Combinatorial Algebraic Geometry pp 353-368 | Cite as

# Towards a Tropical Hodge Bundle

## Abstract

The moduli space \(\mathop{\mathrm{M}}\nolimits _{g}^{\text{trop}}\) of tropical curves of genus *g* is a generalized cone complex that parametrizes metric vertex-weighted graphs of genus *g*. For each such graph *Γ*, the associated canonical linear system | *K*_{ Γ } | has the structure of a polyhedral complex. In this article, we propose a tropical analogue of the Hodge bundle on \(\mathop{\mathrm{M}}\nolimits _{g}^{\text{trop}}\) and study its basic combinatorial properties. Our construction is illustrated with explicit computations and examples.

## MSC 2010 codes:

14T05## Notes

### Acknowledgements

This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. Both authors would like to acknowledge his input. Thanks are also due to the Max-Planck-Institute of Mathematics in the Sciences in Leipzig, Germany, for its hospitality. The second author would like to thank Diane Maclagan for several discussions related to the topic of this note, as well as the Fields Institute for Research in Mathematical Sciences. Finally, many thanks are due to the anonymous referees for several helpful comments and suggestions.

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