arXiv subjects

Terry Dekun Song

Publications and source records attributed to Terry Dekun Song.

8 recordsLinked to original sources

arXiv2026

Motivic quasimap wall-crossing for Grassmannians

We prove a wall-crossing formula for the Euler characteristics, considered as virtual mixed Hodge structures, of moduli spaces of $\varepsilon$-stable quasimaps to the Grassmannian $\mathbb{G}(r, N)$. For each $\varepsilon > 0$, we define a $\mathbb{Q}$-algebra automorphism of the ring of symmetric functions which takes the generating function for the $\mathbb{S}_n$-equivariant Euler characteristics of the moduli spaces of stable maps $\overline{\mathcal{M}}_{g, n}(\mathbb{G}(r, N), d)$ to the corresponding generating function for Toda's moduli spaces of $\varepsilon$-stable quasimaps $\overline{\mathcal{Q}}_{g, n}^{\varepsilon}(\mathbb{G}(r, N), d)$. The automorphism is given by explicit $q$-deformations of the power sum symmetric functions. The $\varepsilon \to 0$ limit of our formula exchanges the spaces of stable maps and the Marian--Oprea--Pandharipande moduli spaces of stable quotients. Our proof uses the geometry of relative Quot schemes to relate the quasimap spaces to moduli spaces of weighted stable maps, for which we obtain wall-crossing formulas via symmetric function theory.

math.AG↗

arXiv2026

Logarithmic Hilbert schemes of curves as weighted blow-ups and their integral Chow rings

The logarithmic Hilbert scheme of a logarithmic curve parametrizes subschemes on the expanded degenerations of the curve that are transverse to the boundary. We prove that the logarithmic Hilbert scheme of points on a smooth pointed curve is an iterated weighted blow-up of the symmetric product of the underlying curve. In doing so, we explicitly identify the blow-up centers, weights, and give them modular interpretations. As applications, we calculate their integral Chow rings in terms of those of the symmetric products. Key ingredients in our work include two recent results: the integral Chow ring formula of weighted blow-ups and a weighted analogue of Castelnuovo's criterion for blow-downs. We recover the folklore result the logarithmic Hilbert scheme of toric $\mathbb{P}^1$ is a toric stack, and the Appendix by Dhruv Ranganathan outlines a complementary approach using Chow quotients.

math.AG↗

arXiv2026

The $S_n$-equivariant Euler characteristic of $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$

We compute the $S_n$-equivariant topological Euler characteristic of the Kontsevich moduli space $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$. Letting $\overline{\mathcal{M}}_{1, n}^{\mathrm{nrt}}(\mathbb{P}^r, d) \subset \overline{\mathcal{M}}_{1, n}(¶^r, d)$ denote the subspace of maps from curves without rational tails, we solve for the motive of $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$ in terms of $\overline{\mathcal{M}}_{1, n}^{\mathrm{nrt}}(\mathbb{P}^r, d)$ and plethysm with a genus-zero contribution determined by Getzler and Pandharipande. Fixing a generic $\mathbb{C}^\star$-action on $\mathbb{P}^r$, we derive a closed formula for the Euler characteristic of $\overline{\mathcal{M}}_{1, n}^{\mathrm{nrt}}(\mathbb{P}^r, d)^{\mathbb{C}^\star}$ as an $S_n$-equivariant virtual mixed Hodge structure, which leads to our main formula for the Euler characteristic of $\overline{\mathcal{M}}_{1,n}(\mathbb{P}^r, d)$. Our approach connects the geometry of torus actions on Kontsevich moduli spaces with symmetric functions in Coxeter types $A$ and $B$, as well as the enumeration of graph colourings with prescribed symmetry.

math.AG↗

arXiv2026

The dual complex of $\mathcal{M}_{1,n}(\mathbb{P}^r,d)$ via the geometry of the Vakil--Zinger moduli space

We study normal crossings compactifications of the moduli space of maps $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$, for $g = 0$ and $g = 1$. In each case we explicitly determine the dual boundary complex, and prove that it admits a natural interpretation as a moduli space of decorated metric graphs. We prove that the dual complexes are contractible when $r \geq 1$ and $d > g$. When $g = 1$, our result depends on a new understanding of the connected components of boundary strata in the Vakil--Zinger desingularization and its modular interpretation by Ranganathan--Santos-Parker--Wise.

math.AG↗

arXiv2026

Pólya enumeration, wreath product symmetric functions, and moduli spaces of curves

We develop a calculus for $S_n$-equivariant Euler characteristics of moduli spaces of stable curves and stable maps. Our approach involves an enrichment of Pólya's cycle index polynomial of a graph to a certain algebra $Λ^{[2]}$ of wreath product symmetric functions. Building on foundational work of Macdonald, we prove that $Λ^{[2]}$ may be viewed as the Grothendieck ring of the category of polynomial functors which map symmetric sequences of vector spaces to vector spaces. This interpretation gives rise to an action of $Λ^{[2]}$ on the ordinary ring of symmetric functions $Λ$, which is described concretely in terms of Adams operations and skewing by power sums. This action lets us deduce appealing formulas, involving only ordinary symmetric functions, for generating functions of $S_n$-equivariant Euler characteristics.

math.CO↗

arXiv2026

Topology of the Vakil--Zinger moduli space

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

math.AG↗

arXiv2026

Virtual Hodge numbers of $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$: stability and calculations

We study $\mathbb{S}_n$-equivariant motivic invariants of the moduli space $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$ of degree-$d$ maps from $n$-pointed curves of genus $g$ to $\mathbb{P}^r$. In particular, we obtain formulas for the Serre characteristic, which specializes to the Hodge--Deligne polynomial. Fixing $g, r \geq 1$, we prove that an explicit invertible transform of the generating function for the Serre characteristics is rational. We use our formula to prove a stability result for the weight-graded compactly-supported Euler characteristics of $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$ as $d \to \infty$. In genus one and two, we reduce the calculation of the Serre characteristic of $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$ to those of the moduli spaces $\mathcal{M}_{g, n}$ of $n$-pointed curves. Formulas for the latter follow from work of Getzler and Petersen, so our formula in particular determines the Serre characteristic of $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$ for arbitrary $n$, $r$, and $d$ when $g = 1$ and $g = 2$.

math.AG↗

arXiv2025

Graph enumeration for moduli spaces of curves and maps

We develop a calculus based on graph enumeration for $S_n$-equivariant motivic invariants of graphically stratified moduli spaces. We apply our theory to the Deligne--Mumford moduli space $\overline{\mathcal{M}}_{g, n}$ and to the space of torus-fixed stable maps $\overline{\mathcal{M}}_{g, n}(X, β)^{\mathbb{C}^\star}$ when the target $X$ admits an appropriate $\mathbb{C}^\star$-action, deriving new formulas in each case. A key role is played by the Pólya--Petersen character of a graph, which enriches Pólya's classical cycle index polynomial. This character is valued in an algebra $Λ^{[2]}$ of wreath product symmetric functions, which we study from combinatorial and representation-theoretic perspectives. We prove that this algebra may be viewed as the Grothendieck ring of the category of polynomial functors which take symmetric sequences of vector spaces to vector spaces, building on foundational work of Macdonald. This leads to a plethystic action of $Λ^{[2]}$ on the ring $Λ$ of ordinary symmetric functions. Using this action, we derive our formulas, which ultimately involve only ordinary symmetric functions and the Grothendieck ring of mixed Hodge structures.

math.AG↗