It is assumed that participants have a basic knowledge of étale cohomology or Galois cohomology of fields, as can be found for example in the textbooks [16], [35]. The most important aspects needed to follow this program are summarized, for example, in [32] Section 2 and 3.
Some of the following lectures (especially lectures 4, 7, 8 and 12) contain a lot of material and are based on more than one main paper. Speakers for these talks are asked to plan their talks particularly carefully to avoid getting lost in just one aspect of the talk.
I. UNRAMIFIED COHOMOLOGY
1. The Artin—Mumford example
Artin—Mumford [1] constructed a unirational threefold with torsion in its third integral cohomology. This was the first example of a unirational variety that is not stably rational. Present Beauville's account to this example in Section 6.1—6.4 of [3] (see also §9 in [2]).
2. Fano varieties with non-trivial Artin—Mumford invariant
It remained open until recently whether there are smooth projective Fano varieties with nontrivial Artin—Mumford obstruction for rationality, i.e. with nonzero torsion in $H^3(X,\mathbb Z)$. Present the recent work of Ottem and Rennemo (with an Appendix by Kollár) where this question is solved, see [23]. This may be seen as a generalization of the interpretation of the Artin—Mumford example in Section 6.3 in [3].
3. The Gersten conjecture for étale cohomology
In [4], Bloch—Ogus prove the Gersten conjecture for étale cohomology, which is an important result on the cohomology of algebraic varieties. A simplified proof, based on ideas of Gabber, can be found in the survey [7]. Present the statement and a sketch of the proof of the Gersten conjecture. At the heart of the statement there is an effacement theorem, which one can think of as a special case of a moving lemma (see Theorem 1.1 and Corollary 1.2 in [34] for this point of view).
4. Milnor K-theory and Pfister quadrics
Introduce Milnor K-theory [17] and explain the norm residue homomorphism, see e.g. the introduction of [37]. Mention that this is an isomorphism by the Milnor conjecture, proven by Voevodsky [37].
Introduce Pfister forms and Pfister neighbours (following one of the standard textbooks or for instance [11]) and show that Pfister neighbours of the same form are stably birational to each other, see Section 3.2 in [28] and the references therein. Explain that a symbol $\alpha=(a_1,\dots ,a_n)\in K^M_n(K)/2$ maps to zero in $K^M_n(K(Q))/2 $, where $Q$ is a smooth projective quadric over $K$, defined by a Pfister neighbour of the Pfister form $ \left\langle \left\langle a_1,\dots ,a_n \right\rangle \right\rangle $. For this result, see e.g. the equivalence of (2) and (4) in Main Theorem 3.2 in [11] and the fact (which goes back to Pfister) that the Pfister quadric pulled back to its own function field is isotropic and hence splits into a sum of hyperbolic planes, see e.g. Corollary 2.3 in [11]. (Another reference for generalizations of this vanishing result are contained in Proposition 1.9 in [15] and Corollary 4.2 in [31].)
If time permits, mention the theorem of Orlov—Vishik—Voevodsky [24], which completely describes the kernel of $K^M_n(K)/2\to K^M_n(K(Q))/2 $.
5. Unramified cohomology and generalizations of the Artin—Mumford example
The torsion in integral cohomology of degree larger than 3 is in general not a birational invariant (blow-up suitable subvarieties of $\mathbb P^n$). Nonetheless, Colliot-Thélène—Ojanguren show in [8] that the Artin—Mumford obstruction can be generalized to higher degree if one thinks about it as unramified Brauer groups (see Sections 3 and 4 in [1] or Section 6.4 in [3]). This leads to the notion of unramified cohomology and to new unirational examples that are not stably rational, see [8]. For surveys on unramified cohomology, see for instance 6 and [32].
6. Unramified cohomology and the integral Hodge conjecture
In [10], Colliot-Thélène—Voisin show that the failure of the integral Hodge conjecture for codimension 2 cycles is related to the third unramified cohomology, see Théorème 3.7 in [10]. This is used to show that the integral Hodge conjecture may fail on unirational varieties (of dimension at least 6). Present these results. For generalizations and an alternative proof, see [33] and in particular Section 3 in [33].
7. Rost cycle modules and Merkurjev's pairing
Rost introduced cycle modules in [27], for a brief summary, see e.g. §2 in [18]. Unramified cohomology is a special case of unramified elements in Rost's cycle modules §2 in [18].
Give a survey on Rost cycle modules [27] and unramified elements in it, see for instance §2 in [18]. Mention that Milnor K-theory and Galois cohomology give examples of cycle modules. Explain Merkurjev's pairing §2.4 in [18], which is a key property that will be used later. (If you want you can restrict to the case of ordinary unramified cohomology which is treated for instance in §5 in [32].)
II. DEGENERATION
8. Decomposition of the diagonal and degeneration
In [38], Voisin uses decompositions of the diagonal (see [5]) and a degeneration method to exhibit the first examples of unirational threefolds with vanishing Artin—Mumford invariant that are not stably rational. Her approach was generalized by Colliot-Thélène—Pirutka in [9], who use it to show that very general quartic threefolds are not stably rational. Surveys on these results are for instance: [25], [26], [39] and Section 7 in [3]. Explain the degeneration method of Voisin and Colliot-Thélène—Pirutka following for instance Section 7 and Theorem 8.2, Corollary 8.3 and Theorem 8.4 in [32]. Sketch the applications to quartic double solids and quartic threefolds in [38] and [9]. To do so, describe the degeneration that is used and argue/sketch that the degeneration method that you described above applies.
9. Fano hypersurfaces that are not ruled
Present the result of Kollár in [13], who uses degeneration to positive characteristic to show that very general hypersurfaces $X \subset \mathbb{P}^{n+1}_{\mathbb C}$ of degree $d\geq 2(n+3)/3 $ are not ruled, hence not rational.
10. Fano hypersurfaces that are not stably rational
Present the result of Totaro in [36], who combines the methods of Kollár and Voisin—Colliot-Thélène—Pirutka to show that very general hypersurfaces $X \subset \mathbb{P}^{n+1}_{\mathbb C}$ of degree $d\geq 2(n+2)/3 $ are not stably rational (and in fact not retract rational).
11. Rationality is not a deformation invariant in smooth families
Present the result of Hassett—Pirutka—Tschinkel who show in [12] that there are smooth projective families of fourfolds such that the very general member is not stably rational, while the locus of rational fibres is dense in the base.
12. Motivic obstructions and quartic fivefolds
Nicaise—Shinder and Kontsevich—Tschinkel showed in [21], [14] that stable rationality, resp. rationality, specialize in smooth families. Sketch the argument for rationality in [14] and note that the same argument works for stable rationality by replacing the free abelian group on birational types by the free abelian group of stable birational types.
Nicaise—Ottem [19] use the motivic obstruction from [21], [14] to show that very general quartic fivefolds are not stably rational over fields of characteristic zero. Sketch this specific result following Example 4.3.2 in [20]. (A generalization to fields of characteristic $\neq 2$ and to retract rationality can be found in [22].)
III. IRRATIONAL HYPERSURFACES UNDER A LOGARITHMIC DEGREE BOUND
13. Degeneration method without resolution
State Theorem 1.1 and Corollary 1.2 in [30], whose proof will be explained in the following talks.
Explain the degeneration method without resolution in Section 3 [30] (see also Section 4 in [28] and Theorem 8.6 in [32]). (If you want, you can assume $\operatorname{char} k=0$ and use resolution of singularities instead of alterations, which may make some arguments slightly easier.)
14. A special quadratic form and a vanishing result
Present the results in [30], Section 4 and 5.
15. Non-Vanishing and hypersurfaces singular along an $r$-plane
Present the non-vanishing result in [30], Section 6. Explain the degeneration to hypersurfaces singular along an $r$-plane in [30], Section 7 and prove Proposition 7.1 in [30].
16. Conclusion of the argument
Finish the proof of Theorem 1.1 in [30] via degeneration to hypersurfaces singular along a linear subspace from [30], Section 8. Explain the application in [30], Theorem 1.5 and Corollay 1.6. If time permits, one could discuss some examples over countable fields as in [30], Section 8.2 and/or discuss Theorem 9.2 in [30], which generalizes one aspect of the vanishing result in Section 4.