On stable rationality of Fano threefolds and del Pezzo fibrations

We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.

Stable Rationality of Cyclic Covers of Projective Spaces

The main aim of this paper is to show that a cyclic cover of ℙn branched along a very general divisor of degree d is not stably rational, provided that n ≥ 3 and d ≥ n + 1. This generalizes the result of Colliot-Thélène and Pirutka. Generalizations for cyclic covers over complete intersections and applications to suitable Fano manifolds are also discussed.

Stable Rationality of del Pezzo Fibrations of Low Degree Over Projective Spaces

The main aim of this article is to show that a very general three-dimensional del Pezzo fibration of degrees 1, 2, and 3 is not stably rational except for a del Pezzo fibration of degree 3 belonging to explicitly described two families. Higher-dimensional generalizations are also discussed and we prove that a very general del Pezzo fibration of degrees 1, 2, and 3 defined over the projective space is not stably rational provided that the anti-canonical divisor is not ample.

Stable rationality of higher dimensional conic bundles

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 $n\geq 3$. Comment: Final version. To appear in Epijournal de Geometrie Algebrique