Torsion Divisors Research Articles

Torsion divisors of plane curves and Zariski pairs

This paper is devoted to the study of the embedded topology of reducible plane curves having a smooth irreducible component. In previous studies, the relationship between the topology and certain torsion classes in the Picard group of degree zero of the smooth component was implicitly considered. Here this relationship is formulated clearly and a criterion is given for distinguishing the embedded topology in terms of torsion classes. Furthermore, a method is presented for systematically constructing examples of curves where this criterion is applicable, and new examples of Zariski N N -tuples are produced.

Torsion divisors of plane curves with maximal flexes and Zariski pairs

AbstractThere is a close relationship between the embedded topology of complex plane curves and the (group‐theoretic) arithmetic of elliptic curves. In a recent paper, we studied the topology of some arrangements of curves that include a special smooth component, via the torsion properties induced by the divisors in the special curve associated to the remaining components, which is an arithmetic property. When this special curve has maximal flexes, there is a natural isomorphism between its Jacobian variety and the degree zero part of its Picard group. In this paper, we consider curve arrangements that contain a special smooth component with a maximal flex and exploit these properties to obtain Zariski tuples, which show the interplay between topology, geometry, and arithmetic.

The number of torsion divisors in a strongly F-regular ring is bounded by the reciprocal of F-signature

Polstra showed that the cardinality of the torsion subgroup of the divisor class group of a local strongly F-regular ring is finite. We expand upon this result and prove that the reciprocal of the F-signature of a local strongly F-regular ring R bounds the cardinality of the torsion subgroup of the divisor class group of R.

Distribution of integral division points on the algebraic torus

Let $ K$ be a number field with algebraic closure $ \overline K$, and let $ S$ be a finite set of places of $ K$ containing all the infinite ones. Let $ {\it\Gamma }_0$ be a finitely generated subgroup of $ {\mathbb{G}}_{\textup {m}} (\overline K)$, and let $ {\it\Gamma } \subset {\mathbb{G}}_{\textup {m}} (\overline K)$ be the division group attached to $ {\it\Gamma }_0$. Here is an illustration of what we will prove in this article. Fix a proper closed subinterval $ I$ of $ [0, \infty )$ and a nonzero effective divisor $ D$ on $ {\mathbb{G}}_{\textup {m}}$ which is not the translate of any torsion divisor on the algebraic torus $ {\mathbb{G}}_{\textup {m}}$ by any point of $ {\it\Gamma }$ with height belonging to $ I$. Then we prove a statement which easily implies that the set of ``integral division points on $ {\mathbb{G}}_{\textup {m}}$ with height near $ I$'', i.e., the set of points of $ {\it\Gamma }$ with (standard absolute logarithmic Weil) height in $ J$ which are $ S$-integral on $ {\mathbb{G}}_{\textup {m}}$ relative to $ D,$ is finite for some fixed subinterval $ J$ of $ [0, \infty )$ properly containing $ I$. We propose a conjecture on the nondensity of integral division points on semi-abelian varieties with prescribed height values, which generalizes some previously known conjectures as well as this finiteness result for $ {\mathbb{G}}_{\textup {m}}$. Finally, we also propose an analogous version for a dynamical system on $ {\mathbb{P}}^1$.

Sequences of Jacobian Varieties with Torsion Divisors of Quadratic Order

A fortuitous intersection of work on periodic continued fraction expansions in hyperelliptic function fields and the study of parametrized families of quadratic number fields with high class number leads us to discover sequences of hyperelliptic curves whose Jacobians contain torsion divisors of order $g^2$. These sequences generalize those earlier constructed by Flynn and by Leprevost.

Teichmüller curves in genus two: Torsion divisors and ratios of sines

Teichmüller curves in genus two: Torsion divisors and ratios of sines

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.

Factorization Sets and Half-Factorial Sets in Integral Domains

Let R be an atomic integral domain. Suppose that H is a nonempty subset of irreducible elements of R, u is a unit of R, and α 1,..., α n , β 1,..., β m are irreducible elements of R such that (1) α 1... α n = u · β 1... β m . Set H α = { i|α i ∈ H } and H β = { j|β j ∈ H }. H is a factorization set ( F-set) of R if for any equality involving irreducibles of the form (1), | H α| ≠ 0 implies that | H β| ≠ 0. H is a half-factorial set ( HF-set) if any equality of the form (1) implies that | H α| = | H β|. In this paper, we explore in detail the structure of the F-sets and HF-sets of an atomic integral domain R. If P is a nonzero prime ideal of R and J ( R) the set of irreducible elements of R, then set H P = P ∩ J ( R). We show that if F is an F-set of R, then there exists a nonempty set X of nonzero prime ideals of R such that F = ∪ P ∈ X H P . We define an F-set to be minimal if it contains no proper subsets which are F-sets. We then show that in R every F-set can be written as a union of minimal F-sets if and only if R satisfies the Principal Ideal Theorem. If, in addition, every height-one prime ideal of R is the radical of a principal ideal, then this representation is unique. We study the structure of the HF-sets of R and concentrate on the case where R is a Krull domain with torsion divisor class group. For such a domain R, we show that if H is an HF-set of R and P is a nonprincipal prime ideal of R with H P ⊆ H , then H Q ⊆ H for each prime ideal Q of R in the same divisor class as P. We also show that if R is a Dedekind domain with prime class number p ≥ 2 and P ( R) is the set of prime elements of R, then H an HF-set of R implies that either H ∪ P ( R) = J ( R) or H ⊆ P ( R).

Restrictions of Torsion Divisor Classes to Hypersurfaces

Restrictions of Torsion Divisor Classes to Hypersurfaces

Curves of genus $2$ with a rational torsion divisor of order $23$

Curves of genus $2$ with a rational torsion divisor of order $23$

Sequences of rational torsions on abelian varieties

We address the question of how fast the available rational torsion on abelian varieties over ℚ increases with dimension. The emphasis will be on the derivation of sequences of torsion divisors on hyperelliptic curves. Work of Hellegouarch and Lozach (and Klein) may be made explicit to provide sequences of curves with rational torsion divisors of orders increasing linearly with respect to genus. The main results in §2) are applications of a new technique which provide sequences of hyperelliptic curves for all torsions in an interval [a g ,a g +b g ]] wherea g is quadratic ing andb g is linear ing. As well as providing an improvement from linear to quadratic, these results provide a wide selection of torsion orders for potential use by those involved in computer integration. We conclude by considering possible techniques for divisors of non-hyperelliptic curves, and for general abelian varieties.

ANDERSON'S CONJECTURE AND THE MAXIMAL MONOID CLASS OVER WHICH PROJECTIVE MODULES ARE FREE

A positive solution is given to a conjecture of D. F. Anderson (Pacific J. Math. 79(1978), 5-17, p. 11) concerning freeness of finitely generated projective modules over normal monoid algebras. In the special case of torsion divisor class groups, or equivalently, the case of an integral extension, this conjecture was proved in 1982 (see Gubeladze, Generalized Serre problem for affine rings generated by monomials, Izdat. Tbiliss. Gos. Univ., Tbilisi, 1982, and Chouinard, MR 84c: 13009). Using that result, the author obtains a description of the maximal class of commutative monoids satisfying the cancellation condition for which all finitely generated projective modules over the corresponding semigroup algebra (with any principal ideal domain as coefficient ring) are free. Namely, this class turns out to be the so-called seminormal monoids. By the same token a complete answer is given to some questions posed by Chouinard in the paper cited above.Bibliography: 16 titles.

Locally almost factorial integral domains

The simpliest, and most important, class of Krull domains are the factorial domains (UFDs), i.e., those Krull domains with trivial divisor class group. Probably the next simpliest class of Krull domains are the Krull domains with torsion divisor class group, i.e., the almost factorial Krull domains [21]. Almost factorial Krull domains occur very frequently: examples include factorial domains, Krull domains with finite divisor class group, and hence any ring of integers of an algebraic number field. Moreover, Goldman [16, Corollary 21 has shown that any Dedekind domain R has an almost factorial overring A with the same unit group as R. The most important property of almost factorial Krull domains is that they are precisely the Krull domains R such that each subintersection of R is a localization of R. Other characterizations of almost factorial Krull domains may be found in [ 131 or [21]. Almost factorial domains have also been called semifactorial [ 191 and prefactorial. Recently there have been several papers on rings each of whose proper localizations or proper overrings satisfy certain ring-theoretic properties, for example [4,5, 8,9]. Following Fossum [ 13, p. 811, in [4] we defined

Torsion divisors on algebraic curves

Torsion divisors on algebraic curves

A criterion for divisors on algebraic varieties to be torsion divisors

A criterion for divisors on algebraic varieties to be torsion divisors