Divisor Class Group Research Articles

Cox rings and combinatorics

Given a variety X X with a finitely generated total coordinate ring, we describe basic geometric properties of X X in terms of certain combinatorial structures living in the divisor class group of X X . For example, we describe the singularities, we calculate the ample cone, and we give simple Fano criteria. As we show by means of several examples, the results allow explicit computations. As immediate applications we obtain an effective version of the Kleiman-Chevalley quasiprojectivity criterion, and the following observation on surfaces: a normal complete surface with finitely generated total coordinate ring is projective if and only if any two of its non-factorial singularities admit a common affine neighbourhood.

Chains of Factorizations and Factorizations with Successive Lengths

ABSTRACT Let H be a commutative cancellative monoid. H is called atomic if every nonunit a ∈ H decomposes (in general in a highly nonunique way) into a product of irreducible elements (atoms) u i of H. The integer n is called the length of (†), and L(a) = {n ∈ ℕ | a decomposes into n irreducible elements of H} is called the set of lengths of a. Two integers k < l are called successive lengths of a if L(a) ∩ {m ∈ ℕ | k ≤ m ≤ l} = {k, l}. For a ∈ H we denote by Z n (a) the set of factorizations of a with length n. Suppose now that H is one of the following monoids: i. A congruence monoid in a Dedekind domain with finite residue fields. ii. H = D\\ {0}, where D is a Noetherian domain having the following properties: is a Krull domain with finite divisor class group, and is a finite ring. iii. H = D\\ {0}, where D is a one-dimensional Noetherian domain with finite normalization and finite Picard group (but possibly infinite residue fields). Let a ∈ H. In the present article, we investigate the structure of concatenating chains in Z n (a) as well as the relation between Z k (a) and Z l (a) if k and l are successive lengths of a. The work continues earlier investigations in Foroutan (2003), Foroutan and Geroldinger (2004), and Hassler (to appear).

L-Class groups of cyclic function fields of degree l

Let l be a prime number and K be a cyclic extension of degree l of the rational function field F q ( T ) over a finite field of characteristic ≠ l. We study the l-part of the ideal class group of the integral closure of F q [ T ] in K, and the l-part of the group of divisor classes of degree 0 of K as Galois modules. Using class field theory, we can describe explicitly part of the structure of these l-class groups. As an application, we get (for l = 2 ) bounds for the order of the 4-torsion on J X ( F q ) , the group of points defined over F q on the Jacobian of a hyperelliptic curve X / F q .

The Grothendieck-Lefschetz theorem for normal projective varieties

We prove that for a normal projective variety X X in characteristic 0 0 , and a base-point free ample line bundle L L on it, the restriction map of divisor class groups Cl ⁡ ( X ) → Cl ⁡ ( Y ) \operatorname {Cl} (X)\to \operatorname {Cl}(Y) is an isomorphism for a general member Y ∈ | L | Y\in |L| provided that dim ⁡ X ≥ 4 \dim {X}\geq 4 . This is a generalization of the Grothendieck-Lefschetz theorem, for divisor class groups of singular varieties.

On L-functions of cyclotomic function fields

We study two criterions of cyclicity for divisor class groups of function fields, the first one involves Artin L-functions and the second one involves “affine” class groups. We show that, in general, these two criterions are not linked.

L1-norm packings from function fields

In this paper, we study some packings in a cube, namely, how to pack n points in a cube so as to maximize the minimal distance. The distance is induced by the L 1 -norm which is analogous to the Hamming distance in coding theory. Two constructions with reasonable parameters are obtained, by using some results from a function field including divisor class group, narrow ray class group, and so on. We also present some asymptotic results of the two packings.

Approximate liftings in local algebra and a theorem of Grothendieck

Let A and B denote local rings such that A = B / tB , where t is a regular nonunit, and let b denote an ideal in B such that the A -ideal a = b / ( t ) has codimension ⩾ 2 . Let F be a reflexive O X -module, where X = Spec A - V ( a ) . Under suitable conditions on A and B and assuming that Ext X 2 ( F , F ) = 0 and E xt X 1 ( F , O X ) = 0 , it is shown in this article that the dual sheaf F v can be extended to a reflexive coherent O Y -module, where Y = Spec B - V ( b ) . The infinitesimal procedure that leads to this sheaf extension makes use of the injective theory of sheaves. Applications to homomorphisms of divisor class groups come about as a consequence of this result, and a strong connection with Grothendieck's theorem on parafactoriality is drawn.

A mean value theorem for orders of degree zero divisor class groups of quadratic extensions over a function field

Let k be a function field of one variable over a finite field with the characteristic not equal to two. In this paper, we consider the prehomogeneous representation of the space of binary quadratic forms over k. We have two main results. The first result is on the principal part of the global zeta function associated with the prehomogeneous vector space. The second result is on a mean value theorem for degree zero divisor class groups of quadratic extensions over k, which is a consequence of the first one.

Normal affine surfaces with C*-actions

A classification of normal affine surfaces admitting a $\bf C^*$-action was given in the work of Bia{\l}ynicki-Birula, Fieseler and L. Kaup, Orlik and Wagreich, Rynes and others. We provide a simple alternative description of such surfaces in terms of their graded rings as well as by defining equations. This is based on a generalization of the Dolgachev-Pinkham-Demazure construction in the case of a hyperbolic grading. As an apllication we determine the structure of singularities, of the orbits and the divisor class groups for such surfaces.

Elasticities of Krull domains with finite divisor class group

Fix any positive integer b , and consider the set ϒ( Z b ) of all values ρ ( R ), where R is a Krull domain with divisor class group Z b , and ρ ( R ) denotes the elasticity of R . We focus on the case b =2 p , showing that 2p−1 p ∈ϒ( Z 2p ) and ϒ( Z p )⊊ϒ( Z 2p ) . We also present a precise description of the set ϒ ⊕ i=1 k Z 2 .

Rees algebras of conormal modules

We deal with classes of prime ideals whose associated graded ring is isomorphic to the Rees algebra of the conormal module. With such prime ideals we describe the divisor class groups of the associated graded rings. Furthermore we study the relationship between the normality of the conormal module and the completeness of components of the associated graded ring.

Elasticities of Krull domains with finite divisor class group

Fix any positive integer b, and consider the set ϒ( Z b) of all values ρ( R), where R is a Krull domain with divisor class group Z b , and ρ( R) denotes the elasticity of R. We focus on the case b=2 p, showing that 2p−1 p ∈ϒ( Z 2p) and ϒ( Z p)⊊ϒ( Z 2p) . We also present a precise description of the set ϒ ⊕ i=1 k Z 2 .

Bunches of Cones in the Divisor Class Group : a new Combinatorial Language for Toric Varieties

As an alternative to the description of a toric variety by a fan in the lattice of one-parameter subgroups, we present a new language in terms of what we call bunches—these are certain collections of cones in the vector space of rational divisor classes. The correspondence between these bunches and fans is based on the classical Gale duality. The new combinatorial language allows a much more natural description of geometric phenomena around divisors of toric varieties than the usual method by fans does. For example, the numerically effective cone and the ample cone of a toric variety can be read off immediately from its bunch. Moreover, the language of bunches appears to be useful for classification problems.

The total coordinate ring of a normal projective variety

The total coordinate ring TC(X) of a normal variety is a generalization of the ring introduced and studied by Cox in connection with a toric variety. Consider a normal projective variety X with divisor class group Cl(X), and let us assume that it is a finitely generated free abelian group. We define the total coordinate ring of X to be TC(X) = oplus_{D} H^0 (X, O_X (D)), where the sum as above is taken over all Weil divisors of X contained in a fixed complete system of representatives of Cl(X). We prove that for any normal projective variety X, TC(X) is a UFD, this is a corollary of a more general theorem that is proved in the paper. (Berchtold and Haussen proved the unique factorization for a smooth variety independently.) We also prove that for X, the blow up of P^2 along a finite number of collinear points, TC(X) is Noetherian. We also give an example that TC(X) is not Noetherian but oplus_n H^0 (X, O(nD)) is Noetherian for any Weil divisor D.

On the p-adic Riemann hypothesis for the zeta function of divisors

In this paper, we continue the investigation of the zeta function of divisors, as introduced by the first author in Wan (in: D. Jungnickel, H. Niederreiter (Eds.), Finite Fields and Applications, Springer, Berlin, 2001, pp. 437–461; Manuscripta Math. 74 (1992) 413), for a projective variety over a finite field. Assuming that the set of effective divisors in the divisor class group forms a finitely generated monoid, then there are four conjectures about this zeta function: p-adic meromorphic continuation, rank and pole relation, p-adic Riemann hypothesis, and simplicity of zeros and poles. This paper proves all four conjectures when the Chow the group of divisors is of rank one. Also, an example with higher rank is provided where all four conjectures hold.

Restriction of divisor classes to hypersurfaces in characteristic p

The injectivity of the restriction homomorphism on divisor class groups to hypersurfaces has been studied by Grothendieck, Danilov, Lipman, and Griffith & Weston, among others. In particular, when A is a Noetherian normal domain of equicharacteristic zero and A/ fA satisfies R 1, Spiroff established a map Cl( A)→Cl(( A/ fA)′), where ( A/ fA)′ represents the integral closure of A/ fA, and gave some conditions for injectivity. In this paper, the authors continue in the same vein, but in the case of characteristic p>0. In addition, when the hypersurface A/ fA is normal, they provide further enlightenment about the kernel of Cl( A)→Cl( A/ fA). Finally, using the second author's previous results, they exhibit a new class of examples for which the kernel is non-trivial.

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

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

On The Asymptotic Values of Length Functions In Krull And Finitely Generated commutative Monoids

AbstractLet M be a commutative cancellative atomic monoid. We consider the behaviour of the asymptotic length functions and on M. If M is finitely generated and reduced, then we present an algorithm for the computation of both and where x is a nonidentity element of M. We also explore the values that the functions and can attain when M is a Krull monoid with torsion divisor class group, and extend a well-known result of Zaks and Skula by showing how these values can be used to characterize when M is half-factorial.

The Affine Class Group of a Normal Scheme

We study the property of a normal scheme, that the complement of every hypersurface is an affine scheme. To this end we introduce the affine class group. It is a factor group of the divisor class group and measures the deviation from this property. We study the behaviour of the affine class group under faithfully flat extensions and under the formation of products, and we compute it for different classes of rings.

Divisorial Linear Algebra of Normal Semigroup Rings

We investigate the minimal number of generators μ and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound C there exist, up to isomorphism, only finitely many divisorial ideals I such that μ(I)≤C. It follows that there exist only finitely many Cohen–Macaulay divisor classes. Moreover, we determine the minimal depth of all divisorial ideals and the behaviour of μ and depth in ‘arithmetic progressions’ in the divisor class group.