Related papers: La connexit\'e rationnelle en arithm\'etique
It is a conjecture of Koll\'ar that a variety $X$ with rational singularities in some open subvariety $U$ has a rationalification; that is, a proper, birational morphism $f: Y \rightarrow X$ such that $Y$ has rational singularities, and…
Non-normal modal logics, interpreted on neighbourhood models which generalise the usual relational semantics, have found application in several areas, such as epistemic, deontic, and coalitional reasoning. We present here preliminary…
A selection of basic results on Borel reducibility of ideals and equivalence relations, especially those with comparably short proofs. This is an unfinished text as yet. Some proofs have missing parts and loose ends. [email protected] and…
Based upon the axiom of choice it is proved that the cardinality of the rational numbers is not less than the cardinality of the irrational numbers. This contradicts a main result of transfinite set theory and shows that the axiom of choice…
Long ago, Fontaine formulated conjectures (now theorems) relating \'etale and de Rham cohomologies of algebraic varieties over $p$-adic fields. In an earlier work we have shown that pro-\'etale and de Rham cohomologies of analytic varieties…
We introduce new motivic invariants of arbitrary varieties over a perfect field. These cohomological invariants take values in the category of one-motives (considered up to isogeny in positive characteristic). The algebraic definition of…
In this paper, we prove a general result computing the number of rational points of bounded height on a projective variety $V$ which is covered by lines. The main technical result used to achieve this is an upper bound on the number of…
In this short note, we show that a construction by Ottem provides an example of a rationally connected variety that is not birationally equivalent to a Mori dream space. This answers in the negative (at least in the category of terminal…
In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the…
We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian…
We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize…
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…
Although conventional logical systems based on logical calculi have been successfully used in mathematics and beyond, they have definite limitations that restrict their application in many cases. For instance, the principal condition for…
We study the multiplicities of pure motives modulo numerical equivalence, which are defined as scalars comparing the tannakian trace with the ring-theoretic trace. Our general set-up is that of a rigid semi-simple tensor category such that…
We extend Hacon--M\textsuperscript{c}Kernan's rational chain connectedness theorem to the complex analytic setting. As a consequence, we prove that the fibers of any resolution of singularities of complex analytic kawamata log terminal…
An observable canonical form is formulated for the set of rational systems on a variety each of which is a single-input-single-output, affine in the input, and a minimal realization of its response map. The equivalence relation for the…
The authors prove that $\Cal M_{15}$ is rationally connected
The article is devoted to the investigation of representation of rational numbers by Cantor series. Necessary and sufficient conditions for a rational number to be representable by a positive Cantor series are formulated for the case of an…
In despair, as Deligne (2000) put it, of proving the Hodge and Tate conjectures, we can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the…
We prove that in a family of projective threefolds defined over an algebraically closed field, the locus of rational fibers is a countable union of closed subsets of the locus of separably rationally connected fibers. When the ground field…