Meeting time and venue: Mondays 1-2pm BST, MR13 and/or online
Motivation:
Log geometry is a natural framework for degenerations. Suppose we have a variety X and a "nice" degeneration of X, i.e. a family \pi: W -> A^1 whose special fibre W_0 = \pi^{-1}(0) is a union of varieties Y_1, Y_2, ..., Y_r such that the singular locus of W_0 is a simple normal crossings divisor D. To study a moduli problem on X, one hopes to compare this moduli problem with the moduli problem on Y_i "relative to D". This is desirable as the moduli on (Y_i, D) can be easier (e.g. when this is a toric pair).
The language to make this precise is log geometry.
Schedule
10/2: Motivation: Simple normal crossings divisors, toroidal compactification, degeneration. Overview of seminar [Siao Chi]
17/2: Geometry of snc divisors [Terry]
24/2: Log structures, Artin fans, tropicalisation [Veronica]
3/3: Log smooth/etale morphisms, log modifications are log etale [Patrick]
6/5: Double ramification cycles [Xuanchun]
12/5: Moduli of stable maps and Gromov--Witten theory [Terry]
26/5: Logarithmic Gromov--Witten theory [Dhruv]
References
LMS lecture notes; https://www.ma.imperial.ac.uk/~rpwt/LMSnotes.pdf
Ulirsch, Functorial tropicalization of logarithmic schemes: The case of constant coefficients; https://arxiv.org/abs/1310.6269
Abramovich, Chen, Gillam, Huang, Olsson, Satriano, and Sun, Logarithmic geometry and moduli; https://arxiv.org/abs/1006.5870
F. Kato, Log Smooth Deformation and Moduli of Log Smooth Curves;
Abramovich, Caporaso, and Payne, The tropicalization of the moduli space of curves; https://arxiv.org/abs/1212.0373
Cavalieri, Chan, Ulirsch and Wise, A moduli stack of tropical curves; https://arxiv.org/abs/1704.03806
Carocci, Nabijou, Rubber tori in the boundary of expanded stable maps; https://arxiv.org/abs/2109.07512
Cavalieri, Markwig and Ranganathan, Tropical and Logarithmic Methods in Enumerative Geometry; https://link.springer.com/book/10.1007/978-3-031-39401-0