conclusion for a hypersurface in PI whose generic plane sections are node curves. Zariski's proof relied on the assertion of Enriques and Severi that any node curve can be degenerated to lines in general position. This assertion remains unproved, however. Abhyankar has proved many special cases of this result. A recent strengthening of Bertini's connectedness theorem, proved jointly with J. Hansen [21, yields a lemma which completes Abhyankar's beautiful argument. Let f: X -+ Y be a finite morphism of projective surfaces over an algebraically closed field, with Y non-singular and X normal. Assume f is a

Bertini theorems for hypersurface sections containing a subscheme

(1979). Bertini theorems for hypersurface sections containing a subscheme. Communications in Algebra: Vol. 7, No. 8, pp. 775-790.

Deformations of Cohen-Macaulay Schemes of Codimension 2 and Non-Singular Deformations of Space Curves

The following work deals with the deformations of embedded affine schemes of codimension 2, which locally have a resolution of length 2. The cases of immediate interest are curves in 3-space and 0-dimensional schemes in the plane. It is first shown that such a scheme X has a global resolution of length 2. Therefore, by a theorem of Burch, the functions defining the ideal of X can be obtained as the maximal minors of a matrix whose columns generate all the relations among these functions. All flat deformations of X can be obtained simply by deforming this matrix, and this permits the construction of the versal deformation space of X. Finally, for X of dimension 3 or less one can construct non-singular deformations of X by taking a parameter space sufficiently large to permit one to change the constant and linear terms of each entry in the matrix. For X of dimension 4, an example is given in XXX [11] in which the scheme not only has no non-singular deformations, but in fact has no non-isomorphic deformations at all. A brief review of the previous literature will help place these results in perspective. It has long been known, by Bertini's theorem, that the generic deformation of a scheme of codimension 1 is non-singular. As a consequence of the work of Fogarty [5], it was also known that every point, or rather, 0-dimensional scheme in the plane has non-singular deformations; Briancon and Galligo [3] give an explicit construction for such a deformation, splitting the scheme into distinct simple points. This led mathematicians interested in algebraic curves to ask if the generic deformation of a space curve is also non-singular, the question which is settled in this paper. Further direct extension of these results in the case of curves is impossible, since the work of Iarrobina [4] permits the construction of a non-reduced curve in affine 4-space which has no non-singular deformations; however, for reduced curves the question is still open as of this writing.