Our seminar is an initiative which gathers up PhD students and postdocs from Low Countries. We meet approximately every 3 months in various places across Belgium and the Netherlands. For the past editions see: https://sites.google.com/view/junior-ag-seminar/home (no longer maintained).

The organizers are Francesca Leonardi (Universiteit Leiden) and Nick Shamaiev (KU Leuven).

Please send an e-mail to Francesca Leonardi if you are interested in giving a talk. You can talk about a result of yours, a work in progress or a paper that you have found interesting.

Next meeting: March 22, 2024 in Antwerpen

Venue: Campus Middelheim, Building G, room G.006

Schedule: 

13:30 - 14:30   Céline Fietz, Universiteit Leiden, Categorical Resolutions of A_2 singularities

15:00 - 16:00  Michaël Maex, KU Leuven, Edixhoven jumps of jacobians, weight functions and tropical geometry

16:30 - 17:30   Timothy De Deyn, University of Glasgow, Categorical resolutions of filtered schemes

Abstracts

Céline Fietz, Categorical Resolutions of A_2 singularities

Recently, A. Kuznetsov and E. Shinder proved that the bounded derived category of a projective variety with an isolated A_1 (nodal) singularity admits a particularly “small” categorical resolution. Moreover, they show that the “categorical exceptional divisor” is generated by a special kind of coherent sheaf which is closely connected to the spinor bundles on a smooth quadric. In this talk I will recall basics about categorical resolutions and discuss some of the results of my master thesis. There I proved that in the case of a four-dimensional variety these results can be generalised to A_2 (cuspidal) singularities and more precisely, we obtain two “special” generators of the “categorical exceptional divisor” which are related to the spinor sheaves on a nodal quadric. This generalisation is expected to be true in any even dimension, which is work in progress. Ultimately, the goal is to understand the derived category of any variety with isolated ADE singularities.

Master thesis of Céline is available under the address: https://www.math.leidenuniv.nl/~duttay/pdfs/thesis-fietz.pdf

Michaël Maex, Edixhoven jumps of jacobians, weight functions and tropical geometry

Let A be an abelian variety over a discrete valuation field K. There are stil many open questions about the behaviour A and its Néron model under extensions of K. B. Edixhoven defined a tuple of real numbers known as the jumps (j_i) of A which measure the behaviour under tame base change. It is unknown whether (j_i) are rational numbers. More is understood when A is the jacobian of a curve C. Here the rationality of (j_i) is known, but the existing proofs are lengthy and technical. 

In this talk I will discuss my recent work joint with E. Kaya and A. Waeterschoot, which shows that the jumps of the Jacobian of a curve C can be computed from weight functions. These functions are associated to canonical forms on C and live on the Berkovich analytification of C. As a corollary we obtain a short proof of the rationality of (j_i). Everything can be made explicit in the tropical framework of Δ_v-regular curves by T. Dokchitser, giving a large class of examples.

Timothy De Deyn, Categorical resolutions of filtered schemes

A. Kuznetsov and V. Lunts showed that over a field of characteristic zero one can always construct a categorical resolution of singularities. Their approach requires a strong version of Hironaka's resolution of singularities, namely that any variety can be resolved by a sequence of blow-ups along smooth centres. In the first part of the talk I will introduce categorical resolutions and Kuznetsov--Lunts' results. Thereafter I will discuss my recent work in which the use of strong Hironaka is circumvented: only the existence of projective resolutions is needed. For this the framework of filtered schemes is paramount. Finally, if time permits, I will explain ongoing work with M. Van den Bergh on generalising the construction to certain mild noncommutative varieties.