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.

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The Noether–Lefschetz Theorem Via Vanishing of Coherent Cohomology

Canadian Mathematical Bulletin ◽

10.4153/cmb-2008-028-x ◽

2008 ◽

Vol 51 (2) ◽

pp. 283-290 ◽

Cited By ~ 1

Author(s):

G. V. Ravindra

Keyword(s):

Picard Number ◽

Generic Hypersurface ◽

Lefschetz Theorem

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.

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On a conjecture by Griffiths and Harris concerning certain Noether–Lefschetz loci

Communications in Contemporary Mathematics ◽

10.1142/s0219199715500029 ◽

2015 ◽

Vol 17 (05) ◽

pp. 1550002 ◽

Cited By ~ 2

Author(s):

Ananyo Dan

Keyword(s):

Irreducible Component ◽

Smooth Surfaces ◽

Picard Number

For any integer d ≥ 5, the Noether–Lefschetz locus, denoted NL d, parametrizes smooth degree d surfaces in ℙ3 with Picard number at least 2. It is well-known (due to works of Voisin, Green and others) that the largest irreducible component of NL d is of codimension (in the space of all smooth surfaces in ℙ3 of degree d) equal to d-3 and parametrizes surfaces containing a line. In this article we study for an integer 3 ≤ r < d, the sub-locus of NL d, denoted NL r,d, parametrizing surfaces with Picard number at least r. A conjecture of Griffiths and Harris states the largest component of NL r,d is of codimension [Formula: see text] and the irreducible component of NL r,d parametrizing the surfaces containing r - 1 coplanar lines is of this codimension. We prove this statement in the case r ≪ d.

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Generalized adjunction and applications

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100064409 ◽

1986 ◽

Vol 99 (3) ◽

pp. 457-472 ◽

Cited By ~ 67

Author(s):

Paltin Ionescu

Keyword(s):

Linear System ◽

Hyperplane Section ◽

Arbitrary Dimension ◽

Adjoint System ◽

Ample Divisor ◽

Projective Surface ◽

Adjoint Systems ◽

Contraction Theorem ◽

New Perspective ◽

Complex Projective

The linear system |K + C| ‘adjoint’ to a curve C on a projective surface was studied by the classical Italian geometers. The adjoint system to a hyperplane section H of smooth projective surface was investigated systematically, in modern terms, by Sommese [22] and Van de Ven [26]. The map associated to the linear system |K + (r−1)H|, where H is a hyperplane section of a smooth variety of arbitrary dimension r, was used to classify submanifolds of ℙn with ‘small invariants’ (e.g. degree, sectional genus, etc.); see [10]. On the other hand, Sommese [23, 24, 25] studied adjoint systems to a smooth ample divisor H on a smooth threefold X and obtained, as applications, many interesting results about the pair (X, H). As noticed independently by several authors (see e.g. [17], [4], [11]) the appearance of Mori's deep contribution [20] (see also [21]) put the subject of adjunction in a new perspective. Accordingly, the present paper–which relies heavily on Mori's results and on the contraction theorem due to Kawamata-Shokurov (see [14])–contains a systematical study of various adjoint systems to an ample (possibly non-effective) divisor on a manifold of arbitrary dimension. More precisely, the main result (which is contained in Section 1) gives the precise description of polarized pairs (X, H), where X is a complex projective mani–fold of dimension r and H an ample divisor on it (not necessarily effective), such that Kx + iH is not semiample (respectively ample) for 1 ≤ i = r + 1, r, r − 1, r − 2 (respectively i = r + 1, r, r − 1).

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On the complement of a nef and big divisor on an algebraic variety

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100074983 ◽

1996 ◽

Vol 120 (3) ◽

pp. 411-422 ◽

Cited By ~ 1

Author(s):

Francesco Russo

Keyword(s):

Algebraic Variety ◽

Cartier Divisor ◽

Ample Divisor ◽

Closed Field ◽

Finitely Generated ◽

Complete Variety ◽

Algebraically Closed Field

Let X be an algebraic (complete) variety over a fixed algebraically closed field k. To every Cartier divisor D on X, we can associate the graded k-algebra . As is known, for a semi-ample divisor D, R(X, D) is a finitely generated k-algebra (see [21] or [9]), while this property is no longer true for arbitrary nef and big divisors (see [21]).

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Vertically of (-1)-Lines in Scrolls Over Smooth Surfaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-1991-038-3 ◽

1991 ◽

Vol 34 (2) ◽

pp. 236-239

Author(s):

Antonio Lanteri

Keyword(s):

Smooth Surface ◽

Ample Divisor ◽

Smooth Surfaces ◽

Conic Bundle ◽

Smooth Complex ◽

Adjoint Bundle ◽

Complex Projective

AbstractLet S be a smooth surface contained as an ample divisor in a smooth complex projective threefold X, which is a P1 -bundle, and assume that induces OP1 (1) on the fibres of X. The following fact is proven. The restriction to S of the bundle projection of X is exactly the reduction morphism of the pair provided that this one is not a conic bundle. The proof is very simple and does not involve any consideration on the nefness of the adjoint bundle Some applications of the proof are given.

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On polarized normal varieties, I

Nagoya Mathematical Journal ◽

10.1017/s0027763000022741 ◽

1986 ◽

Vol 104 ◽

pp. 175-211 ◽

Cited By ~ 4

Author(s):

T. Matsusaka

Keyword(s):

Positive Integer ◽

Linear System ◽

Linear Systems ◽

Cartier Divisor ◽

Normal Variety ◽

Easy Consequence ◽

Normal Varieties ◽

Complete Linear System

In this paper, we deal with the first part of an application of our Riemann-Roch type inequalities (cf. [13], [23]) toward deformations of polarized normal varieties. In Chapter I, we discuss the problem of eliminating fixed components from complete linear systems defined by multiples of a given divisor. Let Un be a complete normal variety and Y0 an ample Cartier divisor on U. The main result of the chapter is that there is a positive integer c0, predicted by the first two leading coefficients of the polynomial X(U, O(m Y0)), such that the complete linear system Λ(rc0Y0) has no fixed component whenever r is a positive integer (which is essentially contained in Theorem 1.1 (cf. [13], Lemma 5.2)). An easy consequence of this result is that when n = 2, we can find another positive integer c1 predicted by the same coefficients as above, such that rc1c0Y0 is very ample on U whenever r is a positive integer. Even though this has been generalized to n = 3 by J. Kollár (cf. [12]), we have included this in Section 3 since it is very simple.

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On threefolds with low sectional genus

Nagoya Mathematical Journal ◽

10.1017/s0027763000000325 ◽

1986 ◽

Vol 101 ◽

pp. 27-36 ◽

Cited By ~ 4

Author(s):

Mauro Beltrametti ◽

Marino Palleschi

Keyword(s):

Linear System ◽

General Problem ◽

Smooth Surface ◽

Point Of View ◽

Ample Divisor ◽

Sectional Genus ◽

Complete Linear System

The general problem of rebuilding the threefolds X endowed with a given ample divisor H, possibly non-effective, is closely related to the study of the complete linear system |KX + H| adjoint to H. Many powerful results are known about |KX + H|, for instance when the linear system | H | contains a smooth surface or, more particularly, when H is very ample (e.g. see Sommese [S1] and [S2]). From this point of view we study some properties of |KX + H |, which turn out to be very useful in the description of the threefolds X polarized by an ample divisor H whose arithmetic virtual genus g(H) is sufficiently low.

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Stereo Electron Microscopy Applied to High Resolution Freeze-Etch Replicas

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100068527 ◽

1970 ◽

Vol 28 ◽

pp. 306-307

Author(s):

L. Andrew Staehelin

Keyword(s):

Electron Microscopy ◽

Plastic Deformation ◽

High Resolution ◽

Smooth Surfaces ◽

Small Particles ◽

Membrane Surfaces ◽

Wide Acceptance ◽

New Information ◽

Number Of Particles ◽

Small Holes

Freeze-etched membranes usually appear as relatively smooth surfaces covered with numerous small particles and a few small holes (Fig. 1). In 1966 Branton (1“) suggested that these surfaces represent split inner mem¬brane faces and not true external membrane surfaces. His theory has now gained wide acceptance partly due to new information obtained from double replicas of freeze-cleaved specimens (2,3) and from freeze-etch experi¬ments with surface labeled membranes (4). While theses studies have fur¬ther substantiated the basic idea of membrane splitting and have shown clearly which membrane faces are complementary to each other, they have left the question open, why the replicated membrane faces usually exhibit con¬siderably fewer holes than particles. According to Branton's theory the number of holes should on the average equal the number of particles. The absence of these holes can be explained in either of two ways: a) it is possible that no holes are formed during the cleaving process e.g. due to plastic deformation (5); b) holes may arise during the cleaving process but remain undetected because of inadequate replication and microscope techniques.

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