\begin{thebibliography}{CTP16b}

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

\bibitem[BC94]{batyrev-cox}
Victor~V. Batyrev and David~A. Cox.
\newblock On the {H}odge structure of projective hypersurfaces in toric
  varieties.
\newblock {\em Duke Math. J.}, 75(2):293--338, 1994.

\bibitem[CDK95]{CDK}
Eduardo Cattani, Pierre Deligne, and Aroldo Kaplan.
\newblock On the locus of {H}odge classes.
\newblock {\em J. Amer. Math. Soc.}, 8(2):483--506, 1995.

\bibitem[CHM88]{CHM}
Ciro Ciliberto, Joe Harris, and Rick Miranda.
\newblock General components of the {N}oether-{L}efschetz locus and their
  density in the space of all surfaces.
\newblock {\em Math. Ann.}, 282(4):667--680, 1988.

\bibitem[CT95]{ct-pure}
Jean-Louis Colliot-Th\'el\`ene.
\newblock Birational invariants, purity and the {G}ersten conjecture.
\newblock In {\em {$K$}-theory and algebraic geometry: connections with
  quadratic forms and division algebras ({S}anta {B}arbara, {CA}, 1992)},
  volume~58 of {\em Proc. Sympos. Pure Math.}, pages 1--64. Amer. Math. Soc.,
  Providence, RI, 1995.

\bibitem[CTO89]{CTO}
Jean-Louis Colliot-Th\'el\`ene and Manuel Ojanguren.
\newblock Vari\'et\'es unirationnelles non rationnelles: au-del\`a de l'exemple
  d'{A}rtin et {M}umford.
\newblock {\em Invent. Math.}, 97(1):141--158, 1989.

\bibitem[CTP16a]{ct-pir-cyclic}
Jean-Louis Colliot-Th\'el\`ene and Alena Pirutka.
\newblock Cyclic covers that are not stably rational.
\newblock {\em Izvestiya RAN, Ser. Math.}, 80(4):35--47, 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{\'e}l{\`e}ne 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[dFF13]{dFF}
Tommaso de~Fernex and Davide Fusi.
\newblock Rationality in families of threefolds.
\newblock {\em Rend. Circ. Mat. Palermo (2)}, 62(1):127--135, 2013.

\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[Has99]{hassett-JAG}
Brendan Hassett.
\newblock Some rational cubic fourfolds.
\newblock {\em J. Algebraic Geom.}, 8(1):103--114, 1999.

\bibitem[HKT15]{HKT-conic}
Brendan Hassett, Andrew Kresch, and Yuri Tschinkel.
\newblock Stable rationality and conic bundles.
\newblock {\em Math. Annalen}, to appear, 2015.
\newblock {\tt arXiv:1503.08497}.

\bibitem[Kah08]{Kahnbook}
Bruno Kahn.
\newblock {\em Formes quadratiques sur un corps}, volume~15 of {\em Cours
  Sp\'ecialis\'es [Specialized Courses]}.
\newblock Soci\'et\'e Math\'ematique de France, Paris, 2008.

\bibitem[Pir16]{pirutka-survol}
Alena Pirutka.
\newblock Varieties that are not stably rational, zero-cycles and unramified
  cohomology, 2016.
\newblock {\tt arXiv:1603.09261}.

\bibitem[Spr52]{Springer}
Tonny~Albert Springer.
\newblock Sur les formes quadratiques d'indice z\'ero.
\newblock {\em C. R. Acad. Sci. Paris}, 234:1517--1519, 1952.

\bibitem[Tot16a]{totaro-JAMS}
Burt Totaro.
\newblock Hypersurfaces that are not stably rational.
\newblock {\em J. Amer. Math. Soc.}, 29(3):883--891, 2016.

\bibitem[Tot16b]{totaro-15}
Burt Totaro.
\newblock Rationality does not specialize among terminal varieties.
\newblock {\em Math. Proc. Cambridge Philos. Soc.}, 161(1):13--15, 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, english edition, 2007.
\newblock Translated from the French by Leila Schneps.

\bibitem[Voi15a]{voisin-stable}
Claire Voisin.
\newblock ({S}table) rationality is not deformation invariant, 2015.
\newblock {\tt arXiv:1511.03591}.

\bibitem[Voi15b]{Voisin}
Claire Voisin.
\newblock Unirational threefolds with no universal codimension {$2$} cycle.
\newblock {\em Invent. Math.}, 201(1):207--237, 2015.

\end{thebibliography}