Equations of tropical varieties
By Jeffrey Giansiracusa, Noah Giansiracusa. 2013
122 citations recorded by the source corpus.
We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of $\mathbb{T}$-points this reduces to Kajiwara-Payne's extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of $\mathbb{T}$-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.
Subfields: Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/a73f7c902fdeb9787b4f19d3c81aa1d6b1c6986a
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In Computational Theory and Mathematics
In Geometry and Topology
Citations in the corpus
- Algebraic Geometry Over Hyperrings (2015, background): “We further note that, in [16], J.Giansiracusa and N.Giansiracusa showed that a scheme theory over semirings can be beautifully linked to tropical geometry by means of the theory of monoid schemes.”