Learning Seminar
Leiden University, Utrecht University
Organized by: Sara Mehidi, Pim Spelier, Amira Tlemsani
Log geometry was introduced in the late 80s by Fontaine-Illusie, Deligne-Faltings and K. Kato. It offers a more general notion of smoothness, which allows to treat certain objects with mild singularities as if they were smooth. It also provides a functorial way to compactify various moduli spaces: smooth objects often degenerate to logarithmically smooth (but non-smooth) objects over the boundary.
We will start the seminar with some motivation and some background on algebraic stacks. We will then learn about log schemes (schemes equipped with a log structure); after which, we introduce the notion of log structures on stacks over the category of schemes. We will also investigate the relation between log stacks (algebraic stacks with a log structure) and algebraic stacks on the category of log schemes. In particular, we will study the stack of log stable curves. We finish off the seminar with an important example of log stacks -Artin fans-, which encode the log structure of a log scheme. The seminar is concluded with some research talks on recent results related to log stacks.
Each session will be 45 min + 15 min break + 30-45 min.
For the prerequisites and references please see the detailed program here.
The talks take place on Mondays 10:15-12:00 (GMT+1 ).
Speakers: Reinier Schmiermann & Amira Tlemsani.
Content: Motivation of the seminar + Overview algebraic stacks.
Room in Utrecht: HFG 707.
Room in Leiden: BM.2.26 (Gorlaeus building).
Speaker: Stijn Velstra.
Content: Quasi-coherent sheaves on algebraic spaces, quasi-coherent sheaves on algebraic stacks.
Room in Utrecht: HFG 707.
Room in Leiden: BM.2.26.
Exceptionally at 9:30-11 am.
Speaker: Boaz Moerman.
Content: Monoids, log structures, log schemes, morphisms of log schemes, examples: divisorial log structure, toric varieties (in particular Spec Z[P ] for a monoid P ), examples of a non-divisorial log structure (pullback of a log structure, example of a log curve (without definition of a log curve)).
Room in Utrecht: HFG 707.
Room in Leiden: BM.2.26.
Speaker: Francesca Leonardi.
Content: Charts (examples from previous talk), definition of fine and saturated log structures, log smoothness étaleness (through log differentials, with examples), chart criterion.
Room in Utrecht: HFG 707.
Room in Leiden: BM 2.26.
Speakers: Céline Fietz & Amira Tlemsani.
Content: The stack of stable curves, log curves, table of log curves, going from stacks with log structure to stacks on log schemes and vice versa: the example of log stable curves.
Room in Utrecht: HFG 707.
Room in Leiden: BW.0.06.
Speaker: Pim Spelier.
Content: Log blowup, idea of the proof that LogX is an algebraic stack, local description of LogX (toric stacks).
Room in Utrecht: HFG 707.
Room in Leiden: BW.2.18a
Speakers: Sara Mehidi & Tom Manopulo.
Content: The category of Artin fans, the Artin fan of a logarithmic scheme, Artin fans and functoriality.
Room in Utrecht: HFG 707.
Room in Leiden: BW.2.18a.
Location: Utrecht University, BBG 065.
Program:
13:15-14:15 : Martin Ulirsch, Goethe University Frankfurt am Main (also AG Seminar in Utrecht).
Title: How to tropicalize algebraic groups (and why we should care): Log edition.
Abstract: Tropical geometry studies a piecewise linear combinatorial shadow of classical algebraic geometry. In many ways, tropical geometry describes the extra information that is added to classical scheme theory in logarithmic geometry in order to study moduli-theoretic problems from a unified perspective. In this talk I will explain what little we currently know about the tropical geometry of algebraic groups. Along the way I will outline how different avenues of progress towards an answer to this question have already led to numerous advantages within and beyond tropical and logarithmic geometry.
This talk will touch upon joint work with Luca Battistella, Desmond Coles, Andreas Gross, Inder Kaur, Kevin Kühn, Arne Kuhrs, Margarido Melo, Sam Molcho, Annette Werner, Alejandro Vargas, Filippo Viviani, and Dmitry Zakharov.
14:45-15:45: George Politopoulos , Cergy University (France)/ Leiden University.
Title: (spin)Strata of differentials and (spin)Double ramification cycles.
Abstract: The strata of k-differentials are spaces parametrizing stable curves with n markings that admit a differential with prescribed zeroes and poles summing to k(2g-2+n). A natural question to ask is whether there exists an explicit formula in the Chow ring of M_{g,n}(bar) for the Chow class of these spaces. When k and all the zeroes and poles are odd numbers, we can define the so-called "spin refinement" of this problem. In the non-spin case, the theory of Double ramification cycles can help to compute all such classes for the various g and n. In this talk we will recall this story and explain how this works in the spin case. This is a joint work in progress with Adrien Sauvaget and David Holmes.
16:00-17:00: Qile Chen , Boston College (via Zoom).
Title: Campana rational connectedness .
Abstract: The notion of Campana points were introduced by Campana and Abramovich, which interpolate between rational points and integral points. In this talk, we will focus on the geometric side and introduce Campana rational connectedness --- a version of rational connectedness for varieties with simple normal crossings boundaries. We further prove that over function fields, weak approximations by Campana points at good places hold assuming Campana rational connectedness of fibers, generalizing a theorem of Hassett and Tschinkel. We further verify Campana rational connectedness for many basic examples. Our approach relies on the theory of stable log maps and their moduli. This is a joint work in progress with Brian Lehmann and Sho Tanimoto.