Moduli Spaces of Curves in Tropical Varieties

Abstract

We describe a framework to construct tropical moduli spaces of rational stable maps to a smooth tropical hypersurface or curve. These moduli spaces will be tropical cycles of the expected dimension, corresponding to virtual fundamental classes in algebraic geometry. As we focus on the combinatorial aspect, we take the weights on certain basic 0-dimensional local combinatorial curve types as input data, and give a compatibility condition in dimension 1 to ensure that this input data glues to a global well-defined tropical cycle. As an application, we construct such moduli spaces for the case of lines in surfaces, and in a subsequent paper for stable maps to a curve [Gathmann et al., Tropical moduli spaces of stable maps to a curve, in Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, ed. by G. Bockle, W. Decker, G. Malle (Springer, Heidelberg, 2018). https://doi.org/10.1007/978-3-319-70566-8_12].