Topology of the Vakil--Zinger moduli space

TL;DR

This paper studies the topology of a desingularized genus one moduli space of stable maps, deriving generators for its rational homology, and proving several conjectures and structural results using tropical geometry and homological techniques.

Contribution

It provides a detailed description of the rational homology generators and relations of the Vakil--Zinger moduli space, extending previous techniques and verifying key conjectures.

Findings

Even homology is tautological and governed by genus-zero and reduced genus-one Gromov--Witten theory.

Hodge and Tate conjectures are verified for the moduli space.

The rational Picard group is completely described.

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 $\overline{\mathcal{M}}_{1,n}(\mathbb{P}^r,d).$