The logarithmic Picard group and its tropicalization

On a preprint of Molcho and Wise in which they prove the representability (in a suitable sense) of the logarithmic Picard functor.

A new version of the paper is here. It corrects an error in the construction of the tropicalization map for LogPic (the earlier construction was only valid locally in the log. etale topology, but TroPic is not a sheaf in the log. etale topology).


David Holmes, Chris Lazda, Adrien Sauvaget and Arne Smeets.

Week 1

Friday 8 March, RU Nijmegen, room HG00.303 in the Huygens building.
When Who What Abstract Videos
13:00-14:00 David Holmes (Leiden) Introduction and overview We begin with an overview of the general problem of forming relative compactifications of families of abelian varieties, with an emphasis on elementary examples. We then outline how logarithmic geometry can (and cannot) help us. Finally, we recall the classical theory of the relative Picard functor and stack, as a `warm up' for the tropical and logarithmic versions we will meet later. Part 1
Part 2
14:30-15:30 Salvatore Floccari (Nijmegen) Basics on log schemes (1) My talk will be a gentle introduction to log schemes. I will start with the definition of log structures, log schemes, morphisms between them, and charts. I will then present some examples. If time permits, I will discuss Kato's notion of log regularity, and explain how a log regular scheme gives rise to a combinatorial object, the Kato fan. Part 1
Part 2
16:00-17:00 Chris Lazda (UvA) Basics on tropical geometry and tropical curves In my talk I will introduce embedded tropical curves, and explain how they arise via the tropicalization of (algebraic) curves embedded in tori. I will then show how these relate to abstract tropical curves, i.e. metrised graphs, and finally to the abstract tropical curves used by Molcho-Wise, where the "lengths" of the edges are allowed to be elements of an arbitrary monoid. Part 1
Part 2
All videos in one playlist: here.

Week 2

Friday 5 April, Utrecht, Room MIN 2.01.
13:00-14:00 Martin Bright (Leiden) Basics on log schemes (2) Part 1
Part 2
14:30-15:30 Rosa Schwarz (Leiden) Log structures on (semi)stable curves Part 1 of 1
16:00-17:00 Reinier Kramer (UvA) Tropical moduli problems Part 1
Part 2
All videos in one playlist: here.

Week 3

Friday 26 April, UvA, room A1.04, Science Park 904. Alas, I (David) forgot the camera today. Latex'd notes are uploaded instead; any errors are most likely due to my bad typing.
13:00-14:00 Thibault Poiret The tropical Picard group and the tropical jacobian Notes
14:30-15:30 Arne Smeets The logarithmic and tropical multiplicative groups Notes
16:00-17:00 Zoe Schroot Log line bundles and the log picard group Notes

Week 4

Friday 7 June, Leiden, room 401 (Snellius).
13:00-14:00 Adrien Sauvaget (Utrecht) Representability of the Logarithmic Picard group
14:30-15:30 Giulio Orecchia (Rennes) Smoothness and Properness of the Logarithmic Picard group Notes
16:00-17:00 Samouil Molcho (Tel Aviv) Structure of the Logarithmic Picard Group: Tropicalization and Schematic Models I will discuss the connection between the logarithmic and tropical Picard groups and the generalized Jacobian. In the process, we will obtain effective presentations which allow us to compute the log Picard group. After computing some examples, I will discuss to what extent log pic fails to be representable by a scheme, and the schematic models that most closely approximate it.

Previous seminar

Last year's seminar was on a preprint of Groechenig, Wyss and Ziegler in which they prove a conjecture of Hausel and Thaddeus on the Hodge numbers of moduli spaces of Higgs bundles. I organised this jointly with Chris Lazda and Arne Smeets. See here for more information.