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Obstructions to Approximating Tropical Curves in Surfaces Via Intersection Theory

  • Erwan Brugallé,
    École polytechnique, Centre Mathématiques Laurent Schwatrz, 91 128 Palaiseau Cedex, France
  • Kristin Shaw,
    Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4
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Abstract

We provide some new local obstructions to approximating tropical curves in smooth tropical surfaces. These obstructions are based on a relation between tropical and complex intersection theories which is also established here. We give two applications of the methods developed in this paper. First we classify all locally irreducible approximable 3-valent fan tropical curves in a fan tropical plane. Secondly, we prove that a generic non-singular tropical surface in tropical projective 3-space contains finitely many approximable tropical lines if it is of degree 3, and contains no approximable tropical lines if it is of degree 4 or more.
Keywords: tropical geometry, amoebas, approximation of tropical varieties, intersection theory tropical geometry, amoebas, approximation of tropical varieties, intersection theory
MSC Classifications: 14T05, 14M25 show english descriptions Tropical geometry [See also 12K10, 14M25, 14N10, 52B20]
Toric varieties, Newton polyhedra [See also 52B20]
14T05 - Tropical geometry [See also 12K10, 14M25, 14N10, 52B20]
14M25 - Toric varieties, Newton polyhedra [See also 52B20]
 

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