Related papers: Rational singularities associated to pairs

We give conditions on a finite set of series of rational numbers to ensure that they are algebraically independent. Specialising our results to polynomials of lower degree, we also obtain new results on irrationality and $mathbb{Q}$-linear…

We study rational streams (over a field) from a coalgebraic perspective. Exploiting the finality of the set of streams, we present an elementary and uniform proof of the equivalence of four notions of representability of rational streams:…

Let $X$ be a finite, 2-dimensional cell complex. The curvature invariants $\rho_\pm(X)$ and $\sigma_\pm(X)$ were defined in [13], and a programme of conjectures was outlined. Here, we prove the foundational result that the quantities…

We study metrical properties of various subsequences associated to the sequence of rational approximants coming from the continued fraction of an irrational number. Our methods build upon Bosma, Jager and Wiedijk's proof of the…

Given an ambient variety $X$ and a fixed subvariety $Z$ we give sufficient conditions for the existence of a boundary $\Delta$ such that $Z$ is a log canonical center for the pair $(X, \Delta)$. We also show that under some additional…

A classical argument was introduced by Khintchine in 1926 in order to exhibit the existence of totally irrational singular linear forms in two variables. This argument was subsequently revisited and extended by many authors. For instance,…

Let $G$ be a semisimple algebraic group defined over an algebraically closed field of characteristic 0 and $P$ be a parabolic subgroup of $G$. Let $M$ be a $P$-module and $V$ be a $P$-stable closed subvariety of $M$. We show in this paper…

In this paper, we discuss a generalization of log canonical singularities in the non-$\mathbb{Q}$-Gorenstein setting. We prove that if a normal complex projective variety has a non-invertible polarized endomorphism, then it has log…

We propose an approach for showing rationality of an algebraic variety $X$. We try to cover $X$ by rational curves of certain type and count how many curves pass through a generic point. If the answer is $1$, then we can sometimes reduce…

For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd…

Given a projective intersection of two quadrics X in at least 9 variables, the quantitative behaviour of the rational points on X is investigated under the assumption that X contains a pair of conjugate singular points defined over the…

The problem to decide whether a given rational function in several variables is positive, in the sense that all its Taylor coefficients are positive, goes back to Szeg\H{o} as well as Askey and Gasper, who inspired more recent work. It is…

Let X be a geometrically rational (or more generally, separably rationally connected) variety over a finite field K. We prove that if K is large enough then X contains many rational curves defined over K. As a consequence we prove that…

We compare some algebras appeared in the recent attempts to prove resolution of singularities in positive characteristic. We also construct an algebra which encodes the same information and it is equivalent, up to integral closure, to the…

This article has the following aims: (1) Extend the notion of fuchsian singularities (of first kind) to base fields of arbitrary characteristic. (2) Discuss their relationship to mathematical objects of a different nature. (3) Provide a…

We formalize the quantum arithmetic, i.e. a relationship between number theory and operator algebras. Namely, it is proved that rational projective varieties are dual to the $C^*$-algebras with real multiplication. Our construction fits all…

This paper demonstrates some connections between the coefficients of a Taylor series $f(z)=\ds\sum_{n=0}^\infty a_n z^n$ and singularities of the function. There are many known results of this type, for example, counting the number of poles…

Normal ideals on regular uncountable cardinals are familiar objects. We investigate ideals that are pleasant--while a normal ideal is closed under arbitrary diagonal unions, a pleasant ideal is closed only under diagonal unions indexed by…

In this work, it is proved that a set of numbers closed under addition and whose representations in a rational base numeration system is a rational language is not a finitely generated additive monoid. A key to the proof is the definition…

We introduce a new variant of tight closure and give an interpretation of adjoint ideals via this tight closure. As a corollary, we prove that a log pair $(X,\Delta)$ is plt if and only if the modulo $p$ reduction of $(X,\Delta)$ is…