Universiteit Leiden

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Dissertation

Dual Complexes of Semistable Varieties

This thesis is comprised of three chapters covering the theme of studying semistable varieties by looking at their dual combinatorial objects.

Author
Akeyr, G.
Date
17 December 2019
Links
Thesis in Leiden Repository

This thesis is comprised of three chapters covering the theme of studying semistable varieties by looking at their dual combinatorial objects.The first chapter defines what it is to be a semistable variety. This is done in such a way as to generalize the case of semistable curves. The dual graph of such a variety has vertices corresponding to the the irreducible components of the variety, with edges between the vertices corresponding to connected components of the non-smooth loci. We show that this definition allows us to reproduce results as in the 1-dimensional case, where a condition on the dual graph called "alignment" is necessary for the existence of a Neron model of the Picard space of the variety.The second chapter is inspired by tropical geometry and seeks to define a good generalization of the tropical curve associated to a logarithmic curve over a logarithmic base scheme.The last chapter shows how, in the case of a semistable curve, the Artin fan associated to the curve has its underlying topological space naturally isomorphic to that of the dual graph. We prove additional results related to the existence of an associated logarithmic morphism.

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