The divisor class group of splittable Zariski surfaces

2018 ◽  
Vol 17 (08) ◽  
pp. 1850150 ◽  
Author(s):  
Jeffrey Lang
Keyword(s):  
Class Group ◽  
Closed Field ◽  
Class Groups ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Algebraically Closed Field ◽  
Normal Variety

Let [Formula: see text] be an algebraically closed field of characteristic [Formula: see text] and [Formula: see text]. Let [Formula: see text] be a normal variety defined by the equation [Formula: see text]. If [Formula: see text] is a product of [Formula: see text] linear factors in [Formula: see text], not necessarily homogeneous, where [Formula: see text] equals the degree of [Formula: see text], we say that [Formula: see text] is splittable. In this paper, we calculate the group of Weil divisors of a splittable [Formula: see text] for a generic [Formula: see text] when the characteristic of [Formula: see text] equals [Formula: see text]. In the process, we describe a general approach to studying class groups of splittable [Formula: see text] that we believe should yield results when [Formula: see text].

scholarly journals Divisor class groups and graded canonical modules of multisection rings

2013 ◽  
Vol 212 ◽  
pp. 139-157
Author(s):  
Kazuhiko Kurano
Keyword(s):  
Mild Condition ◽  
Projective Variety ◽  
Class Group ◽  
Class Groups ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Canonical Module ◽  
Krull Domain

AbstractWe describe the divisor class group and the graded canonical module of the multisection ringT(X;D1,…,Ds) for a normal projective varietyXand Weil divisorsD1,…,DsonXunder a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor.

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 Divisor class groups and graded canonical modules of multisection rings

2013 ◽  
Vol 212 ◽  
pp. 139-157 ◽  
Author(s):  
Kazuhiko Kurano
Keyword(s):  
Mild Condition ◽  
Projective Variety ◽  
Class Group ◽  
Class Groups ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Canonical Module ◽  
Krull Domain

AbstractWe describe the divisor class group and the graded canonical module of the multisection ring T (X;D1,…, Ds) for a normal projective variety X and Weil divisors D1,…, Ds on X under a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor.

scholarly journals Some factorization properties of Krull domains with infinite cyclic divisor class group

1994 ◽  
Vol 96 (2) ◽  
pp. 97-112 ◽  
Author(s):  
David F. Anderson ◽  
Scott T. Chapman ◽  
William W. Smith
Keyword(s):  
Class Group ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Krull Domains

scholarly journals The divisor class group of a Quot scheme

2010 ◽  
Vol 3 (0) ◽  
pp. 1-14
Author(s):  
Rafael Hernández ◽  
Daniel Ortega
Keyword(s):  
Class Group ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Quot Scheme

scholarly journals ON FACTORIZATION IN BLOCK MONOIDS FORMED BY $\{\bar{1},\bar{a}\}$ IN $\mathbb{Z}_{n}$

2003 ◽  
Vol 46 (2) ◽  
pp. 257-267 ◽  
Author(s):  
Scott T. Chapman ◽  
William W. Smith
Keyword(s):  
Class Group ◽  
Subject Classification ◽  
Division Algorithm ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Irreducible Elements ◽  
Complete Set ◽  
Cross Number ◽  
Block Monoids ◽  
Block Monoid

AbstractWe consider the factorization properties of block monoids on $\mathbb{Z}_n$ determined by subsets of the form $S_a=\{\bar{1},\bar{a}\}$. We denote such a block monoid by $\mathcal{B}_a(n)$. In §2, we provide a method based on the division algorithm for determining the irreducible elements of $\mathcal{B}_a(n)$. Section 3 offers a method to determine the elasticity of $\mathcal{B}_a(n)$ based solely on the cross number. Section 4 applies the results of §3 to investigate the complete set of elasticities of Krull monoids with divisor class group $\mathbb{Z}_n$.AMS 2000 Mathematics subject classification: Primary 20M14; 20D60; 13F05

scholarly journals A Theorem About Maximal Cohen–Macaulay Modules

2020 ◽  
Author(s):  
Thomas Polstra
Keyword(s):  
Natural Number ◽  
Regular Ring ◽  
Class Group ◽  
Torsion Subgroup ◽  
Finitely Generated ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Frobenius Endomorphism

Abstract It is shown that for any local strongly $F$-regular ring there exists natural number $e_0$ so that if $M$ is any finitely generated maximal Cohen–Macaulay module, then the pushforward of $M$ under the $e_0$th iterate of the Frobenius endomorphism contains a free summand. Consequently, the torsion subgroup of the divisor class group of a local strongly $F$-regular ring is finite.

scholarly journals The Noether–Lefschetz theorem for the divisor class group

2009 ◽  
Vol 322 (9) ◽  
pp. 3373-3391 ◽  
Author(s):  
G.V. Ravindra ◽  
V. Srinivas
Keyword(s):  
Class Group ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Lefschetz Theorem

On the structure of generalized Albanese varieties

1992 ◽  
Vol 111 (2) ◽  
pp. 267-272
Author(s):  
Hurit nsiper
Keyword(s):  
Algebraic Group ◽  
Class Group ◽  
Effective Divisor ◽  
Closed Field ◽  
Projective Surface ◽  
Mapping Property ◽  
Rational Map ◽  
Algebraically Closed Field ◽  
Albanese Map ◽  
Albanese Variety

Given a smooth projective surface X over an algebraically closed field k and a modulus (an effective divisor) m on X, one defines the idle class group Cm(X) of X with modulus m (see 1, chapter III, section 4). The corresponding generalized Albanese variety Gum and the generalized Albanese map um:X|m|Gum have the following universal mapping property (2): if :XG is a rational map into a commutative algebraic group which induces a homomorphism Cm(X)G(k) (1, chapter III, proposition 1), then factors uniquely through um.

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