Related papers: Rationality does not specialize among terminal var…

In characteristic zero, quotient singularities are log terminal. Moreover, we can check whether a quotient variety is canonical or not by using only the age of each element of the relevant finite group if the group does not have…

Let $X$ be a nonsingular rational variety. We prove that $X\times \mathbb{C}^2$ is uniformly rational. It follows that nonsingular stably rational varieties are stably uniformly rational.

Let G be a connected linear algebraic group over an algebraically closed field k, and let H be a connected closed subgroup of G. We prove that the homogeneous variety G/H is a rational variety over k whenever H is solvable, or when dim(G/H)…

A resolution-free definition of rational singularities is introduced, and it is proved that for a variety admitting a resolution of singularities, so in particular in characteristic zero, this is equivalent to the usual definition. It is…

We prove that a finite group is rational if and only if it has a set of permutation characters which separate conjugacy classes. It follows from this that a finite group is rational if and only if it has a representation as a permutation…

In this paper we look for necessary and sufficient conditions for a genus one fibration to have rational curves. We show that a projective variety with log terminal singularities that admits a relatively minimal genus one fibration…

Nonsingular projective varieties which are both convex and rationally connected are considered. We ask whether such varieties must be algebraic homogeneous spaces G/P. In case X is a complete intersection, an affirmative answer is obtained…

We study the problem of counting the number of varieties in families which have a rational point. We give conditions on the singular fibres that force very few of the varieties in the family to contain a rational point, in a precise…

We prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities. Our proof is based on motivic specialization techniques and the criterion of Larsen and…

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…

We show that there do not exist semistable varietes defined over the rationals with good reduction outside one prime p if p = 2, 3, 5 or 7.

(On the fundamental group of rationnally connected varieties.) I show that the fundamental group of a normal variety which is rationally chain connected is finite. The proof holds in non-zero characteristic. Je d\'emontre que le groupe…

Rationality has been an intriguing topic for several decades. Even the scope of definition of rationality across different subjects varies. Several theories (e.g., game theory) initially evolved on the basis that agents (e.g., humans) are…

The splitting principle states that morphisms in a derived category do not "split" accidentally. This has been successsfully applied in several characterizations of rational, DB, and other singularities. In this article I prove a general…

We prove that there exist rational but not uniformly rational smooth algebraic varieties. The proof is based on computing a certain numerical obstruction developed in the case of compactifications of affine spaces. We show that for some…

A variety is unirational if it is dominated by a rational variety. A variety is rationally connected if two general points can be joined by a rational curve. This paper aims to show that the two notions can cooperate and, building on…

The theory of regularity is a philosophical perspective in which laws of nature are just descriptions, that is to say, laws of nature do not govern the world. Moreover, according to the theory of regularity, the number of laws of nature…

Let $X$ be a projective variety with log terminal singularities and vanishing augmented irregularity. In this paper we prove that if $X$ admits a relatively minimal genus one fibration then it does contain a subvariety of codimension one…

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 exhibit families of smooth projective threefolds with both stably rational and non stably rational fibers.