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

1996 ◽  
Vol 120 (3) ◽  
pp. 411-422 ◽  
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]).

Fundamental, Picard, and Class Groups of Rings of Invariants

1976 ◽  
Vol 28 (3) ◽  
pp. 659-664 ◽  
Author(s):  
Andy R. Magid
Keyword(s):  
Fundamental Group ◽  
Algebraic Group ◽  
Algebraic Variety ◽  
Closed Field ◽  
Affine Variety ◽  
Class Groups ◽  
Finitely Generated ◽  
Divisor Class ◽  
Algebraically Closed Field ◽  
Affine Normal

Let G be a n affine algebraic group over the algebraically closed field k, and let V be an affine, normal algebraic variety over k on which G acts. Suppose that the ring of invariants k [F]G is finitely generated over k, and let W be the affine variety with k[W] = k[V]G. The purpose of this paper is to show that the induced homomorphism from the étale fundamental group of V to that of W is surjective, and to examine the consequences of this observation in terms of the relations between the Picard and divisor class groups of k[V] and k[W],

scholarly journals A remark on the Grothendieck-Lefschetz theorem about the Picard group

1978 ◽  
Vol 71 ◽  
pp. 169-179 ◽  
Author(s):  
Lucian Bădescu
Keyword(s):  
Algebraic Variety ◽  
Picard Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Lefschetz Theorem

Let K be an algebraically closed field of arbitrary characteristic. The term “variety” always means here an irreducible algebraic variety over K. The notations and the terminology are borrowed in general from EGA [4].

Ideals with Trivial Conormal Bundle

1980 ◽  
Vol 32 (1) ◽  
pp. 210-218 ◽  
Author(s):  
A. V. Geramita ◽  
C. A. Weibel
Keyword(s):  
Polynomial Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Question Answering ◽  
General Theorem ◽  
Closed Field ◽  
Natural Question ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Conormal Bundle

Throughout this paper all rings considered will be commutative, noetherian with identity. If R is such a ring and M is a finitely generated R-module, we shall use v(M) to denote that non-negative integer with the property that M can be generated by v(M) elements but not by fewer.Since every ideal in a noetherian ring is finitely generated, it is a natural question to ask what v(I) is for a given ideal I. Hilbert's Nullstellensatz may be viewed as the first general theorem dealing with this question, answering it when I is a maximal ideal in a polynomial ring over an algebraically closed field.More recently, it has been noticed that the properties of an R-ideal I are intertwined with those of the R-module I/I2.

scholarly journals Corrections to “Seminormal rings and weakly normal varieties”

1987 ◽  
Vol 107 ◽  
pp. 147-157 ◽  
Author(s):  
Marie A. Vitulli
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Regular Function ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Regular Functions ◽  
Normal Varieties

In “Seminormal rings and weakly normal varieties” we introduced the notion of a c-regular function on an algebraic variety defined over an algebraically closed field of characteristic zero. Our intention was to describe those k-valued functions on a variety X that become regular functions when lifted to the normalization of X, but without any reference to the normalization in the definition. That is, we aspired to identify the c-regular functions on X with the regular functions on the weak normalization of X

scholarly journals Algebraic Barth-Lefschetz theorems

1996 ◽  
Vol 142 ◽  
pp. 17-38 ◽  
Author(s):  
Lucian Bădescu
Keyword(s):  
Positive Integer ◽  
Projective Space ◽  
Algebraic Variety ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Basic Properties ◽  
Positive Integers ◽  
Lefschetz Theorems ◽  
Algebraic Scheme

We shall work over a fixed algebraically closed field k of arbitrary characteristic. By an algebraic variety over k we shall mean a reduced algebraic scheme over k. Fix a positive integer n and e = (e0, el,…, en) a system of n + 1 weights (i.e. n + 1 positive integers e0, el,…, en). If k[T0, Tl,…, Tn] is the polynomial k-algebra in n + 1 variables, graded by the conditions deg(Ti) = ei i = 0, 1,…, n, denote by Pn(e) = Proj(k[T0, T1,…, Tn]) the n-dimensional weighted projective space over k of weights e. We refer the reader to [3] for the basic properties of weighted projective spaces.

scholarly journals On ample divisors

1982 ◽  
Vol 86 ◽  
pp. 155-171 ◽  
Author(s):  
Lucian Bădescu
Keyword(s):  
Cartier Divisor ◽  
Closed Field ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Smooth Projective Varieties ◽  
Ample Divisors

In this paper we are dealing with the following problem: determine all normal (or smooth) projective varieties X over an algebraically closed field k supporting a given variety Y as an ample Cartier divisor.

scholarly journals Every Curve of Genus not Greater Than Eight Lies on a K3 Surface

2008 ◽  
Vol 190 ◽  
pp. 183-197 ◽  
Author(s):  
Manabu Ide
Keyword(s):  
Smooth Curve ◽  
Ample Divisor ◽  
Closed Field ◽  
K3 Surface ◽  
Algebraically Closed Field ◽  
Double Points ◽  
Smooth Locus ◽  
Double Coverings

Let C be a smooth irreducible complete curve of genus g ≥ 2 over an algebraically closed field of characteristic 0. An ample K3 extension of C is a K3 surface with at worst rational double points which contains C in the smooth locus as an ample divisor.In this paper, we prove that all smooth curve of genera. 2 ≤ g ≤ 8 have ample K3 extensions. We use Bertini type lemmas and double coverings to construct ample K3 extensions.

Generators of Ideals Defining Certain Surfaces in Projective Space

1996 ◽  
Vol 48 (3) ◽  
pp. 585-595 ◽  
Author(s):  
Sandeep H. Holay
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Ample Divisor ◽  
Plane Curves ◽  
Closed Field ◽  
Divisor Class ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Generating Sets ◽  
Blowing Up

AbstractWe consider the surface obtained from the projective plane by blowing up the points of intersection of two plane curves meeting transversely. We find minimal generating sets of the defining ideals of these surfaces embedded in projective space by the sections of a very ample divisor class. All of the results are proven over an algebraically closed field of arbitrary characteristic.

Moduli spaces of algebraic curves with rational maps

1975 ◽  
Vol 78 (2) ◽  
pp. 283-292 ◽  
Author(s):  
Herbert Lange
Keyword(s):  
Algebraic Variety ◽  
Moduli Spaces ◽  
Algebraic Curves ◽  
Rational Maps ◽  
Closed Field ◽  
Rational Map ◽  
Algebraically Closed Field

Let ℳg be the coarse moduli scheme of curves of genus g. For an algebraically closed field k define is a quasiprojective algebraic variety over k, its dimension being 3g – 3 for g ≥ 2, 1 for g = 1, and 0 for g = 0. It can be considered as the moduli variety for the classes of birationally equivalent curves of genus g over k. For 0 < g, g′ and n ≥ 1 let be the subset of those points of whose corresponding curves possess a rational map of degree n into a curve of genus g′ over k.

Constructing ω-stable structures: rank 2 fields

2000 ◽  
Vol 65 (1) ◽  
pp. 371-391 ◽  
Author(s):  
John T. Baldwin ◽  
Kitty Holland
Keyword(s):  
General Framework ◽  
Saturated Model ◽  
Closed Field ◽  
Morley Rank ◽  
Natural Numbers ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Fast Growing ◽  
Rank 2 ◽  
Stable Structures

AbstractWe provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion of separation of quantifiers which is a condition on the class of expansions of finitely generated models for the expanded theory to have a countable ω-saturated model. We apply these results to construct for each sufficiently fast growing finite-to-one function μ from ‘primitive extensions’ to the natural numbers a theory Tμ of an expansion of an algebraically closed field which has Morley rank 2. Finally, we show that if μ is not finite-to-one the theory may not be ω-stable.

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