Stable rationality and the decomposition of the diagonal, Spring 2016

Organizational:

1. Please email Johan if you want to be on the associated mailing list.
2. The talks will be 2x45 minutes with a short break.
3. Time and place: 10:30am–noon in room 407.
4. First organizational meeting: Jan, 22. If you are a student and interested in actively participating, then take a look at the lectures and the references.

Dates

Lectures

0. Introduction to rationality questions.
   Outline of the lectures.
1. Introduction to Chow groups.
   1. Define Chow groups and state their properties: proper pushforward, flat pullback, pullback along morphisms of smooth varieties, intersection product for smooth varieties, projection formula, localization exact sequence, cycle class maps.
   2. Recall rational connectedness and show that it implies that Ch₀ is trivial.
   3. Show that Ch₀ is a stable birational invariant of smooth projective varieties [1, Lemma 2.11].
   4. In all of the above, clearly state the setting and hypotheses (varities over a field, smooth varieties over a field, schemes of finite type etc.)
   Reference: [1, Section 2.2] and [2] for general background.
2. Decomposition of the diagonal.
   1. Define Chow and cohomological decomposition of the diagonal (dd).
   2. Relate decomposition of the diagonal to Ch₀ (Bloch-Srinivas).
   3. Define what it means for Ch₀ to be universally trivial and show that Chow dd is equivalent to the universal triviality of Ch₀.
   4. Prove that having Chow or cohomological dd is a stable birational property.
   References: [1, Section 4.1]
3. The specialization theorem.
   1. Define what it means for a morphism to be universally Ch₀ trivial [4, Definition 1.1]. Characterize this in terms of the fibers [4, Proposition 1.8].
   2. Show some examples and non-examples of the above, for example for resolutions of some simple singularities.
   3. Prove that (under appropriate hypotheses), the property of having Chow dd specializes from the geometric generic fiber of a family over a DVR to the desingularization of the special fiber [4, Theorem 1.14].
   4. Show that in a family of varieties, the property of having a Chow dd holds on a countable union of closed subsets [5, Proposition 1.4].
   Reference: [4] (French). This subsumes [5, Theorem 1.1].
4. The Artin-Mumford invariant.
   1. Recall the specialization theorem from last time and highlight that to use it, we must have some examples of non stably rational varieties.
   2. Define the Artin-Mumford invariant and show that it is a stable birational invariant.
   3. Show that cohomological dd implies that this invariant vanishes.
   4. Describe Artin and Mumford's examples of unirational threefolds with nontrivial Artin-Mumford invariant.
   References: [1, Section 2.1.3] and [3].
5. Quartic double solids.
   1. Discuss the overall shape of the proof: We want to prove that a very general member of a family is not stably rational. We exhibit a particular X₀ (possibly singular) and show that its desingularization does not have Chow/cohomological dd. For this, we use a classical method like Artin-Mumford. We conclude that a very general member does not have Chow/cohomological dd. Highlight that the classical method might fail for the very general member.
   2. Define quartic double solids.
   3. Use this method to show that a very general quartic double solid is not stably rational.
   4. Amplify the above result to show that a very general quartic double solid with at most 7 nodes is not stably rational.
   References: [5, Section 1].
6. Hypersurfaces.
   Explain Totaro's paper [6]. The main theorem is that a large class of hypersurfaces does not admit Chow dd and hence is not stably rational. The argument goes by degenerating a hypersurface to an inseparable double cover (in characteristic 2). To show that the special member is not stably rational, he uses another classical obstruction, namely the non vanishing of a space of pluri-canonical forms.

   Additional fun topics

   1. Mumford, Rational equivalence of 0-cycles on surfaces.
   2. Beauville, Voisin, On the Chow ring of a K3 surface.
   3. Clemens, Griffiths, The intermediate Jacobian of the cubic threefold.
   4. Murre, Algebraic equivalence modulo rational equivalence on a cubic threefold.
   5. Murre, Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of mumford.
   6. Iskovskih, On the rationality problem for conic bundles and Towards the problem of rationality of conic bundles.
   7. Voisin, Unirational threefolds with no universal codimension 2 cycle (the part about codimension 2 cycles).
   8. Serre, On the fundamental group of a unirational variety.
   9. Hassett, Some rational cubic fourfolds.
   10. Beauville, Donagi, La variété des droites d'une hypersurface cubique de dimension 4.
   11. Barth, Van de Ven, Fano-Varieties of lines on hypersurfaces.
   12. Galkin, Shinder, The Fano variety of lines and rationality problem for a cubic hypersurface.
   13. Larsen, Lunts, Motivic measures and stable birational geometry.
   14. Something else of your choice.

   References

   1. Voisin, Stable birational invariants and the Luroth problem.
   2. Fulton, Intersection theory (book).
   3. Artin, Mumford, Some elementary examples of unirational varieties which are not rational.
   4. Colliot-Thélène, Pirutka Hypersurfaces quartiques de dimension 3 : non rationalité stable.
   5. Voisin, Unirational threefolds with no universal codimension 2 cycle.
   6. Totaro, Hypersurfaces that are not stably rational.
   7. Shen On relations among 1-cycles on cubic hypersurfaces
   8. Shen Hyperkahler manifolds of Jacobian type
   9. de Fernex, Fusi, Rationality in families of threefolds.