Related papers: The rationality problem for conic bundles

We classify all positive integers n and r such that (stably) non-rational complex r-fold quadric bundles over rational n-folds exist. We show in particular that for any n and r, a wide class of smooth r-fold quadric bundles over projective…

We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional…

This mainly expository text translates into stack language the proof of King and Schofield for the rationality of moduli schemes of vector bundles on a curve in the coprime case. An appendix summarizes some basic properties of the relevant…

We present here some conjectures on the diagonalizability of uniform principal bundles on rational homogeneous spaces, that are natural extensions of classical theorems on uniform vector bundles on the projective space, and study the…

We consider the algebraization problem for principal bundles with reductive structure group, defined on the complement of a closed subset Z in a proper formal scheme. We show that, when Z is of codimension at least 3, an algebraization…

Refereed version to appear in Michigan Mathematical Journal. A mistake in the last section of the previous version has been corrected. The new title exactly describes the main result obtained. Building on the geometry of cubic surfaces and…

We survey some recent progress in the theory of vector bundles on algebraic varieties and related questions in algebraic K-theory.

We study the rationality properties of the moduli space $\mathcal{A}_g$ of principally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the results to arithmetic questions. In particular we show that any principally polarised abelian…

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…

We construct smooth rational real algebraic varieties of every dimension $\ge$ 4 which admit infinitely many pairwise non-isomorphic real forms.

The paper adresses the problem of reasoning with ambiguities. Semantic representations are presented that leave scope relations between quantifiers and/or other operators unspecified. Truth conditions are provided for these representations…

We consider rationally connected complex projective manifolds M and show that their loop spaces--infinite dimensional complex manifolds--have properties similar to those of M. Furthermore, we give a finite dimensional application concerning…

We show how quiver representations and their invariant theory natu- rally arise in the study of some moduli spaces parametrizing bundles dened on an algebraic curve, and how they lead to ne results regarding the geometry of these spaces.

Rational pairs generalize the notion of rational singularities to reduced pairs $(X,D)$. In this paper we deal with the problem of determining whether a normal variety $X$ has a rationalizing divisor, i.e. a reduced divisor $D$ such that…

We give a method to construct stable vector bundles whose rank divides the degree over curves of genus bigger than one. The method complements the one given by Newstead. Finally, we make some systematic remarks and observations in…

We study Brauer-Severi surface bundles over smooth projective varieties via root stacks, with a view towards applications to failure of stable rationality.

A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between…

For sufficiently ample linear systems on rational surfaces we show that a very general associated Brauer-Severi surface bundle is not stably rational.

In this note, motivated by the recent preprint of W. Ou, we pursue three main objectives. The first is to make progress towards the positivity of the relative canonical bundle in the K\"ahler setting. In the second part, we provide a proof…

This note studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one restrict such a function to a subvariety? Can one extend such a function from a…