Tropical Curves and Covers and Their Moduli Spaces
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Tropical Curves and Covers and Their Moduli Spaces Hannah Markwig1
© The Author(s) 2020
Abstract Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and covers of tropical curves maps between metric graphs satisfying certain conditions. In this short survey, we offer an introduction to the combinatorial theory of abstract tropical curves and covers of curves, and their moduli spaces, and we showcase three results demonstrating how this theory can be applied in algebraic geometry. Keywords Tropical geometry · Moduli spaces · Tropical covers · Mirror symmetry · Hurwitz numbers · Enumerative geometry Mathematics Subject Classification (2010) Primary 14T05 · secondary 14N10
1 Introduction Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. A degeneration referred to as tropicalization associates a combinatorial object to a given algebraic variety which surprisingly captures many important properties of the algebraic variety. For that reason, tropical geometry allows to incorporate methods from discrete mathematics into algebraic geometry. Vice versa, it is also possible to use methods from algebraic geometry in discrete mathematics using tropical geometry, a prime example is the recent proof of Rota’s conjecture for characteristic polynomials of matroids by Adiprasito, Huh and Katz [2].
B H. Markwig
[email protected]
1
Fachbereich Mathematik, Eberhard Karls Universität Tübingen, Tübingen, Germany
H. Markwig
For algebraic varieties which come embedded into projective space, tropicalization can be expressed using methods from the theory of Gröbner bases and initial ideals. The book [39] by Maclagan and Sturmfels offers a broad and friendly introduction for this approach. But also for abstract varieties, it is possible to study tropicalizations. In particular for the case of curves, abstract tropical geometry has in recent years become an attractive combinatorial theory uniting several important perspectives such as semistable reduction, Berkovich theory resp. non-Archimedean analytic geometry, graph theory, topology and representation theory [4–6, 33, 34]. In this article, we offer an introduction to the combinatorial theory of abstract tropical curves and covers of curves, and their moduli spaces, and we survey recent results demonstrating how this theory can be applied in algebraic geometry. An (abstract) tropical curve is, roughly, just a metrized graph, possibly with unbounded half-edges called ends. The process of tropicalization that produces a tropical curve from an algebraic curve can be pictured topologically: we think of a smooth algebraic curve as a Riemann surface and let the tubes become so thin that they end up being edges of a graph. For example, an elliptic curve, which is a torus when viewed as a Riemann surface, becomes a circle. Moduli space of tropical curves are thus essentially spaces parametrizing certa
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