The purpose of this working seminar is to learn how to use logarithmic structures in the sense of Fontaine, Illusie and Kato to construct, compactify, and sometime improve the behavior of certain moduli spaces.

We will start on safe ground and hopefully end up discovering unknown territory.

After a brief introduction, we will proceed with the first lecture, given by Qile Chen.

The basic paper is Kato, Kazuya Logarithmic structures of Fontaine-Illusie. Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 191--224, Johns Hopkins Univ. Press, Baltimore, MD, 1989.

There is also a book in preparation by Arthur Ogus: http://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf

The rough plan for the beginning is almost parallel to Olsson's lectures by the same title in http://math.berkeley.edu/~anton/written/AspectsModuli/MO.pdf:

1. definitions and basic properties
See Kato's paper above

2. differentials, smoothness, and deformations
See Kato's paper above, also see Kato, Kazuya Toric singularities. Amer. J. Math. 116 (1994), no. 5, 1073--1099.

3. Log smooth curves
see the setup in Fumiharu Kato. Log smooth deformation and moduli of log smooth curves. Internat. J. Math., 11(2):215-232, 2000.

4. D semistability and log structures
Kawamata, Yujiro; Namikawa, Yoshinori Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Invent. Math. 118 (1994), no. 3, 395--409.

5. Stacks of logarithmic structures
Olsson, Martin C. Logarithmic geometry and algebraic stacks. Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 5, 747--791.

Then we will try to address questions of specific interest to organizers and participants, such as:

6. log twisting and root constructions
Olsson, Martin C. (Log) twisted curves. Compos. Math. 143 (2007), no. 2, 476--494.

7. Moduli of stable curves as moduli of log smooth curves
Fumiharu Kato. Log smooth deformation and moduli of log smooth curves. Internat. J. Math., 11(2):215-232, 2000.

8. Gluing log smooth curves and their morphisms
Needs to be done

9. The main component of moduli spaces
Olsson, Martin Semi-stable degenerations and period spaces for polarized K3 surfaces. Duke Math. J. 125 (2004), 121-203.
Olsson, Martin Logarithmic interpretation of the main component in toric Hilbert schemes, in Proceedings of the International Conference on Curves and Abelian Varieties Edited by: Valery Alexeev, Arnaud Beauville, Herb Clemens, and Elham Izadi.

10. Log stable maps according to J. Li and B. Kim (with expanded degenerations)
Li, Jun, Stable morphisms to singular schemes and relative stable morphisms. J. Differential Geom. 57 (2001), no. 3, 509--578.
Bumsig Kim Logarithmic Stable Maps arXiv:0807.3611

11. Log stable maps of J. Li without expanded degenerations, and comparison
Needs to be done

12. Log stable maps in general
needs to be done. There is work in progress by mark Gross and Bernd Siebert

13. Is there a meaningful relation between the minimal model program and logarithmic structures of Fontaine-Illusie-Kato?
needs to be done

Exercises
Basic set
First set
second set