A minicourse within the semester Rational Points and Algebraic Cycles at the Centre Interfacultaire Bernoulli (EPFL)

**Instructor:** Anthony Várilly-Alvarado

**Lectures:** Tuesdays 10-11am and 11:30-12:30pm (July 10th - August 21st, 2012) in AAC006

**Office:** AAC040, e-mail: varilly "usual symbol here" rice.edu

**Syllabus:** In this course we will survey classical and recent advances in the explicit study of failures of local-to-global principles on del Pezzo and K3 surfaces. Topics include: a review of geometric facts about these surfaces; (potential) density of rational points and unirationality results; explicit techniques for the computation of Picard and Brauer groups of these surfaces; Brauer-Manin obstructions. Time permitting, we will discuss these topics for Enriques surfaces (including étale-Brauer obstructions), by focusing on their K3 double covers.

**Grading:** There will be no exams. Grades will be based on a final project. If you are registered for this course, please come talk to me to discuss a topic for the project.

**Lecture Notes:** For the first three weeks, I will be (mostly) following a set of
notes I wrote for a similar mini-course I gave at the workshop ``Arithmetic of Surfaces'' at the Lorenz Center, Leiden.
Bjorn Poonen (along with René Pannekoek for the fifth lecture) took real-time notes.

**Exercise Sheets:**

**What we covered:**

**July 10th:** Motivating questions for rational points on varieties over global fields: existence of points, failures of local-to global principles, (potential) density. Birational invariance of these questions (Lang-Nishimura lemma; v-adic implicit function theorem), and Iskovskikh's classification of geometrically rational surfaces. Geometry of del Pezzo surfaces: classification over separably closed fields, Picard groups and Galois actions, exceptional curves and root systems, introduction to anticanonical models.

**July 17th:** Finished discussion on Anticanonical models. Arithmetic of
high degree del Pezzo surfaces (i.e., degree at least 6): Severi-Brauer
varieties satisfy the Hasse principle, del Pezzo surfaces of degree 8 are
either k-birational to the plane or are forms of P1xP1, del Pezzo surfaces
of degree 7 always have a point, del Pezzo surfaces of degree 6 satisfy the
Hasse principle and are k-birational to the plane if they have a point. In all
cases, these surfaces satisfy weak approximation (assuming they have a point).

**July 24th:** Showed that del Pezzo surfaces of degree 5 always have a rational point and are
birational to the plane, over the ground field. Introduced the Azumaya Brauer group and the cohomological
Brauer group (which coincide for nice varieties). Discussed the Brauer-Manin obstruction, the filtration on
the Brauer group of a variety by constant and algebraic elements, and how to use a Leray spectral sequence
to compute all possible Brauer groups modulo constant classes (abstractly as a finite abelian group) for del
Pezzo surfaces. Indicated a technique to construct cyclic Azumaya algebras on a nice variety from elements
of H^1(Gal(L/k),Pic X_L) for L/k cyclic. Gave a slide presentation for a counter-example to weak
approximation on a minimal del Pezzo surface of degree 1 following this mechanism.

**July 31st:** Briefly dicussed uniqueness of Brauer-Manin obstructions on del Pezzo surfaces of
degree 4: stated Colliot-Thélène and Sansuc's conjecture, as well as results by Salberger and
Skorobogatov on weak approximation and Wittenberg on the Hasse principle. Discussed unirationality of del
Pezzo surfaces of degrees 2, 3, and 4 over infinite fields (following ideas of Segre, Manin and
Kollár) as well as some preliminary work (joint with Salgado and Testa) on unirationality of dP2s
over finite fields.

**August 7th:** Discussed the geometry of complex K3 surfaces. Computed their singular cohomology
groups and determined the lattice structure of H^2(X,Z). Went over the Hodge structure on this lattice
(all K3 surfaces being Kaehler), the period domain, and stated the weak and global Torelli theorems for
algebraic K3s, as well as the surjectivity theorem. Gave an example (which I learned from lectures of
Morrison) on how to use these theorems to prove the existence of K3 surfaces with certain divisors having
specified geometric properties/invariants.

**August 14th:** Discussed potential density of rational points on varieties over number fields. Examples: curves of genus
0 or 1, geometrically rational and unirational varieties (including most families of Fano 3-folds), Brauer-Severi varieties,
abelian varieties, bielliptic surfaces. Counter examples and obstructions to potential density: curves of genus at least 2 don't
satisfy potential
density; Chevalley-Weil theorem: if X satisfies potential density then it has no étale covers that dominate a curve of
genus at least 2 (which can be used to construct surfaces of Kodaira dimension 1 that don't satisfy potential density);
Bombieri-Lang conjecture. Went over work of Bogomolov and Tschinkel: K3 surfaces with an elliptic fibration or an infinite
automorphism group satisfy potential density (overview of proof of the latter). Introduced saliently ramified multisections of
elliptic fibrations and showed how to use these to show that potential density holds on surfaces with two different elliptic
fibrations to P^1. Concluded Enriques surfaces satisfy potential density.

**August 21st:** Discussed Picard numbers on K3 surfaces X over number fields. At a place of good reduction, the
geometric Néron-Severi group injects into the corresponding group of the reduction. The rank of the latter is bounded
above by the number of Eigenvalues v of Frobenius acting on H_et^2(Xbar,Q_l) such that v/q is a root of unity (q = cardinality
of residue field). Combining Newton's identities with the Lefschetz trace formula, this reduces giving an upper bound for the
Picard Number of X to point counting. This bound, however, is always even. To improve on it we discussed work of van
Luijk, Kloosterman, Elsenhans and Jahnel. Also discussed transcendental Brauer groups of K3 surfaces via the transcendental
lattice of their middle cohomology. Introduced results of van Geemen, Mukai and Voisin, as well as recent arithmetic
applications to Brauer-Manin obstructions by Hassett and myself.

**Resources:** (These will be updated as we go along)

- Manin: Cubic Forms (especially sections 24-29)
- Kollár: Rational curves on Algebraic varieties (especially III.3)
- Demazure: Surfaces de del Pezzo II, III, IV, and V.
- Gille and Szamuely: Central Simple Algebras and Galois Cohomology
- Colliot-Thélène and Sansuc: La descente sur les variétés rationnelles (appeared in the Angers proceedings, 1980).
- Salberger and Skorobogatov: Weak approximation for surfaces defined by two quadratic forms (appeared in Duke, 1991).
- Wittenberg: Intersections de deux quadriques et pinceaux de courbes de genre 1 (Lecture Notes in Math. vol. 1901).
- Kollár: Unirationality of cubic hypersurfaces (appeared in JIMJ 2002).
- Barth, Peters, Hulek and van de Ven: Compact complex surfaces.
- Morrison: The Geometry of K3 surfaces.
- Pjateckii-Shapiro and Shafarevich: A Torelli theorem for algebraic surfaces of type K3 (Izvestiya Akademii Nauk SSSR, 1971). The proof given here contains some errors. See item below.
- Looijenga and Peters: Torelli Theorems for Kaehler K3 surfaces.
- Hassett: Potential density of rational points on algebraic varieties
- Colliot-Thélène, Skorobogatov and Swinnerton-Dyer: Double fibres and double covers: paucity of rational points (appeared in Acta Arithmetica, 1997).
- Bogomolov and Tschinkel: Density of rational points on Enriques surfaces (appeared in Math Research Letters, 1998).
- Bogomolov and Tschinkel: Density of rational points on elliptic K3 surfaces (appeared in Asian Journal of Mathematics, 2000).
- Bogomolov and Tschinkel: On the density of rational points on elliptic fibrations (appeared in Crelle, 1999).
- van Luijk: K3 surfaces with Picard number one and infinitely many rational points (appeard in Algebra and Number Theory, 2007)
- Kloosterman: Elliptic K3 surfaces with geometric Mordell-Weil rank 15 (appeared in Canadian Math Bulletin, 2007)
- Elsenhans and Jahnel: The Picard group of a K3 surface and its reduction modulo p (appeared in Algebra and Number Theory, 2011)
- van Geemen: Some remarks on Brauer groups of K3 surfaces (appeared in Advances in Mathematics, 2005).
- Hassett, Várilly-Alvarado and Varilly: Transcendental obstructions to weak approximation on general K3 surfaces (appeared in Advances in Mathematics, 2011).
- Hassett, Várilly-Alvarado: Failure of the Hasse principle on general K3 surfaces (preprint, 2011).

Acknowledgements: This course was made possible by the Centre Interfacultaire Bernoulli. Preparation for the course was also supported by National Science Foundation Grant DMS-1103659. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author, and do not necessarily reflect the views of the National Science Foundation.