Special elliptic fibrations

We construct examples of elliptic fibrations of orbifold general type (in the sense of Campana) which have no etale covers dominating a variety of general type.Comment: 13 pages, LaTe

Lagrangian subvarieties of abelian fourfolds

We construct examples of algebraic surfaces with interesting fundamental groups.Comment: 26 pages, LaTe

Hodge theory and Lagrangian planes on generalized Kummer fourfolds

We analyze the intersection properties of projective planes embedded in generalized Kummer fourfolds, with a view toward classifying the homology classes represented by these submanifolds.Comment: 28 page

Models of Brauer-Severi surface bundles

We study Brauer-Severi surface bundles over smooth projective varieties via root stacks, with a view towards applications to failure of stable rationality.Comment: 50 page

Curves in abelian varieties over finite fields

We study the distribution of algebraic points on curves in abelian varieties over finite fields.Comment: 7 pages, to appear in IMR

Height zeta functions of toric bundles over flag varieties

We investigate analytic properties of height zeta functions of toric bundles over flag varieties.Comment: 64 pages, LaTe

Universal torsors over Del Pezzo surfaces and rational points

We discuss Manin's conjecture concerning the distribution of rational points of bounded height on Del Pezzo surfaces, and its refinement by Peyre, and explain applications of universal torsors to counting problems. To illustrate the method, we provide a proof of Manin's conjecture for the unique split singular quartic Del Pezzo surface with a singularity of type D4. (Lectures at the summer school "Equidistribution in number theory", Montreal, August 2005)Comment: 33 pages, 2 figure

Potential density of rational points for K3 surfaces over function fields

We give examples of non-isotrivial K3 surfaces over complex function fields with Zariski-dense rational points and N'eron-Severi rank one

Geometry of equivariant compactifications of GanG^n_a

Equivariant compactifications of reductive groups can be described by combinatorial data. On the other hand, equivariant compactifications of the additive group G^n_a are more complicated in at least two respects. First, they often admit moduli. Second, even simple varieties (like projective spaces) admit many different structures as equivariant compactifications of the additive group. We give a dictionary relating Artinian local rings, certain systems of partial differential equations, and equivariant compactifications of G^n_a. As an application, we classify the structures on varieties like projective spaces, ruled surfaces, and certain threefolds. We consider these results as a first step in a systematic study of G^n_a-equivariant birational geometry.Comment: 22 pages, LaTe

On stable rationality of Fano threefolds and del Pezzo fibrations

We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.Comment: 21 page