Related papers: Potentially stably rational conic bundles over non…

We study stable rationality properties of conic bundles over rational surfaces.

This is a survey on (lack of) stable rationality over arbitrary fields (including algebraically closed fields). Topics addressed include: Rationality and unirationality, R-equivalence on rational points, Chow groups of zero-cycles, Galois…

We classify Galois actions on Picard lattices of del Pezzo surfaces of degrees 1,2, and 3 giving rise to minimal surfaces with no cohomological obstructions to stable rationality.

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 prove that a very general nonsingular conic bundle $X\rightarrow\mathbb{P}^{n-1}$ embedded in a projective vector bundle of rank $3$ over $\mathbb{P}^{n-1}$ is not stably rational if the anti-canonical divisor of $X$ is not ample and…

We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree…

We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers.

A double cover $Y$ of $\mathbb{P}^1 \times \mathbb{P}^2$ ramified over a general $(2,2)$-divisor will have the structure of a geometrically standard conic bundle ramified over a smooth plane quartic $\Delta \subset \mathbb{P}^2$ via the…

Yanchevski\u{i} had asked whether conic bundle surfaces over $\mathbf{P}^1_k$ are unirational when $k$ is a finite field. We give a partial answer to his question by showing that for quasi-finite fields $k$ (e.g. finite fields) a regular…

In this paper, it is proved that certain stable rank-3 vector bundles can be written as extensions of line bundles and stable rank-2 bundles. As an application, we show the rationality of certain moduli spaces of stable rank-3 bundles over…

Let $k$ be a field with char $k \not= 2$, $X$ be an affine surface defined by the equation $z^2=P(x)y^2+Q(x)$ where $P(x), Q(x) \in k[x]$ are separable polynomials. We will investigate the rationality problem of $X$ in terms of the…

We prove a general specialization theorem which implies stable irrationality for a wide class of quadric surface bundles over rational surfaces. As an application, we solve with the exception of two cases, the stable rationality problem for…

We show that the property of being rationally $K$-stable passes from the fibers of a continuous $C(X)$-algebra to the ambient algebra, under the assumption that the underlying space $X$ is compact, metrizable, and of finite covering…

This paper considers the links between the geometry of the various flag manifolds of loop groups and bundles over families of rational curves. Aa an application, a stability result for the moduli on a rational ruled surface of G-bundles…

Let $p: Y \ra X$ be a Galois cover of smooth projective curves over $\CC$ with Galois group $\Gamma$. This paper is devoted to the study of principal orthogonal and symplectic bundles $E$ on $Y$ to which the action of $\Gamma$ on $Y$ lifts.…

We study whether the norm one torus associated with a finite separable non-Galois field extension $K/k$ is $p$-retract rational over $k$ for a prime $p$, focusing on the case where the Galois group of the Galois closure of $K/k$ is either…

We study the arithmetic of del Pezzo surfaces $Y$ of degree 2 over a function field, and in particular, the cokernel of the homomorphism from the Picard group to the Galois-invariants of the geometric Picard group $\operatorname{Pic} Y…

For a semisimple complex algebraic group $G$ we determine the rational cohomology and the Hodge-Tate structure of the moduli stack ${\mathscr B}un_{G,X}$ of principal $G$-bundles over a connected smooth complex projective variety $X$ of…

This expository paper is concerned with the rationality problems for three-dimensional algebraic varieties with a conic bundle structure. We discuss the main methods of this theory. We sketch the proofs of certain principal results, and…

We give a complete answer to the rationality problem (up to stable $k$-equivalence) for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ of $K/k$ whose Galois closures $L/k$ are $A_5\simeq {\rm PSL}_2(\mathbb{F}_4)$ and ${\rm…