article Renzo Cavalieri author Melody Chan author Martin Ulirsch author 114697 Jonathan Wise author We contribute to the foundations of tropical geometry with a view towards formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves. Our approach to tropical geometry permits tropical moduli problems---moduli of curves or otherwise---to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms. 2017 eng 1704.0380610.1017/fms.2020.16 R. Cavalieri, M. Chan, M. Ulirsch, J. Wise, ArXiv:1704.03806 (2017). Cavalieri, Renzo, Melody Chan, Martin Ulirsch, and Jonathan Wise. “A Moduli Stack of Tropical Curves.” <i>ArXiv:1704.03806</i>, 2017. <a href="https://doi.org/10.1017/fms.2020.16">https://doi.org/10.1017/fms.2020.16</a>. R. Cavalieri, M. Chan, M. Ulirsch, and J. Wise, “A moduli stack of tropical curves,” <i>arXiv:1704.03806</i>, 2017, doi: <a href="https://doi.org/10.1017/fms.2020.16">10.1017/fms.2020.16</a>. Cavalieri, R., Chan, M., Ulirsch, M., &#38; Wise, J. (2017). A moduli stack of tropical curves. <i>ArXiv:1704.03806</i>. <a href="https://doi.org/10.1017/fms.2020.16">https://doi.org/10.1017/fms.2020.16</a> @article{Cavalieri_Chan_Ulirsch_Wise_2017, title={A moduli stack of tropical curves}, DOI={<a href="https://doi.org/10.1017/fms.2020.16">10.1017/fms.2020.16</a>}, journal={arXiv:1704.03806}, author={Cavalieri, Renzo and Chan, Melody and Ulirsch, Martin and Wise, Jonathan}, year={2017} } Cavalieri R, Chan M, Ulirsch M, Wise J. A moduli stack of tropical curves. <i>arXiv:170403806</i>. Published online 2017. doi:<a href="https://doi.org/10.1017/fms.2020.16">10.1017/fms.2020.16</a> Cavalieri, Renzo, et al. “A Moduli Stack of Tropical Curves.” <i>ArXiv:1704.03806</i>, 2017, doi:<a href="https://doi.org/10.1017/fms.2020.16">10.1017/fms.2020.16</a>. 663362026-07-08T07:07:13Z2026-07-08T08:26:17Z