Jonathan Wise: The logarithmic Picard group and its tropicalization

Abstract: I will describe work in progress with W. Gillam and S. Molcho on the logarithmic Picard group of a family of nodal curves. The logarithmic Picard group is a moduli problem on logarithmic schemes, originally proposed by L. Illusie, and has several features that are impossible to obtain simultaneously within the category of schemes: it is a group, it has a logarithmically etale cover by a logarithmic scheme, it coincides with the usual Picard group when the curve is smooth, and it is complete. The logarithmic Picard group of a curve X also admits a morphism to the Picard group of the tropicalization of X, using which one can recover toroidal compactifications of the Picard group from subdivisions of the tropical Picard group.