Discriminant loci of ample and spanned line bundles

Let (X,L,V) be a triplet where X is an irreducible smooth complex projective variety, L is an ample and spanned line bundle on X and V⊆H⁰(X,L) spans L. The discriminant locus D(X,V)⊂|V| is the algebraic subset of singular elements of |V|. We study the components of D(X,V) in connection with the jumping sets of (X,V), generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of (X,L,V)) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided.     

Mahler measures and computations with regulators

In this work we apply the techniques that were developed in [M.N. Lalín, An algebraic integration for Mahler measure, Duke Math. J. 138 (2007), in press] in order to study several examples of multivariable polynomials whose Mahler measure is expressed in terms of special values of the Riemann zeta function or Dirichlet L-series. The examples may be understood in terms of evaluations of regulators. Moreover, we apply the same techniques to the computation of generalized Mahler measures, in the sense of Gon and Oyanagi [Y. Gon, H. Oyanagi, Generalized Mahler measures and multiple sine functions, Internat. J. Math. 15 (5) (2004) 425-442].     

Class numbers with many prime factors

Here, we construct infinitely many number fields of any given degree d>1 whose class numbers have many prime factors.     

Between compactness and completeness

Call a sequence in a metric space cofinally Cauchy if for each positive ε there exists a cofinal (rather than residual) set of indices whose corresponding terms are ε-close. We give a number of new characterizations of metric spaces for which each cofinally Cauchy sequence has a cluster point. For example, a space has such a metric if and only each continuous function defined on it is uniformly locally bounded. A number of results exploit a measure of local compactness functional that we introduce. We conclude with a short proof of Romaguera's Theorem: a metrizable space admits such a metric if and only if its set of points having a compact neighborhood has compact complement.     

α‐Tetraphosphorus Trichalcogenide Diamides and Imides containing both Sulfur and Selenium

Reaction of a mixture of bicyclic phosphorus sulfide selenide iodides α‐P₄S_nSe_3−nI₂ (n = 0–3) with PrⁱNH₂ and Et₃N gave corresponding diamides α‐P₄S_nSe_3−n(NHPrⁱ)₂ (n = 0–3) and imides α‐P₄S_nSe_3−n(μ‐NPrⁱ) (n = 2–3), identified in solution by ³¹P NMR. In one isomer of α‐P₄S₂Se(μ‐NPrⁱ), the C₂ symmetry of imides such as α‐P₄S₃(μ‐NPrⁱ) was broken, allowing relative assignment of ²J NMR couplings to the PNP bridge and the PSP bridge opposite to it. The coupling through the sulfur bridge was found to be reduced to ca. zero, in contrast to previous assumptions for this class of compounds. Ab initio models were calculated at the MPW1PW91/svp level for the sulfide selenide imides and for a selection of bond rotamers of the diamides, and at the MPW1PW91/LanL2DZ(d) level for the sulfide selenide diiodides. Different skeletal isomers were prevalent for the mixed chalcogenide diamides than for the diiodides, showing that exchange of chalcogen between skeletal positions took place in the amination reaction even at room temperature. Similar differences to those observed were predicted by the models, suggesting that equilibrium was attained.     

A relative version of Webb’s theorem on the equivariant chain complex of a subgroup complex

We prove a relative version of the theorem of Webb that the augmented chain complex of the p-subgroup complex of a finite group, considered as a complex of modules for the group, is homotopy equivalent to a complex of projectives. This allows us to take into account the group of automorphisms of the group.     

Reduction principles in the theory of selections

In the present article many theorems about existence of continuous selections for a set-valued mapping into the metrizable spaces are extended to mappings into non-metrizable spaces.     

Rigidity in motivic homotopy theory

We show that extensions of algebraically closed fields induce full and faithful functors between the respective motivic stable homotopy categories with finite coefficients.     

Bézout rings with almost stable range 1

Elementary divisor domains were defined by Kaplansky [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949) 464-491] and generalized to rings with zero-divisors by Gillman and Henriksen [L. Gillman, M. Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. 82 (1956) 362-365]. In [M.D. Larsen, W.J. Lewis, T.S. Shores, Elementary divisor rings and finitely presented modules, Trans. Amer. Math. Soc. 187 (1) (1974) 231-248], it was also proved that if a Hermite ring satisfies (N), then it is an elementary divisor ring. The aim of this article is to generalize this result (as well as others) to a much wider class of rings. Our main result is that Bézout rings whose proper homomorphic images all have stable range 1 (in particular, neat rings) are elementary divisor rings.     

Intermediary rings in normal pairs

We establish in this paper a result that gives the number of intermediary rings between R and S where (R,S) is a normal pair of rings. This result answers in particular a question which was left open in [A. Jaballah, Finiteness of the set of intermediary rings in a normal pair, Saintama Math. J. 17 (1999) 59-61]. Further applications are also given.