Divisor Class Group Research Articles

On local divisor class groups of complete intersections

Samuel conjectured in 1961 that a (Noetherian) local complete intersection ring that is a UFD in codimension at most three is itself a UFD. It is said that Grothendieck invented local cohomology to prove this fact. Following the philosophy that a UFD is nothing else than a Krull domain (that is, a normal domain, in the Noetherian case) with trivial divisor class group, we take a closer look at the Samuel–Grothendieck Theorem and prove the following generalization: Let A be a local Cohen–Macaulay ring.(1)A is a normal domain if and only if A is a normal domain in codimension at most 1.(2)Suppose that A is a normal domain and a complete intersection. Then the divisor class group of A is a subgroup of the projective limit of the divisor class groups of the localizations Ap, where p runs through all prime ideals of height at most 3 in A. We use this fact to describe for an integral Noetherian locally complete intersection scheme X the gap between the groups of Weil and Cartier divisors, generalizing in this case the classical result that these two concepts coincide if X is locally a UFD.

On the rational cuspidal divisor class groups of Drinfeld modular curves X0(𝔭r)

Let [Formula: see text] be the rational cuspidal divisor class group of the Drinfeld modular curve [Formula: see text] for a prime power level [Formula: see text]. We relate the rational cuspidal divisors of degree 0 on [Formula: see text] with [Formula: see text]-quotients, where [Formula: see text] is the Drinfeld discriminant function. As a result, we are able to determine explicitly the structure of [Formula: see text] for arbitrary prime [Formula: see text] and [Formula: see text].

The Picard index of a surface with torus action

AbstractWe consider normal rational projective surfaces with torus action and provide a formula for their Picard index, that means the index of the Picard group inside the divisor class group. As an application, we classify the log del Pezzo surfaces with torus action of Picard number one up to Picard index $$ 10\,000 $$ 10 000 .

The divisor class group of a discrete polymatroid

In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete polymatroid, its toric ring is studied deeply for several classes of polymatroids.

Torsionfreeness for divisor class groups of toric rings of integral polytopes

In the present paper, we give some sufficient conditions for Cl(k[P]) to be torsionfree, where Cl(k[P]) denote the divisor class group of the toric ring k[P] of an integral polytope P. We prove that Cl(k[P]) is torsionfree if P is compressed, and Cl(k[P]) is torsionfree if P is a (0,1)-polytope which has at most dim⁡P+2 facets. Moreover, we characterize the toric rings of (0,1)-polytopes in the case Cl(k[P])≅Z.

The rational cuspidal subgroup of J0(p2M)$J_0(p^2M)$ with M squarefree

AbstractFor a positive integer N, let be the modular curve over and its Jacobian variety. We prove that the rational cuspidal subgroup of is equal to the rational cuspidal divisor class group of when for any prime p and any squarefree integer M. To achieve this, we show that all modular units on can be written as products of certain functions , which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on under a mild assumption.

The rational torsion subgroup of J0(N)

Let N be a positive integer and let J0(N) be the Jacobian variety of the modular curve X0(N). For any prime p≥5 whose square does not divide N, we prove that the p-primary subgroup of the rational torsion subgroup of J0(N) is equal to that of the rational cuspidal divisor class group of X0(N), which is explicitly computed in [33]. Also, we prove the same assertion holds for p=3 under the extra assumption that either N is not divisible by 3 or there is a prime divisor of N congruent to −1 modulo 3.

Some Properties of n-semidualizing Modules

Let R be a commutative noetherian ring. The n-semidualizing modules of R are generalizations of its semidualizing modules. We will prove some basic properties of n-semidualizing modules. Our main result and example shows that the divisor class group of a Gorenstein determinantal ring over a field is the set of isomorphism classes of its 1-semidualizing modules. Finally, we pose some questions about n-semidualizing modules.

Orbits of the Automorphism Group of Horospherical Varieties, and Divisor Class Group

В 2013 г. Бажов доказал критерий того, что две точки полного торического многообразия лежат в одной орбите связной компоненты единицы группы автоморфизмов. Этот критерий сформулирован в терминах группы классов дивизоров. В том же году Аржанцев и Бажов получили аналогичный критерий для аффинного торического многообразия. В работе доказывается необходимое условие, аналогичное этому критерию, в случаях аффинного и проективного орисферических многообразий.

Orbits of the Automorphism Group of Horospherical Varieties, and Divisor Class Group

Orbits of the Automorphism Group of Horospherical Varieties, and Divisor Class Group

Class groups of open Richardson varieties in the Grassmannian are trivial

We prove that the divisor class group of any open Richardson variety in the Grassmannian is trivial. Our proof uses Nagata’s criterion, localizing the coordinate ring at a suitable set of Plücker coordinates. We prove that these Plücker coordinates are prime elements by showing that the subscheme they define is an open subscheme of a positroid variety. Our results hold over any field and over the integers.

The rational cuspidal divisor class group of X0(N)

For any positive integer N, we completely determine the structure of the rational cuspidal divisor class group of X0(N), which is conjecturally equal to the rational torsion subgroup of J0(N). More specifically, for a given prime ℓ, we construct a rational cuspidal divisor Zℓ(d) for any non-trivial divisor d of N. Also, we compute the order of the linear equivalence class of Zℓ(d) and show that the ℓ-primary subgroup of the rational cuspidal divisor class group of X0(N) is isomorphic to the direct sum of the cyclic subgroups generated by the linear equivalence classes of Zℓ(d).

On the distribution of prime divisors in Krull monoid algebras

In the present work, we prove that every class of the divisor class group of a Krull monoid algebra contains infinitely many prime divisors. Several attempts to this result have been made in the literature so far, unfortunately with open gaps. We present a complete proof of this fact.

Canonical Modules and Class Groups of Rees-Like Algebras

Rees-like algebras have played a major role in settling the Eisenbud–Goto conjecture. This paper concerns the structure of the canonical module of the Rees-like algebra and its class groups. Via an explicit computation based on linkage, we provide an explicit and surprisingly well-structured resolution of the canonical module in terms of a type of double-Koszul complex. Additionally, we give descriptions of both the divisor class group and the Picard group of a Rees-like algebra.

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.

Derived categories of singular surfaces

We develop an approach that allows one to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite-dimensional algebras. We first explain how to induce a semiorthogonal decomposition of a surface X with rational singularities from a semiorthogonal decomposition of its resolution. In the case when X has cyclic quotient singularities, we introduce the condition of adherence for the components of the semiorthogonal decomposition of the resolution that allows one to identify the components of the induced decomposition of X with derived categories of local finite-dimensional algebras. Further, we present an obstruction in the Brauer group of X to the existence of such a semiorthogonal decomposition, and show that in the presence of the obstruction a suitable modification of the adherence condition gives a semiorthogonal decomposition of the twisted derived category of X . We illustrate the theory by exhibiting a semiorthogonal decomposition for the untwisted or twisted derived category of any normal projective toric surface depending on whether its Weil divisor class group is torsion-free or not. For weighted projective planes we compute the generators of the components explicitly and relate our results to the results of Kawamata based on iterated extensions of reflexive sheaves of rank 1.

Pullbacks of 𝜅 classes on \overline{ℳ}_{0,𝓃}

The moduli space M ¯ 0 , n \overline {\mathcal {M}}_{0,n} carries a codimension- d d Chow class κ d \kappa _{d} . We consider the subspace K n d \mathcal {K}^{d}_{n} of A d ( M ¯ 0 , n , Q ) A^d(\overline {\mathcal {M}}_{0,n},\mathbb {Q}) spanned by pullbacks of κ d \kappa _d via forgetful maps. We find a permutation basis for K n d \mathcal {K}^{d}_{n} , and describe its annihilator under the intersection pairing in terms of d d -dimensional boundary strata. As an application, we give a new permutation basis of the divisor class group of M ¯ 0 , n \overline {\mathcal {M}}_{0,n} .

Ladder determinantal rings over normal domains

We explicitly describe the divisor class groups and semidualizing modules for ladder determinantal rings with coefficients in an arbitrary normal domain for arbitrary ladders, not necessarily connected, and all sizes of minors.

Grothendieck groups, convex cones and maximal Cohen–Macaulay points

Let R be a commutative noetherian ring. Let $$\mathsf {H}(R)$$ be the quotient of the Grothendieck group of finitely generated R-modules by the subgroup generated by pseudo-zero modules. Suppose that the $$\mathbb {R}$$ -vector space $$\mathsf {H}(R)_\mathbb {R}=\mathsf {H}(R)\otimes _\mathbb {Z}\mathbb {R}$$ has finite dimension. Let $$\mathsf {C}(R)$$ (resp. $$\mathsf {C}_r(R)$$ ) be the convex cone in $$\mathsf {H}(R)_\mathbb {R}$$ spanned by maximal Cohen–Macaulay R-modules (resp. maximal Cohen–Macaulay R-modules of rank r). We explore the interior, closure and boundary, and convex polyhedral subcones of $$\mathsf {C}(R)$$ . We provide various equivalent conditions for R to have only finitely many rank r maximal Cohen–Macaulay points in $$\mathsf {C}_r(R)$$ in terms of topological properties of $$\mathsf {C}_r(R)$$ . Finally, we consider maximal Cohen–Macaulay modules of rank one as elements of the divisor class group $${\text {Cl}}(R)$$ .

Divisor class group arithmetic on C3,4 curves

Divisor class group arithmetic on C3,4 curves