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... Show moreThis 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. Show less
In the first part, we introduce a new condition, called toric-additivity, on a family of abelian varieties degenerating to a semi-abelian scheme over a normal crossing divisor. The condition... Show moreIn the first part, we introduce a new condition, called toric-additivity, on a family of abelian varieties degenerating to a semi-abelian scheme over a normal crossing divisor. The condition depends only on the l-adic Tate module of the generic fibre, for a prime l invertible on S. We show that toric-additivity is strictly related to the property of existence of Neron models. In the second part, we consider the case of a family of nodal curves over a discrete valuation ring, having split singularities. We say that such a family is semi-factorial if every line bundle on the generic bre extends to a line bundle on the total space. We give a necessary and sufficient condition for semifactoriality, in terms of combinatorics of the dual graph of the special fibre. Show less
