Topology of the Vakil--Zinger moduli space

Abstract

We derive a set of generators for the rational homology of the desingularised genus one mapping space $\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^r,d)$ constructed by Vakil--Zinger and qualitatively describe the relations among the generators. The results build on a detailed study of the stratifications of the moduli spaces coming from tropical geometry and the constraints coming from the weight filtration on the Borel--Moore homology groups of strata, extending the techniques from the previous study on $\overline{\mathcal{M}}_{g,n}.$ Our results imply that the even homology of $\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^r,d)$ is tautological and controlled by genus-zero and reduced genus-one Gromov--Witten theory. We verify the Hodge and Tate conjectures for $\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^r,d),$ completely describe its rational Picard group, and recover known results on the vanishing of odd cohomology. Our techniques also apply to the pure weight homology groups of genus one stable maps $\overline{\mathcal{M}}_{1,n}(\mathbb{P}^r,d).$