Schedule  [PDF]

The talks will be held in the lecture hall 5F, building 25.21 (map), according to the following time table.

Time	MON July 22	TUE July 23	WED July 24	THU July 25	FRI July 26
09:30-10:30	Registration    and coffee	M1: Testa	M2: Castravet	M2: Castravet	M3: Cascini
10:30-11:00		Coffee break
11:00-12:00	M1: Testa	M1: Testa	M2: Castravet	M3: Cascini	M3: Cascini
12:00-14:00	Lunch break
14:00-15:00	Casagrande	Smith	Free afternon	Tschinkel	Dolgachev
15:15-16:15	Corti	Derenthal		Waldron
16:15-16:45	Coffee break			Coffee break
16:45-17:45	Petracci	Shepherd-Barron		Tanaka
19:30-		Social dinner   at Kurhaus





Titles and abstracts

Mini-courses

Paolo Cascini: Vanishing and non-vanishing on Fano varieties in positive characteristic.


Many results in the minimal model program depends on Kawamata-Viewheg vanishing on some Fano varieties.
I will survey some results of vanishing and show some examples of non-vanishing on del Pezzo surfaces and higher dimensional Fano varieties.
I will then show some applications on the study of the singularities that appear in the minimal model program.





Ana-Maria Castravet: Birational Geometry and Moduli Spaces.


I will start these lectures by discussing Mukai's counterexamples to Hilbert's 14'th problem.
The key observation is that the Nagata invariant ring for the action of a certain g-dimensional vector group can be identified with the total coordinate ring of a blow-up of an (r-g-1)-dimensional projective space at r general points (r>=g+3). This ring is finitely generated if and only if 1/2+1/g+1/(r-g)>1.
When the inequality holds, such blow-ups are natural generalizations of del Pezzo surfaces and their birational geometry is very explicit: they can be realized as moduli spaces of stable (often parabolic) vector bundles, and the steps of the Minimal Model Program are achieved by varying stability conditions.





Damiano Testa: Arithmetic properties of del Pezzo surfaces.


In this series of lectures, I will talk about del Pezzo surfaces with an emphasis on arithmetic properties.
I will touch upon the combinatorial structure of the set of (-1)-curves, the existence of rational points, the construction of (uni-)rational parameterisations.
If time permits, I may also talk about Cox rings.

NOTES





Talks

Cinzia Casagrande: Fano 4-folds with rational fibrations.


Smooth, complex Fano 4-folds are not classified, and we still lack a good understanding of their general properties. Here we focus on Fano 4-folds with large second Betti number b_2, studied via birational geometry and the detailed analysis of their contractions and rational contractions. We recall that a contraction is a morphism with connected fibers onto a normal projective variety, and a rational contraction is given by a sequence of flips followed by a (regular) contraction. The main result that we want to present is the following: let X be a Fano 4-fold having a rational contraction X --> Y of fiber type (with dim Y > 0). Then either b_2(X) is at most 18, with equality only for a product of surfaces, or Y is P^1 or P^2.





Alessio Corti: Smoothing toric Fano 3-folds I: conjectures


I will state and motivate some conjectures on smoothings of toric Fano 3-folds. This has applications to the Fano/LG correspondence and the classification of Fanos.
The talk by Andrea will present the state of the art of what we can prove at this time. This is joint work with several people of the London group and Paul Hacking.





Ulrich Derenthal: On Manin's conjecture for certain spherical Fano varieties.


Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded anticanonical height on Fano varieties over the rational numbers. We discuss a proof of this conjecture for certain spherical Fano varieties, which combines Cox rings with methods of analytic number theory. This is joint work in progress with Valentin Blomer, Jörg Brüdern and Giuliano Gagliardi.





Igor Dolgachev: Quintic del Pezzo surfaces and 15-nodal quartic surfaces.


I will discuss a classical connection between del Pezzo surfaces and congruences of lines in three-dimensional projective space. In the case of quintic del Pezzo surfaces, it relates them to 15-nodal quartic surfaces and gives a simple proof of unirationality of the moduli space of such surfaces.





Andrea Petracci: Smoothing toric Fano 3-folds II: theorems.


This talk continues the one by Alessio Corti. I will show how to construct smoothings of toric Fano 3-folds with Gorenstein singularities starting from some combinatorial input on their associated polytopes. This is joint work with Alessio Corti and Paul Hacking.





Nick Shepherd-Barron: Asymptotic period relations and alkanes.


The branches of the hyperelliptic locus in Siegel space along the locus of diagonal matrices correspond exactly to the alkanes of elementary chemistry: each branch has a first-order ("asymptotic") description as a vector bundle over the diagonal locus, and elliptic curves, with their four 2-torsion points, correspond to quadrivalent carbon atoms.

This picture persists for the image of the moduli space of Jacobian elliptic surfaces: for each alkane there is one branch of the moduli space through the locus that parametrizes unions of special Kummer surfaces, and each branch has an explicit description as a vector bundle over a period domain.





Karen Smith: Fano and globally F-regular varieties in prime characteristic.


In this talk, we will explain the close relationship between positivity of the anti-canonical bundle and Frobenius splitting. After reviewing F-regularity, which is a strong form of Frobenius splitting along every effective divisor, we'll show that a globally F-regular variety X of prime characteristic always admit an effective Q-divisor D such that the pair (X, D) has kit singularities and -K_X-D is ample. The converse false in char p, but holds in char 0, suitably interpreted.





Hiromu Tanaka: Del Pezzo fibrations in positive characteristic.


We naturally encounter del Pezzo fibrations in minimal model program. In this talk, we summarise some results on del Pezzo fibrations in positive characteristic. We also discuss some open problems related to this topic.





Yuri Tschinkel: Rationality of del Pezzo surfaces and Fano threefolds over nonclosed fields.


I will discuss recent results and constructions concerning (stable) rationality of Del Pezzo surfaces and Fano threefolds.





Joe Waldron: Tate's genus change formula and del Pezzo surfaces.


We discuss a generalisation of Tate's genus change formula, and applications to regular del Pezzo surfaces over non-closed fields. This is joint with Zsolt Patakfalvi.