Ivan Arzhantsev (Moscow State University): Flexible varieties and automorphism groups
Given an affine algebraic variety X of dimension at least 2, we let SAut(X) denote the special automorphism group of X, i.e., the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus of X then it is infinitely transitive on this locus. In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point x of X the tangent space at x is spanned by the velocity vectors of one-parameter unipotent subgroups of Aut(X). We provide also different variations and applications. (This is a joint work with H. Flenner, S. Kaliman, F. Kutzschebauch, and M. Zaidenberg.
David Bourqui (Université de Rennes 1): Moduli spaces of curves and Cox rings
We will explain why, under certain circumstances, Cox rings might prove useful to understand the properties of the moduli space of morphisms from a curve to a Fano variety.
Hélène Esnault (Universität Duisburg-Essen): Index of varieties over strictly henselian fields
(Joint with Marc Levine and Olivier Wittenberg)
Walter Gubler (Universität Regensburg): Canonical measures and the geometric Bogomolov conjecture
Chambert-Loir measures occur as equidistribution measures of non-archimedean dynamical systems. In the case of a subvariety X of an abelian variety, we get canonical measures on the Berkovich space associated to X. In this talk, we will give a precise description of these canonical measures and we will show that this has applications to the geometric Bogomolov conjecture.
Brendan Hassett (Rice University): Fibrations in rational surfaces and their sections
A fibration is a surjective morphism from a smooth projective variety to a smooth curve, all defined over a field k. Assume the fibers are rational surfaces. Then every fibration admits a section, when k is algebraically closed. There is a conjectural framework for deciding whether there is a section when k is finite, expressed in terms of the Brauer group and the existence of local sections.
Our approach to these questions hinges on understanding the geometry of the scheme parametrizing all sections of our fibration, especially in contexts where the rational surfaces are relatively simple, e.g., quadric surfaces and intersections of two quadric hypersurfaces. The main application is the existence of sections provided the fibration is sufficiently general, in a sense that can be made precise. (Joint with Yuri Tschinkel)
Jürgen Hausen (Universität Tübingen): Rational varieties with a complexity one torus action
We present an explicit approach via the Cox ring to n-dimensional rational varieties X coming with an effective action of an (n-1)-dimensional algebraic torus T. As examples, we discuss resolution of singularities and the automorphism group of X from this point of view.
Norbert Hoffmann (FU Berlin): Counting Arakelov bundles over arithmetic curves
Under the classical analogy between function fields of algebraic curves and number fields, vector bundles over curves correspond to Arakelov bundles over arithmetic curves. After recalling this, I will explain an arithmetic analogue of Faltings' criterion, which relates stability of vector bundles to the existence of tensor products without global sections. In the arithmetic case, this existence of bundles without sections follows from a counting argument, which shows that an avarage number of global sections is less than one.
Emmanuel Kowalski (ETH Zürich): Rational points and expander graphs
In recent work with J. Ellenberg and C. Hall, we have shown that rather general combinatorial properties of a family of coverings of an algebraic curve are enough to ensure that the gonality of the curves tends to infinity. This has consequences for the finiteness of rational points of bounded degree on the curves, and of other interesting sets of algebraic points of bounded degree which can be be controlled using such families. We will survey the graph-theoretic aspects as well as the applications, and discuss open problems and questions suggested by our results.
Vladimir Lazić (Universität Bayreuth): Geography of models
Given an algebraic variety, finding its birational model which has good geometric properties is the crucial question in birational geometry. There are two classes of algebraic varieties where this can be achieved, and whose birational geometry can be very well understood: one is that where a classical Minimal Model Program can be run, and the other is varieties called Mori Dream Spaces. I will formulate a general (and in some sense maximal) setup where we can answer the above question in a satisfactory way. This generalises works of Birkar-Cascini-Hacon-McKernan (by using recent results of Cascini-Lazic) and of Hu-Keel. (This is joint work with A.-S. Kaloghiros and A. Küronya)
Christian Liedtke (Universität Bonn): On the birational nature of lifting
Whenever a smooth projective variety lifts to characteristic zero, many characteristic-p-"pathologies" cannot happen. Now, lifting results are difficult to establish (and in general, starting from dimension 2, lifting does not hold in general), but sometimes it is easier, and even more natural, to lift a birational model, maybe even a slightly singular one, of a given variety. Thus, it is natural to study to what extent lifting is a birational invariant for varieties. We will see that it is not a birational invariant even for smooth projective and rational varieties if the dimension is large. Trivially, it holds for normal curves, and it is also not difficult to see that it holds for smooth surfaces. For surfaces with canonical singularities, we will see that this question is surprisingly subtle. We address this question from the point of view of classical resolutions of singularities, as well as "stacky" resolutions of singularities. This is joint work with Matthew Satriano.
Emmanuel Peyre (Université de Grenoble): Asymptotics for curves
Inspired by the analogy with the program of Batyrev-Manin for rational points, it is natural to ask whether the moduli space of very free morphisms from a given curve to an almost Fano variety converges, after renormalisation. This talk shall give some evidence for this question.
Per Salberger (Chalmers University of Technology): On the Manin-Peyre conjecture for a certain singular cubic fourfold
In this talk, based on joint work with V. Blomer and J. Brüdern, we present new evidence for the Manin-Peyre conjecture. There has been a number of papers on this conjecture for singular cubic surfaces, but very few such papers for cubic threefolds or fourfolds. We discuss in our talk how one can use the universal torsor over a crepant resolution to prove a strong form of the Manin-Peyre conjecture for a certain singular cubic fourfold.
Alexei Skorobogatov (Imperial College): Applications of additive combinatorics to conic and quadric bundles
Methods of Green and Tao can be used to prove the Hasse principle and weak approximation for some special intersections of quadrics defined over Q. This implies that the Brauer-Manin obstruction controls weak approximation on conic bundles with an arbitrary number of degenerate fibres, all defined over Q, and some similar varieties. This is a joint work with T. Browning and L. Matthiesen.
Jason Starr (Stony Brook University): Rational points of "rationally simply connected" varieties over global function fields
This is joint work with Chenyang Xu building on earlier work with de Jong and He and using work of Esnault in a crucial manner. A variety of Picard number 1 is "rationally simply connected" if, roughly, the parameter spaces of rational curves in the variety are themselves rationally connected (this has been extended to higher Picard number by Yi Zhu). We prove that such a variety -- or a degeneration thereof -- defined over a global function field has a rational point if it has vanishing "elementary obstruction" ala Colliot-Thélène and Sansuc. This gives new proofs for these fields of the Tsen-Lang theorem, the Period-Index theorem (originally due to Brauer-Hasse-Noether), and the split case of Serre's "Conjecture II" (originally due to Harder). Combined with results of Robert Findley, this also implies some new existence results over global function fields for hypersurfaces in projective homogeneous spaces (not only projective space). One interesting feature of these new proofs is that we get height bounds which are independent of the characteristic.
Michael Stoll (Universität Bayreuth): Naive and canonical heights on Jacobians of hyperelliptic curves of genus 3
Based on an explicit model of the Kummer variety of the Jacobian J of a hyperelliptic curve C of genus 3 over a number field k, which gives rise to a hanive height on J, we show how the corresponding canonical height can be computed and how the difference between the two heights can be bounded. This makes it possible for the first time to show that generators of a finite-index subgroup of J(k) actually generate J(k). This in turn is needed for applications, for example for the determination of the set of integral points on C (if k is the field of rational numbers).
Yuri Tschinkel (New York University): Unramified cohomology
I will describe a construction of universal spaces for birational invariants of algebraic varieties (joint with F. Bogomolov).
Yi Zhu (Stony Brook University): Families of Homogeneous Spaces over Curves
A basic question in arithmetic geometry is whether a given variety defined over a non-closed field admits a rational point. When the base field is of geometric nature, i.e., function fields of varieties, one naturally hopes to solve the problem via purely geometric methods. In this talk, I will discuss the geometry of the moduli space of sections of a projective homogeneous space fibration over an algebraic curve. By studying its MRC quotient, we prove that either over a function field of a complex algebraic surface or over a global function field, a projective homogeneous space admits a rational point if the elementary obstruction vanishes.