Flat ring epimorphisms and universal localizations of commutative rings

We study different types of localizations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localization in the sense of Cohn and Schofield; and (b) when such universal localizations are classical rings of fractions. In order to find such criteria, we use the theory of support and we analyse the specialization closed subset associated to a flat ring epimorphism. In case the underlying ring is locally factorial or of Krull dimension one, we show that all flat ring epimorphisms are universal localizations. Moreover, it turns out that an answer to the question of when universal localizations are classical depends on the structure of the Picard group. We furthermore discuss the case of normal rings, for which the divisor class group plays an essential role to decide if a given flat ring epimorphism is a universal localization. Finally, we explore several (counter)examples which highlight the necessity of our assumptions.

Improved divisor arithmetic on generic hyperelliptic curves

The divisor class group of a hyperelliptic curve defined over a finite field is a finite abelian group at the center of a number of important open questions in algebraic geometry, number theory and cryptography. Many of these problems lend themselves to numerical investigation, and as emphasized by Sutherland [14, 13], fast arithmetic in the divisor class group is crucial for their efficiency. Besides, implementations of these fundamental operations are at the core of the algebraic geometry packages of widely-used computer algebra systems such as Magma and Sage.

A Theorem About Maximal Cohen–Macaulay Modules

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.

Rational Picard Group of Moduli of Pointed Hyperelliptic Curves

We determine the rational divisor class group of the moduli spaces of smooth pointed hyperelliptic curves and of their Deligne–Mumford compactification, over the field of complex numbers.

Computing the invariants of intersection algebras of principal monomial ideals

We continue the study of intersection algebras [Formula: see text] of two ideals [Formula: see text] in a commutative Noetherian ring [Formula: see text]. In particular, we exploit the semigroup ring and toric structures in order to calculate various invariants of the intersection algebra when [Formula: see text] is a polynomial ring over a field and [Formula: see text] are principal monomial ideals. Specifically, we calculate the [Formula: see text]-signature, divisor class group, and Hilbert–Samuel and Hilbert–Kunz multiplicities, sometimes restricting to certain cases in order to obtain explicit formulæ. This provides a new class of rings where formulæ for the [Formula: see text]-signature and Hilbert–Kunz multiplicity, dependent on families of parameters, are provided.

The divisor class group of splittable Zariski surfaces

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].

Hochster's theta pairing and numerical equivalence

Let (A, ) be a local hypersurface with an isolated singularity. We show that Hochster's theta pairing θA vanishes on elements that are numerically equivalent to zero in the Grothendieck group of A under the mild assumption that Spec A admits a resolution of singularities. This extends a result by Celikbas-Walker. We also prove that when dimA = 3, Hochster's theta pairing is positive semi-definite. These results combine to show that the counter-example of Dutta-Hochster-McLaughlin to the general vanishing of Serre's intersection multiplicity exists for any three dimensional isolated hypersurface singularity that is not a UFD and has a desingularization. We also show that, if A is three dimensional isolated hypersurface singularity that has a desingularization, the divisor class group is finitely generated torsion-free. Our method involves showing that θA gives a bivariant class for the morphism Spec (A/) → Spec A.