For any affine variety equipped with coordinates, there is a surjective, continuous map from its Berkovich space to its tropicalisation. Exploiting torus actions, we develop techniques for finding an explicit, continuous section of this map. In particular, we prove that such a section exists for linear spaces, Grassmannians of planes (reproving a result due to Cueto, Häbich, and Werner), matrix varieties defined by the vanishing of 3 × 3-minors, and for the hypersurface defined by Cayley’s hyperdeterminant.

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Journal Manuscripta Mathematica
Draisma, J, & Postinghel, E. (2016). Faithful tropicalisation and torus actions. Manuscripta Mathematica, 149(3-4), 315–338. doi:10.1007/s00229-015-0781-3