Enriched tropical intersection

Abstract: Tropical geometry has been proven to be a powerful computational tool in enumerative geometry over the complex and real numbers. In this talk we present an example of a quadratic refinement of this tool, namely a proof of the quadratically refined Bézout’s theorem for tropical curves. We recall the necessary notions of enumerative geometry over arbitrary fields valued in the Grothendieck-Witt ring. We will mention the Viro’s patchworking method that served as inspiration to our construction based on the duality of the tropical curves and the refined Newton polytope associated to its defining polynomial. We will prove that the quadratically refined multiplicity of an intersection point of two tropical curves can be computed combinatorially. We will use this new approach to prove an enriched version of the Bézout theorem and of the Bernstein–Kushnirenko theorem, both for enriched tropical curves. Based on a joint work with S. Pauli.

algebraic geometrycombinatorics

Audience: researchers in the topic

(LAGARTOS) Latin American Real and Tropical Geometry Seminar

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