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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. Whiel 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.
REREFENCES: O. Lorsheid: Scheme theoretic tropicalization. Preprint arXiv:1508.07949. J. Giansiracusa, N. Giansiracusa: Equations of tropical varieties. Preprint arxiv:1308.0042. O. Lorsheid: Blue schemes as relative schemes after Toen and Vaquie. Preprint arxiv:1212.3261. O. Lorsheid, 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 semas 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) |