# Math 260X: Rationality questions in algebraic geometry (Spring 2022)

**Instructor:** Dori Bejleri (bejleri [at] math [dot] university [dot] edu)

**Time and place:** Monday + Wednesday 12:00pm - 1:15pm in Science Center 304.

**Office hours:** By appointment, Science Center 525.

### Lecture notes (under construction).

## Syllabus

### Course description

Given a collection of algebraic equations, can we find a parametrization of the set of solutions by rational functions? This is one of the most fundamental questions in algebraic geometry and leads to the notion of rationality and its various generalizations (stable rationality, unirationality, rational connectedness). In this class we will survey some classical results, recent breakthroughs and open problems in the study of rationality.### Overview of lectures

1. Introduction and overview - 01/24/2022.

2. Overview (cont.) and plurigenera - 01/26/2022.

3. Castelnuovo's theorem - 01/31/2022.

4. Deformation invariance of rationality for surfaces - 02/02/2022.

5. Hypersurfaces of low degree I - 02/07/2022.

6. Hypersurfaces of low degree II - 02/09/2022.

7. Rational curves on varieties - 02/14/2022.

8. Uniruled varieties - 02/16/2022.

9. Rationally connected varieties I - 02/28/2022.

10. Rationally connected varieties II - 03/02/2022.

11. Fano varieties I - 03/07/2022.

12. Fano varieties II - 03/09/2022.

13. Birational superrigidity I - 03/21/2022.

14. Birational superrigidity II - 03/23/2022.

15. Intermediate Jacobians I - 03/28/2022.

16. Intermediate Jacobians II - 03/30/2022.

17. Intermediate Jacobians III - 04/04/2022.

18. Intermediate Jacobians IV - 04/06/2022.

19. Conic bundles, irrational stably rational varieties - 04/11/2022.

20. The Artin-Mumford invariant and Brauer groups - 04/13/2022.

21. Brauer groups (cont.) - 04/18/2022.

22. The Artin-Mumford example, rationality of quotients - 04/20/2022.

23. Measures of irrationality (guest lecture by Nathan Chen) - 04/25/2022.

24. Decomposition of the diagonal - 04/27/2022.

### References

#### Books and lectures:

Kollár–Smith–Corti, *Rational and nearly rational varieties.* Cambridge University Press, 2004.

Pardini–Pirola, eds., *Rationality problems in algebraic geometry.* Springer, 2016.

Mustaţă, Lecture notes on rationality of algebraic varieties.

Tschinkel, Rationality and specialization.

Kollár, *Rational curves on algebraic varieties.* Springer, 1996.

Debarre, *Higher-dimensional algebraic geometry.* Springer, 2001.

Voisin, Stable birational invariants
and the Lüroth problem.

#### Non-exhaustive list of papers:

Artin–Mumford, Some elementary examples of unirational varieties. *Proc. Londom Math. Soc.,* 1972.

Beauville–Colliot-Thélène–Sansuc–Swinnerton-Dyer, Variétés stablement rationnelles non rationnelles. *Ann. of Math.,* 1985.

Bouckson–Demailly–Păun–Peternell, The pseudoeffective cone of a compact K¨ahler manifold and varieties of negative Kodaira dimension. *J. Algebraic Geom.,* 2013.

Campana. Connexité rationnelle des variétés de Fano. *Ann. Sci. École Norm. Sup.,* 1992.

Clemens–Griffiths. The intermediate Jacobian of the cubic threefold. *Ann. of Math.,* 1972.

Colliot-Thélène–Pirutka, Hypersurfaces quartiques de dimension 3: non-rationalité stable. *Ann. Sci. École Norm. Sup.,* 2016.

Graber–Harris–Starr, Families of rationally connected varieties. *J. Amer. Math. Soc.,* 2003.

Harris–Mazur–Pandharipande, Hypersurfaces of low degree. *Duke Math. J.,* 1998.

Hassett–Pirutka–Tschinkel, Stable rationality of quadric surface bundles over surfaces. *Acta Math.* 2018.

Iskovskikh–Manin, Three-dimensional quartics and counterexamples to the Lüroth problem. *Mat. Sb.,* 1971.

Kollár–Miyaoka–Mori, Rationally connected varieties. * J. Algebraic Geom.,* 1992.

Kontsevich–Tschinkel, Specialization of birational types. *Invent. Math.,* 2019.

Nicaise–Shinder, The motivic nearby fiber and degeneration of stable rationality. *Invent. Math.,* 2019.

Schreieder, Stably irrational hypersurfaces of small slope. *J. Amer. Math. Soc.,* 2019.

Voisin, Unirational threefolds with no universal codimension 2 cycle. *Invent. Math.,* 2015.

### Prerequisites

Algebraic geometry at the level of a first year graduate course (e.g. chapters 2 and 3 of Hartshorne’s Algebraic Geometry). Some algebraic topology may also be useful.

### Grades

The grade for undergraduate students will be based off of course participation and a final presentation on a topic related to the course material.