Motivic and $$\ell $$-adic realizations of the category of singularities of the zero locus of a global section of a vector bundle

We study the motivic and $$\ell $$ ℓ -adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D. Orlov and J. Burke–M. Walker to give a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n.

Projective toric codes

Any integral convex polytope [Formula: see text] in [Formula: see text] provides an [Formula: see text]-dimensional toric variety [Formula: see text] and an ample divisor [Formula: see text] on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on [Formula: see text], obtained by evaluating global section of the line bundle corresponding to [Formula: see text] on every rational point of [Formula: see text]. This work presents an extension of toric codes analogous to the one of Reed–Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with nonzero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope [Formula: see text] and an algorithmic technique to get a lower bound on the minimum distance is described.

Koszul complexes and spectral sequences associated with Lie algebroids

We study some spectral sequences associated with a locally free $${{\mathscr {O}}}_X$$ O X -module $${{\mathscr {A}}}$$ A which has a Lie algebroid structure. Here X is either a complex manifold or a regular scheme over an algebraically closed field k. One spectral sequence can be associated with $${{\mathscr {A}}}$$ A by choosing a global section V of $${{\mathscr {A}}}$$ A , and considering a Koszul complex with a differential given by inner product by V. This spectral sequence is shown to degenerate at the second page by using Deligne’s degeneracy criterion. Another spectral sequence we study arises when considering the Atiyah algebroid $${{{\mathscr {D}}}_{{{\mathscr {E}}}}}$$ D E of a holomolorphic vector bundle $${{\mathscr {E}}}$$ E on a complex manifold. If V is a differential operator on $${{\mathscr {E}}}$$ E with scalar symbol, i.e, a global section of $${{{\mathscr {D}}}_{{{\mathscr {E}}}}}$$ D E , we associate with the pair $$({{\mathscr {E}}},V)$$ ( E , V ) a twisted Koszul complex. The first spectral sequence associated with this complex is known to degenerate at the first page in the untwisted ($${{\mathscr {E}}}=0$$ E = 0 ) case.

A note on transition in discrete gauge groups in F-theory

We observe a new puzzling physical phenomenon in F-theory on the multisection geometry, wherein a model without a gauge group transitions to another model with a discrete [Formula: see text] gauge group via Higgsing. This phenomenon may suggest an unknown aspect of F-theory compactification on multisection geometry lacking a global section. A possible interpretation of this puzzling physical phenomenon is proposed in this note. We also propose a possible interpretation of another unnatural physical phenomenon observed for F-theory on four-section geometry, wherein a discrete [Formula: see text] gauge group transitions to a discrete [Formula: see text] gauge group via Higgsing as described in the previous literature.

Transcendental Liouville Inequalities on Projective Varieties

Let $p$ be an algebraic point of a projective variety $X$ defined over a number field. Liouville inequality tells us that the norm at $p$ of a non-vanishing integral global section of a hermitian line bundle over $X$ is zero or it cannot be too small with respect to the $\sup $ norm of the section itself. We study inequalities similar to Liouville’s for subvarietes and for transcendental points of a projective variety defined over a number field. We prove that almost all transcendental points verify a good inequality of Liouville type. We also relate our methods to a (former) conjecture by Chudnovsky and give two applications to the growth of the number of rational points of bounded height on the image of an analytic map from a disk to a projective variety.

IMAGEN DE LA CIENCIA Y LA TECNOLOGÍA EN LA SECCIÓN ALDEA GLOBAL DEL PERIÓDICO LA NACIÓN

El presente artículo aborda la construcción de discurso periodístico sobre ciencia y tecnología en la sección “Aldea Global” del periódico La Nación de Costa Rica. Se realizó un aná- lisis de contenido de 35 textos periodísticos basado en la teoría del encuadre y la literatura sobre periodismo científico. Los resultados evidencian que la producción periodística de “Aldea Global” se centra en temáticas como la salud y la tecnología comercial, lo cual deja por fuera un gran espectro del quehacer científico. Este fenómeno dificulta una alfabetización y cultura científica, que permita a la población conocer su entorno y tomar mejores decisiones.----This article discusses the construction of journalistic discourse on science and technology in “Aldea Global”, section of the newspaper La Nación of Costa Rica. A content analysis of 35 newspaper articles based on the framing theory and literature on science journalism was performed. The results show that the journalistic production on “Aldea Global” focuses on topics such as health and commercial technology, which leaves a wide spectrum of scientific work out. This phenomenon hinders scientific literacy that enables people to know their environment and make better decisions.

SINGULAR RATIONALLY CONNECTED THREEFOLDS WITH NON-ZERO PLURI-FORMS

This paper is concerned with singular projective rationally connected threefolds $X$ which carry non-zero pluri-forms, that is the reflexive hull of $({\rm\Omega}_{X}^{1})^{\otimes m}$ has a non-zero global section for some positive integer $m$. If $X$ has $\mathbb{Q}$-factorial terminal singularities, then we show that there is a fibration $p$ from $X$ to $\mathbb{P}^{1}$. Moreover, we give a formula for the numbers of $m$-pluri-forms as a function of the ramification of the fibration $p$.

Cohomology bounds for sheaves of dimension one

We find sharp bounds on h0(F) for one-dimensional semistable sheaves F on a projective variety X. When X is the projective plane ℙ2, we study the stratification of the moduli space by the spectrum of sheaves. We show that the deepest stratum is isomorphic to a closed subset of a relative Hilbert scheme. This provides an example of a family of semistable sheaves having the biggest dimensional global section space.

Queens

As the wives, mothers, and daughters of kings, medieval queens acquired their status in one of two ways, either through marriage or, less commonly, through inheritance. The experience of being a queen, in particular as partner to the king, the development of the office of the queen, and the role of queen regent evolved over time with medieval monarchy, and queenship varied across regions as different legal codes and customs informed female inheritance. Women who became queens through marriage often shared the experience of straddling two cultures and two families (natal and marital), and, thus, they were alien outsiders who simultaneously had the greatest access to the center of power, the king. Often women who became queens were not native to the territory with which they became associated and, thus, the names by which they are known, for example, Blanche of Castile, may be misleading: Blanche, who was from Castile, was queen of France through marriage. Queens thus served as intercessors, patrons, and cultural innovators as well as operated as great lords, as rulers, and often, but not always, as mothers. The historiography of medieval queenship is equally varied, beginning with positivist-inspired biographies of the 19th century and subsequently influenced by developments in social history during the 1960s and 1970s and by interdisciplinary and feminist approaches in recent decades. Currently, scholarship simultaneously seeks to recover the histories of individual queens, to understand the specifics of the queen’s office within the institution of the monarchy, and to understand how gender operated at the highest levels of political, cultural, and economic power in the Middle Ages. The first principle of organization for this article is chronological, with sections on Early Medieval Queens (Anglo-Saxon, Celtic, Germanic) and Merovingian Queens and Carolingian Queens. Because queens were always queens of a realm, however, and because the extent (and number) of European monarchies on both the continent and in Britain changed radically in the post-Carolingian era, the remainder of the article is organized both geographically and chronologically, with sections on England (General, Anglo-Norman Queens, Plantagenet Queens, and Lancastrian, York, and Early Tudor England); Scotland, France (sections on Capetian France and Valois France), Germany and Early Medieval Italy, Scandinavia, and the kingdoms of the Iberian Peninsula (sections on Iberia generally as well as Crown of Aragon, León-Castile, and Portugal). In some instances, queens who have merited extensive scholarship are treated in separate sections. The article concludes with sections on the liminal but comparatively important queens/empresses of Byzantium, and the Crusader Kingdom of Jerusalem. While the general focus of this bibliography is on European queens and queenship, it is important to recognize the experience and lives of royal women and queens, or their equivalents, beyond Europe, which are featured in the Global section.