Date: Wednesday, February 01, 2017
Location: 4096 East Hall (4:10 PM to 5:30 PM)
Title: The moduli stack of tropical curves
Abstract: The moduli space of tropical curves (and its variants) are some of the moststudied objects in tropical geometry. So far this moduli space has only been considered as an essentially settheoretic coarse moduli space (sometimes with additional structure). As a consequence of this restriction, the tropical forgetful map does not define a universal curve (at least in the positive genus case). The classical work of DeligneKnudsenMumford has resolved a similar issue for the algebraic moduli space of curves by considering the fine moduli stacks instead of the coarse moduli spaces.
In this talk I am going to give an introduction to these fascinating moduli spaces and report on ongoing work with Renzo Cavalieri, Melody Chan, and Jonathan Wise, where we propose the notion of a moduli stack of tropical curves as a geometric stack over the category of rational polyhedral cones. Using this $2$categorical framework one can give a natural interpretation of the forgetful morphism as a universal curve. Moreover, I will propose two different ways of describing the process of tropicalization: one via logarithmic geometry in the sense of KatoIllusie and the other via nonArchimedean analytic geometry in the sense of Berkovich.
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Speaker: Martin Ulirsch
Institution: UM
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