Topology of the Vakil--Zinger moduli space

1. Problem Statement

The paper addresses the topological structure of the Vakil–Zinger moduli space, denoted as $\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^r, d)$. This space is a specific desingularization of the main component of the moduli space of degree $d$ stable maps from genus one curves with $n$ marked points to projective space $\mathbb{P}^r$.

• Context: The standard moduli space of stable maps $\overline{\mathcal{M}}_{1,n}(\mathbb{P}^r, d)$ is reducible and singular. The "main component" $\mathcal{M}_{1,n}^{\text{main}}(\mathbb{P}^r, d)$ is the closure of the locus of smooth domain curves, but it remains singular.
• The Object: Vakil and Zinger (2008) constructed a smooth, proper Deligne–Mumford stack $\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^r, d)$ that resolves the singularities of the main component.
• The Gap: While the geometry of this space is understood via its construction, its rational homology (specifically generators and relations) was not fully characterized. Understanding this homology is crucial for computing Gromov–Witten invariants and verifying deep conjectures in arithmetic geometry (Hodge and Tate conjectures).

2. Methodology

The author employs a multi-faceted approach combining stratification theory, tropical geometry, and mixed Hodge structures.

A. Stratification via Centrally Aligned Graphs

The core of the methodology relies on the stratification of $\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^r, d)$ by centrally aligned (1, n, d)-graphs.

• Centrally Aligned Graphs: Introduced by Ranganathan–Santos-Parker–Wise, these graphs decorate the dual graph of a stable map. When the genus-one "core" has degree zero, the graph is equipped with a "central alignment" $\rho$ that assigns levels to vertices, effectively blowing up the locus where the core is contracted.
• Strata Structure: Each stratum $\widetilde{\mathcal{M}}_{[G, \rho]}$ is described as a fiber product of three components:
  1. A stratum in the moduli space of centrally aligned genus-one curves ($\mathcal{M}_{1,n}^{\text{cen}}$).
  2. A space of genus-zero maps satisfying a "factorization property" (linear dependency of tangent vectors at attaching points).
  3. Standard genus-zero stable maps (rational tails).

B. Spectral Sequence and Weight Filtration

The author utilizes the stratification spectral sequence in Borel–Moore homology:
$$E_1^{p,q} = \bigoplus_{\dim \widetilde{\mathcal{M}}_{[G, \rho]}=p} H^{BM}_{p+q}(\widetilde{\mathcal{M}}_{[G, \rho]}) \implies H_{p+q}(\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^r, d))$$

• Weight Truncation: Since $\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^r, d)$ is smooth and proper, its homology carries a pure Hodge structure. The author truncates the spectral sequence at the pure weight level.
• Key Insight: The pure weight homology of the total space is generated by the pure weight homology of the strata. The relations among these generators are determined by the "off-by-one" weight graded pieces (i.e., weight $-(p+q)+1$) of the strata, which correspond to the differentials in the spectral sequence.

C. Building Blocks Analysis

The paper decomposes the problem by analyzing the homology of the fiber product factors:

1. Genus-One Core: Analyzed via a partial compactification by linear systems ($\widehat{Q}_{1,n}$), relating homology to the Picard stack and using excision sequences.
2. Genus-Zero Factorization Spaces: Analyzed using the geometry of parametrized maps with prescribed tangent vectors. The author proves these spaces have trivial cohomology except for classes generated by $\psi$-classes (from automorphism groups) and specific basepoint classes.
3. Rational Tails: Standard genus-zero stable maps, whose homology is generated by tautological cycles.

3. Key Contributions and Results

A. Generators of Homology (Theorem A)

The paper provides a complete set of generators for the rational homology $H_*(\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^r, d))$. These generators are pushforwards of classes from strata closures, constructed from:

1. Pullbacks from Moduli of Curves: Classes from $W_{-\star}H^{BM}_*(\mathcal{M}_{1,m})$ (related to cusp forms).
2. Tautological Classes: The relative polarization $\Theta$ (from the Picard stack) and the hyperplane class $\widehat{H}$ (from the linear system).
3. $\psi$-Classes: From the moduli of genus-zero curves with 1 or 2 marked points ($\mathcal{M}_{0,1}, \mathcal{M}_{0,2}$).
4. Rational Tails: Classes from genus-zero stable maps attached to the core.

B. Relations (Theorem B)

The relations among these generators are multiplicatively generated by:

1. WDVV and Getzler's Relations: Pulled back from the moduli of centrally aligned curves $\mathcal{M}_{1,n}^{\text{cen}}$.
2. Basepoint Relations: New relations derived from the geometry of maps with basepoints. These arise when rational tails are contracted to basepoints on the genus-one core. The author explicitly computes these relations using the excision sequence of the quasimap space.
3. $\psi$-Class Relations: Relations of the form $\psi_v = \psi_w$ for vertices on the same level of the central alignment.

C. Vanishing and Hodge Structure

• Odd Cohomology Vanishing: The paper recovers and generalizes Fontanari's result: $H^k(\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^r, d)) = 0$ for all odd $k < 11$.
• Hodge Structure: The non-vanishing odd cohomology is entirely controlled by Eichler–Shimura isomorphisms, specifically products of Tate structures $L$ and spaces of cusp forms $S_{k+1}$ (where $k \ge 11$).
• Lowest Non-Vanishing Degree: For $r \ge 11$, the first non-vanishing odd cohomology appears at degree $k=123$ (for $d=11$) or $k=13$ (for specific high-degree configurations).

D. Picard Group and Conjectures

• Rational Picard Group: The paper completely describes $H^2(\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^r, d))$. It is generated by $\Theta$, $\widehat{H}$, and boundary divisors.
• Cycle Class Map: The map from the rational Chow group to homology is an isomorphism in degree 2.
• Hodge and Tate Conjectures: The author verifies that the Hodge and Tate conjectures hold for $\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^r, d)$. This is a direct consequence of the fact that the even homology is generated by algebraic cycles (tautological classes and boundary strata).

4. Significance

1. Structural Clarity: This work provides the first comprehensive algebraic description of the homology of the Vakil–Zinger space, moving beyond geometric construction to explicit algebraic topology.
2. Gromov–Witten Theory: The results imply that the even homology of the space is tautological and controlled by genus-zero and reduced genus-one Gromov–Witten theory. This simplifies the computation of invariants.
3. Arithmetic Geometry: By verifying the Hodge and Tate conjectures, the paper establishes a strong link between the topology of these moduli spaces and number theory (specifically the Galois representations associated with cusp forms).
4. Methodological Advancement: The technique of using weight filtrations on stratified spaces to isolate generators and relations is refined here. It offers a blueprint for studying other moduli spaces of maps, including those for higher genus or different target varieties.
5. Basepoint Relations: The explicit derivation of "basepoint relations" fills a gap in understanding how rational tails interact with the core in the desingularized space, a phenomenon unique to the genus-one setting compared to genus-zero.

In summary, Song's paper successfully decodes the complex topology of the Vakil–Zinger moduli space, proving that its rational homology is generated by standard tautological classes and boundary strata, with relations governed by classical curve moduli relations and new basepoint constraints, thereby confirming deep conjectures regarding its arithmetic and geometric nature.