Moduli spaces of elliptic curves in toric varieties

Abstract: The moduli spaces of stable maps to toric varieties occur naturally in enumerative geometry and mirror symmetry. While they have several pleasing properties, they are often quite singular, reducible, and non-equidimensional. When the source curves have genus 0, the situation is markedly improved by adding logarithmic structure to the moduli problem. This produces irreducible and non-singular moduli spaces of rational curves in toric varieties, whose geometry is tightly controlled by a tropical moduli problem. When the source curves have genus 1, logarithmic structures alone do not suffice to produce a smooth and compact moduli space. However, by combining modern advances in logarithmic Gromov-Witten theory (due to Abramovich, Chen, Gross, Siebert, and Wise) with heuristics from a 2005 theorem of Speyer in tropical geometry, these moduli spaces can be desingularized "combinatorially". The result is a smooth and irreducible compactification of the space of elliptic curves in any toric variety, carrying an "honest" fundamental class, and whose intersection theory encodes "honest" counts of elliptic curves. This generalizes work of Vakil and Zinger.