Noether-Lefschetz Theorem Research Articles

The Noether–Lefschetz theorem in arbitrary characteristic

We show that if X ⊂ P k N X\subset \mathbb P^N_k is a normal variety of dimension ≥ 3 \geq 3 and H ⊂ P k N H\subset \mathbb P^N_k a very general hypersurface of degree d = 4 d=4 or ≥ 6 \geq 6 , then the restriction map Cl ⁡ ( X ) → Cl ⁡ ( X ∩ H ) \operatorname {Cl}(X)\to \operatorname {Cl}(X\cap H) is an isomorphism up to torsion. If dim ⁡ X ≥ 4 \dim X\geq 4 , the result holds for d ≥ 2 d\geq 2 . The proof uses the relative Jacobian of a curve fibration, together with a specialization argument, and the result holds over fields of arbitrary characteristic.

Moishezon's theorem and degeneration

Let L be a tensor product of two very ample line bundles on the smooth projective complex threefold X. Under the hypothesis that H0(X,KX(L))≠0, we show that the restriction map r:PicX→PicY is an isomorphism for very general Y in the linear system |L|. For such L this result recovers the Noether-Lefschetz theorem of Moishezon, who extended the original topological and Hodge-theoretic arguments of Lefschetz. We give a degeneration argument which is almost completely algebraic in nature.

The Noether–Lefschetz locus of surfaces in toric threefolds

The Noether–Lefschetz theorem asserts that any curve in a very general surface [Formula: see text] in [Formula: see text] of degree [Formula: see text] is a restriction of a surface in the ambient space, that is, the Picard number of [Formula: see text] is [Formula: see text]. We proved previously that under some conditions, which replace the condition [Formula: see text], a very general surface in a simplicial toric threefold [Formula: see text] (with orbifold singularities) has the same Picard number as [Formula: see text]. Here we define the Noether–Lefschetz loci of quasi-smooth surfaces in [Formula: see text] in a linear system of a Cartier ample divisor with respect to a [Formula: see text]-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether–Lefschetz loci which contain a line, defined as a rational curve which is minimal in a suitable sense.

An infinitesimal Noether–Lefschetz theorem for Chow groups

Let X be a smooth, complex projective variety, and Y be a very general, sufficiently ample hypersurface in X. A conjecture of M.V. Nori states that the natural restriction map CHp(X)Q→CHp(Y)Q is an isomorphism for all p<dim⁡Y and an injection for p=dim⁡Y. This is the generalized Noether–Lefschetz conjecture. We prove an infinitesimal version of this conjecture.

EXTENSIONS OF VECTOR BUNDLES WITH APPLICATION TO NOETHER–LEFSCHETZ THEOREMS

Given a smooth, projective variety Y over an algebraically closed field of characteristic zero, and a smooth, ample hyperplane section X ⊂ Y, we study the question of when a bundle E on X, extends to a bundle [Formula: see text] on a Zariski open set U ⊂ Y containing X. The main ingredients used are explicit descriptions of various obstruction classes in the deformation theory of bundles, together with Grothendieck–Lefschetz theory. As a consequence, we prove a Noether–Lefschetz theorem for higher rank bundles, which recovers and unifies the Noether–Lefschetz theorems of Joshi and Ravindra–Srinivas.

A Noether-Lefschetz theorem for varieties of r-planes in complete intersections

AbstractWe prove a Noether-Lefschetz type theorem for varieties ofr-planes in complete intersections. We then use it to study the Abel-Jacobi map of planes on a smooth cubic fivefold.

A Noether-Lefschetz theorem for varieties of r-planes in complete intersections

AbstractWe prove a Noether-Lefschetz type theorem for varieties of r-planes in complete intersections. We then use it to study the Abel-Jacobi map of planes on a smooth cubic fivefold.

Noether-Lefschetz Theorem with Base Locus

We compute the class groups of very general normal surfaces in complex projective three-space containing an arbitrary base locus $Z$, thereby extending the classic Noether-Lefschetz theorem (the case when $Z$ is empty). Our method is an adaptation of Griffiths and Harris' degeneration proof, simplified by a cohomology and base change argument. We give applications to computing Picard groups, which generalize several known results.

The Noether–Lefschetz theorem for the divisor class group

Let X be a normal projective threefold over a field of characteristic zero and | L | be a base-point free, ample linear system on X. Under suitable hypotheses on ( X , | L | ) , we prove that for a very general member Y ∈ | L | , the restriction map on divisor class groups Cl ( X ) → Cl ( Y ) is an isomorphism. In particular, we are able to recover the classical Noether–Lefschetz theorem, that a very general hypersurface X ⊂ P C 3 of degree ⩾4 has Pic ( X ) ≅ Z .

The Noether–Lefschetz Theorem Via Vanishing of Coherent Cohomology

AbstractWe prove that for a generic hypersurface in ℙ2n+1 of degree at least 2 + 2/n, the n-th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing.

Noether-Lefschetz Problems for Vector Bundles

In this paper we generalize two Noether-Lefschetz theorems for surfaces by L. Ein to varieties of arbitrary even dimension. Our results also generalize the classical Noether-Lefschetz theorem for complete intersections of even dimension in ℙn.

A Noether-Lefschetz theorem for vector bundles

In this note we use the monodromy argument to prove a Noether-Lefschetz theorem for vector bundles.

Noether-Lefschetz Locus for Surfaces

We generalize M. Green’s Explicit Noether-Lefschetz Theorem to the family of smooth complete intersection surfaces in the higher dimensional projective spaces. Moreover, we give a new proof of the Density Theorem due to C. Ciliberto, J. Harris, and R. Miranda [5].

Noether-Lefschetz locus for surfaces

We generalize M. Green's Explicit Noether-Lefschetz Theorem to the family of smooth complete intersection surfaces in the higher dimensional projective spaces. Moreover, we give a new proof of the Density Theorem due to C. Ciliberto, J. Harris, and R. Miranda [5].

On a conjecture of Griffiths-Harris generalizing the Noether-Lefschetz theorem

On a conjecture of Griffiths-Harris generalizing the Noether-Lefschetz theorem

A new proof of the explicit Noether-Lefschetz theorem

A new proof of the explicit Noether-Lefschetz theorem

On the Noether-Lefschetz theorem and some remarks on codimension-two cycles

Here, "of general moduli" means that there is a countable union V of subvarieties of the space pN of surfaces of degree d in p3, such that the statement Pie(S) = Z holds for S ~ p N _ V. Noether, it would seem, stated this theorem but never completely proved it. Instead, he gave a plausibility argument, based on the following construction 1 : let ,~.g be the Hilbert scheme of curves of degree n and genus 9 in p3, l(gr3(d)l -~ IP N the space of surfaces of degree d in p3, and ZC~t~.,~• the incidence correspondence S. ,o ,a = ( (C, S) : C z S } .