10th Belgian-Dutch Algebraic Geometry Day

Location: Arenberg Castle, KU Leuven Campus Heverlee (map)

Speakers

13:45 – 14:45: Sho Tanimoto (Copenhagen): Towards a refinement of Manin's conjecture

Manin's conjecture predicts the generic distribution of rational points on Fano varieties, and it has the explicit asymptotic formula in terms of geometric invariants of the underlying variety. However the original version which predicts asymptotic formulae after removing proper closed sets is wrong due to covering families of subvarieties violating compatibility of Manin's conjecture, and a possible refinement, suggested by Peyre, removes thin sets instead of closed sets. One natural question is how to choose the exceptional thin set. In this talk, I would like to address this issue using birational geometry, e.g., the minimal model program and the boundedness of log Fano varieties. This is joint work with Brian Lehmann and Yuri Tschinkel.

15:00 – 16:00: John Christian Ottem (Oslo): Effective cones of cycles on blow-ups of projective space.

While the cones of curves and effective divisors are important tools in algebraic geometry, the cones of subvarieties of intermediate codimension remain much more mysterious, mostly due to the lack of examples. In this talk, I will outline the two explicit computations of such cones; on blow-ups of projective space and on certain hyperkahler fourfolds. In the first case, we determine bounds on the number of points for which these cones are generated by the classes of linear cycles, and for which these cones are finitely generated. Surprisingly, we discover that in some cases, the higher codimension cones behave better than the cones of divisors. In the case of hyperkahler fourfolds, we show the cone of nef 2-cycles can be strictly larger than the cone of pseudoeffective 2-cycles, showing that the usual intuition from divisors do not generalize to higher codimension.

16:00 - 16:30: Coffee break

16:30 – 17:30: Bruno Chiarellotto (Padova): Monodromy action and special fiber for semistable schemes over a DVR

We plan to discuss some results on the action of the monodromy on the (unipotent) de Rham fundamental group of the generic fiber of some type of semistable schemes over a DVR in mixed characteristic. The action will be interpreted in terms of structure of the graph of the  special fiber. This is joint work with Ch. Lazda.