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Tropicalization of Classical Moduli Spaces

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Abstract

The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our primary example is the Burkhardt quartic, whose tropicalization is a 3-dimensional fan in 39-dimensional space. This effectuates a synthesis of concrete and abstract approaches to tropical moduli of genus 2 curves.

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Authors and Affiliations

  1. Google Inc., Mountain View, CA, 94043, USA

    Qingchun Ren

  2. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Steven V. Sam & Bernd Sturmfels

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  1. Qingchun Ren
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  2. Steven V. Sam
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  3. Bernd Sturmfels
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Correspondence to Bernd Sturmfels.

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Ren, Q., Sam, S.V. & Sturmfels, B. Tropicalization of Classical Moduli Spaces. Math.Comput.Sci. 8, 119–145 (2014). https://doi.org/10.1007/s11786-014-0185-x

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