22 feb 2016 -- 11:00
Aula 311, Dip. Matematica, Università "Roma Tre", Roma
Abstract.
After Kajiwara and Payne's works in the late 2000's, it was well-understood how to tropicalize a closed subvariety of a toric variety. In recent years, this process got generalized in different direcetions. While Jeff and Noah Giansiracusa enhanced tropical varieties with an underlying scheme structure, employing the theory of so-called semiring schemes, Thuillier and Ulirsch replaced the ambient toric variety by toroidal embeddings, or more general, a log structure. In this series of three lectures, we will show how all of these theories can be understood on a common basis by using the language of so-called blueprints. After reviewing the above mentioned concepts, we will introduce bluprints and blue schemes. We redefine the tropicalization of a variety as the solution to a certain moduli problem and consruct the corresponding moduli space under some ambient hypothesis. Finally, we show how to recover the above mentioned theories within the languange of blue schemes.
TIMETABLE:
Monday 22 February, hour 11-13 (Room 311)
Tuesday 23 February, hour 10:30-12:30 (Room 311)
Thursday 25 February, hour 10:30-12:30 (Room 311)
REFERENCES:
O. Lorscheid: Scheme theoretic tropicalization. Preprint arXiv:1508.07949.
J. Giansiracusa, N. Giansiracusa: Equations of tropical varieties. Preprint arxiv:1308.0042.
O. Lorscheid: Blue schemes as relative schemes after Toen and Vaquie. Preprint arxiv:1212.3261.
O. Lorscheid, C. Salgado: Schemes as functors on topological rings. J. Number Theory 159, 193-201, 2016.
D. Maclagan, F. Rincon: Tropical schemes, tropical cycles, and valuated matroids. Preprint arXiv:1401.4654.
A. Thuillier: Geometrie toroidale et geometrie analytique non archimedienne. Application au type de homotopie de certains schemas formels. Manuscripta Math. 123 (2007), 381-51.
M. Ulirsch: Functorial tropicalization of logarithmic schemes: the case of constant coefficients. Preprint availble at www.math.uni-bonn.de/people/ulirsch/troplog.pdf.
PREREQUISITES:
A solid understanding of scheme theory (e.g. Chapter 2 of Hartshorne's book).