### Arakelov heights in toric varieties

Algebra and Geometry Seminar

13th February 2019, 2:30 pm – 3:30 pm

Howard House, 4th Floor Seminar Room

Starting from the classical definition of height given by Northcott and Weil, I will present the main philosophy and guidelines of adelic Arakelov geometry. In the case of toric varieties, I will show how it is possible to translate such a theory in terms of convex geometry, following the work of Burgos Gil, Philippon and Sombra. I will then present a combinatorial formula relating the height of a hypersurface in a toric variety with the collection of the v-adic Ronkin functions of its defining Laurent polynomial. If time permits, I will sketch results and obstacles for a higher codimensional analogue.

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