\begin{thebibliography}{APBvB16}

\bibitem[APBvB16]{auel-co}
Asher Auel, Alena Pirutka, Christian B\"{o}hning, and Hans-Christian von
  Bothmer.
\newblock Conic bundles with nontrivial unramified {B}rauer group over
  threefolds, 2016.
\newblock {\tt arXiv:1610.04995}.

\bibitem[APS15]{APS}
Asher Auel, R.~Parimala, and V.~Suresh.
\newblock Quadric surface bundles over surfaces.
\newblock {\em Doc. Math.}, (Extra vol.: Alexander S. Merkurjev's sixtieth
  birthday):31--70, 2015.

\bibitem[Ara75]{arason}
J{\'o}n~Kr. Arason.
\newblock Cohomologische invarianten quadratischer {F}ormen.
\newblock {\em J. Algebra}, 36(3):448--491, 1975.

\bibitem[BCP97]{Magma}
Wieb Bosma, John Cannon, and Catherine Playoust.
\newblock The {M}agma algebra system. {I}. {T}he user language.
\newblock {\em J. Symbolic Comput.}, 24(3-4):235--265, 1997.
\newblock Computational algebra and number theory (London, 1993).

\bibitem[Bea77]{Beau77}
Arnaud Beauville.
\newblock Vari\'et\'es de {P}rym et jacobiennes interm\'ediaires.
\newblock {\em Ann. Sci. \'Ecole Norm. Sup. (4)}, 10(3):309--391, 1977.

\bibitem[Bea16]{beau-6}
Arnaud Beauville.
\newblock A very general sextic double solid is not stably rational.
\newblock {\em Bull. Lond. Math. Soc.}, 48(2):321--324, 2016.

\bibitem[CTP16a]{ct-pir-cyclic}
Jean-Louis Colliot-Th\'el\`ene and Alena Pirutka.
\newblock Cyclic covers that are not stably rational.
\newblock {\em Izv. Ross. Akad. Nauk Ser. Mat.}, 80(4):35--48, 2016.

\bibitem[CTP16b]{ct-pirutka}
Jean-Louis Colliot-Th\'el\`ene and Alena Pirutka.
\newblock Hypersurfaces quartiques de dimension 3 : non rationalit\'e stable.
\newblock {\em Ann. Sci. \'Ec. Norm. Sup\'er. (4)}, 49(2):371--397, 2016.

\bibitem[CTV12]{CTV}
Jean-Louis Colliot-Th\'el\`ene and Claire Voisin.
\newblock Cohomologie non ramifi\'ee et conjecture de {H}odge enti\`ere.
\newblock {\em Duke Math. J.}, 161(5):735--801, 2012.

\bibitem[GS]{M2}
Daniel~R. Grayson and Michael~E. Stillman.
\newblock Macaulay2, a software system for research in algebraic geometry.
\newblock Available at \url{http://www.math.uiuc.edu/Macaulay2/}.

\bibitem[HKT16]{HKT-conic}
Brendan Hassett, Andrew Kresch, and Yuri Tschinkel.
\newblock Stable rationality and conic bundles.
\newblock {\em Math. Ann.}, 365(3-4):1201--1217, 2016.

\bibitem[HPT16a]{HPT}
Brendan Hassett, Alena Pirutka, and Yuri Tschinkel.
\newblock Stable rationality of quadric surface bundles over surfaces, 2016.
\newblock {\tt arXiv:1603.09262 }.

\bibitem[HPT16b]{HPT-double}
Brendan Hassett, Alena Pirutka, and Yuri Tschinkel.
\newblock A very general quartic double fourfold is not stably rational, 2016.
\newblock {\tt arXiv:1605.03220}.

\bibitem[HT16]{HT-fano}
Brendan Hassett and Yuri Tschinkel.
\newblock On stable rationality of {F}ano threefolds and del {P}ezzo
  fibrations.
\newblock {\em J. Reine Angew. Math.}, to appear, 2016.
\newblock {\tt arXiv:1601.07074}.

\bibitem[KO17]{krylov}
Igor Krylov and Takuzo Okada.
\newblock Stable rationality of del {P}ezzo fibrations of low degree over
  projective spaces, 2017.
\newblock {\tt arXiv:1701.08372}.

\bibitem[Las89]{Las}
Yves Laszlo.
\newblock Th\'eor\`eme de {T}orelli g\'en\'erique pour les intersections
  compl\`etes de trois quadriques de dimension paire.
\newblock {\em Invent. Math.}, 98(2):247--264, 1989.

\bibitem[Oka16]{okada}
Takuzo Okada.
\newblock Stable rationality of cyclic covers of projective spaces, 2016.
\newblock {\tt arXiv:1604.08417}.

\bibitem[Pir16]{pirutka-survol}
Alena Pirutka.
\newblock Varieties that are not stably rational, zero-cycles and unramified
  cohomology.
\newblock In {\em Algebraic geometry---{S}alt {L}ake {C}ity 2015}, volume to
  appear. 2016.
\newblock {\tt arXiv:1603.09261}.

\bibitem[Ter90]{Ter}
Tomohide Terasoma.
\newblock Infinitesimal variation of {H}odge structures and the weak global
  {T}orelli theorem for complete intersections.
\newblock {\em Ann. of Math. (2)}, 132(2):213--235, 1990.

\bibitem[Tot16]{totaro-JAMS}
Burt Totaro.
\newblock Hypersurfaces that are not stably rational.
\newblock {\em J. Amer. Math. Soc.}, 29(3):883--891, 2016.

\bibitem[Voi07]{voisin-book}
Claire Voisin.
\newblock {\em Hodge theory and complex algebraic geometry. {II}}, volume~77 of
  {\em Cambridge Studies in Advanced Mathematics}.
\newblock Cambridge University Press, Cambridge, 2007.

\bibitem[Voi15]{Voisin}
Claire Voisin.
\newblock Unirational threefolds with no universal codimension {$2$} cycle.
\newblock {\em Invent. Math.}, 201(1):207--237, 2015.

\bibitem[Voi17]{VoisinJEMS}
Claire Voisin.
\newblock On the universal {CH{$_0$}} group of cubic hypersurfaces.
\newblock {\em J. Eur. Math. Soc. (JEMS)}, 19(6):1619--1653, 2017.

\end{thebibliography}