Oct 21

Recent progress on the Diophantine geometry of curves

Minhyong Kim (Warwick)

The study of rational or integral solutions to polynomial equations is among the oldest subjects in mathematics. After a brief description of the history, we will review some recent geometric approaches to describing sets of solutions when the number of variables is 2.
Please click on the link to the “abstract” to view the slides.

Oct 28

Derived categories and birationality

Martin Olsson (UC Berkeley )

I will discuss expectations and results around the following question: If $X$ and $Y$ are two smooth projective varieties with equivalent derived categories, when can one conclude that $X$ and $Y$ are birational? The study of FourierMukai equivalences yields many examples of nonbirational varieties with equivalent derived categories. On the other hand, it appears that by considering slightly more structure than just the derived categories one can conclude birationality in many cases. This is joint work with Max Lieblich.
Recording available here:
https://us02web.zoom.us/rec/share/U7Zp8zsHQrL4WGyhHAx9sSLRNwEPoFAp2AnvK5_lvC4M0_5aByj6YMYM00_zdsiG.Pr97s6WdDsEx0qM

Nov 4

Quadratic Euler characteristics of hypersurfaces and hypersurface singularities

Marc Levine (Essen)

This is a report on joint work with V. Srinivas and Simon Pepin Lehalleur. Recently, with Arpon Raksit, we have shown that for a smooth projective variety X over a field k, the quadratic Euler characteristic of X, an element of the GrothendieckWitt ring of quadratic forms over k, can be computed via the cup product on Hodge cohomology followed by the canonical trace map. Following work of CarlsonGriffiths, this leads to an explicit formula for the quadratic Euler characteristic of a smooth projective hypersurface defined by a homogeneous polynomial F in terms of the Jacobian ring of F, as well as a similar formula for a smooth hypersurface in a weighted projective space. In some special cases, this leads to quadratic versions of classical conductor formulas with some mysterious and unexpected correction terms, even in characteristic zero.

Nov 11

No meeting

No meeting


Nov 18

Rigidity, Residues and Duality: Recent Progress

Amnon Yekutieli (Be'er Sheva)

Let K be a regular noetherian ring. I will begin by explaining what is a rigid dualizing complex over an essentially finite type (EFT) Kring A. This concept was introduced by Van den Bergh in the 1990’s, in the setting of noncommutative algebra. It was imported to commutative algebra by Zhang and myself around 2005, where it was made functorial, and it was also expanded to the arithmetic setting (no base field). The arithmetic setting required the use of DG ring resolutions, and in this aspect there were some major errors in our early treatment. These errors have recently been corrected, in joint work with Ornaghi and Singh.
Moreover, we have established the forward functoriality of rigid dualizing complexes w.r.t essentially etale ring homomorphisms, and their backward functoriality w.r.t. finite ring homomorphisms. These results mean that we have a twisted induction pseudofunctor, constructed in a totally algebraic way (rings only, no geometry).
Looking to the future, we plan to study a more refined notion: rigid residue complexes. These are complexes of quasicoherent sheaves in the big etale site of EFT Krings, and they admit backward functoriality, called indrigid traces, w.r.t. arbitrary ring homomorphisms.
Rigid residue complexes can be easily glued on EFT Kschemes, and they still have the indrigid traces w.r.t. arbitrary scheme maps. The twisted induction now becomes the geometric twisted inverse image pseudofunctor f \mapsto f^!. We expect to prove the Rigid Residue Theorem and the Rigid Duality Theorem for proper maps of EFT Kschemes, thus recovering almost all of the theory in the original book “Residues and Duality”, in a very explicit way.
The etale functoriality implies that every finite type DeligneMumford (DM) K‑stack admits a rigid residue complex. Here too we have the f \mapsto f^! pseusofunctor. For a map of DM stacks there is the indrigid trace. Under a mild technical condition, we expect to prove the Rigid Residue Theorem for proper maps of DM stacks, and the Rigid Duality Theorem for such maps that are also tame.
Lecture notes will be available at
http://www.math.bgu.ac.il/~amyekut/lectures/RRD2020/notes.pdf .
(November 2020)

Nov 25

Bad reduction and fundamental groups

Netan Dogra (Oxford)

This talk will be about two related results concerning Galois actions on prop fundamental groups of curves over mixed characteristic local fields, with applications to the algorithmic resolution of Diophantine equations. The first result is joint with Alex Betts, and gives a description of how the Galois action on the fundamental group varies with the choice of basepoint in terms of harmonic analysis on the dual graph of the special fibre of a stable model (when p is different from the residue characteristic). The second result is joint with Jan Vonk, and gives a description of how to compute the Galois action (in a padic Hodge theoretic sense) when the residue characteristic is p and the curve has semistable reduction.
