Abstract: Given a cyclic subgroup G of GL(2,C) acting on C^2, it was first noticed by Wunram in the 80s that there is a correspondence between certain special representations of G and the exceptional curves appearing in the minimal resolution Y of the surface singularity C^2/G. In modern terms, this was reformulated by Ishii and Ueda as…

We investigate the hierarchy between rationality, unirationality, and rational connectedness, discussing the various implications among these concepts. We then examine specific cases, addressing quadric bundles and del Pezzo surfaces over arbitrary fields, as well as complex 3-folds.

This talk asks which tropicalisations of subvarieties of the torus know the cohomology of the original variety? A motivating example are linear embeddings of complements of hyperplane arrangements. We prove that the tropicalisation knows the cohomology of the variety in a strong sense if and only if it satisfies local tropical Poincaré duality and the original variety is so-called “…

A classical result due to Clebsch from the mid-nineteenth century confirms that every complex space sextic curve (given as an intersection of a quadric and a cubic surface in projective 3-space) has exactly 120 tritangent planes. In this talk we will show how to use combinatorial methods arising from tropical geometry to revisit this classical problem and perform the analogous…

We construct affine charts of a smooth projective toric variety which contain its non-
negative points, and which admit a closed embedding into the total coordinate space
of Cox’s quotient construction. We show that such positive charts arise from smooth
subcones of the nef cone. To each positive chart we associate an algebraic moment map,<…