Voronoi tilings and loop groups
-- Pablo Solis, March 25, 2016

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I would like to describe a partial compactification of the loop group
LT of a torus. All the ingredients are infinite dimensional but the
final result is essentially described by a finite dimensional toric
variety. In the case of T= C<sup>*</sup> the compactification recovers the Tate
curve which has a central fiber which is an infinite chain of
projective lines which is closely related to the moduli of line
bundles on a genus 0 nodal curve. A similar modular interpretation is
available for higher rank tori. It seems likely that there is also a
connection with Aleexev and Nakamura's work on degenerations of
Abelian varieties.