arXiv subjects

Siddarth Kannan

Publications and source records attributed to Siddarth Kannan.

17 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.

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arXiv2026

Automorphisms of the boundary complex of $\overline{\mathcal{M}}_{0, n}(\mathbb{P}^r, d)$

We compute the automorphism group of the dual complex $\mathsf{T}_{d, n}$ of the boundary divisor in the Kontsevich moduli space $\overline{\mathcal{M}}_{0, n}(\mathbb{P}^r, d)$. When $d \geq 2$, we find that $\mathrm{Aut}(\mathsf{T}_{d, n}) \cong \mathbb{S}_{n}$, while $\mathrm{Aut}(\mathsf{T}_{1, n}) \cong \mathbb{S}_{n + 1}$ for all $n \geq 4$. The complex $\mathsf{T}_{1, n}$ is also the dual complex of the boundary divisor in the Fulton--MacPherson compactification of the configuration space of $n$ points on $X$, if $X$ is any smooth, proper, and connected algebraic variety over $\mathbb{C}$. Following work of Massarenti, this implies that $\mathsf{T}_{1, n}$ admits automorphisms which in general do not extend to $X[n]$.

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arXiv2026

P\'olya 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 P\'olya's cycle index polynomial of a graph to a certain algebra $\Lambda^{[2]}$ of wreath product symmetric functions. Building on foundational work of Macdonald, we prove that $\Lambda^{[2]}$ 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 $\Lambda^{[2]}$ on the ordinary ring of symmetric functions $\Lambda$, 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.

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arXiv2026

Euler characteristics of the universal Picard stack

We study $\mathbb{S}_n$-equivariant weight-graded and topological Euler characteristics of the universal Picard stack $\mathrm{Pic}_{g, n}^d \to \mathcal{M}_{g, n}$ of degree-$d$ line bundles over $\mathcal{M}_{g, n}$. We prove that in the weight-zero and topological cases, the generating function for Euler characteristics of $\mathrm{Pic}_{g, n}^d$ is obtained from the corresponding one for $\mathcal{M}_{g, n}$ by an extremely simple combinatorial transformation. This lets us deduce closed formulas for the two generating functions, taking as input the Chan--Faber--Galatius--Payne formula in the weight-zero case and Gorsky's formula in the topological case. As an immediate corollary, we obtain closed formulas for the weight-zero and topological Euler characteristics of $\mathrm{Pic}^d_g$. Our weight-zero calculations follow from a general result passing from the weight-graded Euler characteristics of $\mathcal{M}_{g, n}$ to those of $\mathrm{Pic}_{g,n}^d$.

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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 $g = 2$.

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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.

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arXiv2025

Representation stability for moduli spaces of admissible covers

We prove a representation stability result for the sequence of spaces $\overline M_{g, n}^A$ of pointed admissible $A$-covers of stable $n$-pointed genus-$g$ curves, for an abelian group $A$. For fixed genus $g$ and homology degree $i$, we give the sequence of rational homology groups $H_i(\overline{M}_{g, n}^A;\mathbb Q)$ the structure of a module over a combinatorial category, a la Sam--Snowden, and prove that this module is generated in degree at most $g + 5 i$. This implies that the generating function for the ranks of the homology groups is rational, with poles in the set $\left\{-1, -\frac{1}{2}, \ldots, -\frac{1}{|A|^2\cdot(g + 5i)}\right\}$. In the case where $A$ is the trivial group, our work significantly improves on previous representation stability results on the Deligne--Mumford compactification $\overline M_{g, n}$.

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arXiv2025

The $\mathbb{S}_n$-equivariant Chow polynomial of the braid matroid

We determine the generating function for the $\mathbb{S}_n$-equivariant Chow polynomials of the braid matroid $B_n$. The Chow polynomial of $B_n$ is the Poincar\'e polynomial of the wonderful compactification of the complement of the braid arrangement, with respect to the maximal building set. A key input to our result is the identification of this wonderful compactification with a moduli space of multiscale differentials, recently established by Devkota, Robotis, and Zahariuc. In this way, our work also contributes to the literature on topology of moduli spaces of multiscale differentials. To prove our main formula, we study the classes of these moduli spaces in the Grothendieck ring of varieties via the formalism of $\mathbb{S}_n$-spaces developed by Getzler and Pandharipande. We also give a new interpretation of the numerical Chow polynomial of $B_n$ as the Poincar\'e polynomial of a moduli space of genus-zero relative stable maps to $\mathbb{P}^1$.

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arXiv2024

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

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arXiv2024

On the weight zero compactly supported cohomology of $\mathcal{H}_{g, n}$

For $g\ge 2$ and $n\ge 0$, let $\mathcal{H}_{g,n}\subset \mathcal{M}_{g,n}$ denote the complex moduli stack of $n$-marked smooth hyperelliptic curves of genus $g$. A normal crossings compactification of this space is provided by the theory of pointed admissible $\mathbb{Z}/2\mathbb{Z}$-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of $\mathcal{H}_{g, n}$. Using this graph complex, we give a sum-over-graphs formula for the $S_n$-equivariant weight zero compactly supported Euler characteristic of $\mathcal{H}_{g, n}$. This formula allows for the computer-aided calculation, for each $g\le 7$, of the generating function $\mathsf{h}_g$ for these equivariant Euler characteristics for all $n$. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissible $G$-covers of genus zero curves, when $G$ is abelian, as a symmetric $Δ$-complex. We use these complexes to generalize our formula for $\mathsf{h}_g$ to moduli spaces of $n$-pointed smooth abelian covers of genus zero curves.

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arXiv2024

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.

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arXiv2024

Equivariant Hodge polynomials of heavy/light moduli spaces

Let $\bar{\mathcal{M}}_{g, m|n}$ denote Hassett's moduli space of weighted pointed stable curves of genus $g$ for the heavy/light weight data $\left(1^{(m)}, 1/n^{(n)}\right)$, and let $\mathcal{M}_{g, m|n} \subset \bar{\mathcal{M}}_{g, m|n}$ be the locus parameterizing smooth, not necessarily distinctly marked curves. We give a change-of-variables formula which computes the generating function for $(S_m\times S_n)$-equivariant Hodge-Deligne polynomials of these spaces in terms of the generating functions for $S_{n}$-equivariant Hodge-Deligne polynomials of $\bar{\mathcal{M}}_{g,n}$ and $\mathcal{M}_{g,n}$.

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arXiv2022

Topology of tropical moduli spaces of weighted stable curves in higher genus

Given integers $g \geq 0$, $n \geq 1$, and a vector $w \in (\mathbb{Q} \cap (0, 1])^n$ such that ${2g - 2 + \sum w_i > 0}$, we study the topology of the moduli space $Δ_{g, w}$ of $w$-stable tropical curves of genus $g$ with volume 1. The space $Δ_{g, w}$ is the dual complex of the divisor of singular curves in Hassett's moduli space of $w$-stable genus $g$ curves $\overline{\mathcal{M}}_{g, w}$. When $g \geq 1$, we show that $Δ_{g, w}$ is simply connected for all values of $w$. We also give a formula for the Euler characteristic of $Δ_{g, w}$ in terms of the combinatorics of $w$.

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arXiv2022

Moduli of relative stable maps to $\mathbb{P}^1$: cut-and-paste invariants

We study constructible invariants of the moduli space $\overline{\mathcal{M}}(\boldsymbol{x})$ of stable maps from genus zero curves to $\mathbb{P}^1$, relative to $0$ and $\infty$, with ramification profiles specified by ${\boldsymbol{x}\in \mathbb{Z}^n}$. These spaces are central to the enumerative geometry of $\mathbb{P}^1$, and provide a large family of birational models of the Deligne--Mumford--Knudsen moduli space $\overline{\mathcal{M}}_{0,n}$. For the sequence of vectors $\boldsymbol{x}$ corresponding to maps which are maximally ramified over $0$ and unramified over $\infty$, we prove that a generating function for the topological Euler characteristics of these spaces satisfies a differential equation which allows for its recursive calculation. We also show that the class of the moduli space in the Grothendieck ring of varieties is constant as $\boldsymbol{x}$ varies within a fixed chamber in the resonance decomposition of $\mathbb{Z}^n$. We conclude by suggesting several further directions in the study of these spaces, giving conjectures on (1) the asymptotic behavior of the Euler characteristic and (2) a potential chamber structure for the Chern numbers.

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arXiv2021

Symmetries of tropical moduli spaces of curves

We compute the automorphism group $\mathrm{Aut}(Δ_{g, n})$ for all $g, n \geq 0$ such that $3g - 3 + n > 0$, where $Δ_{g, n} \subset M_{g, n}^\mathrm{trop}$ is the moduli space of stable $n$-marked tropical curves of genus $g$ and volume one. In particular, we show that $\mathrm{Aut}(Δ_{g})$ is trivial for $g \geq 2$, while $\mathrm{Aut}(Δ_{g, n}) \cong S_n$ when $n \geq 1$ and $(g, n) \neq (0, 4), (1, 2)$. The space $Δ_{g, n}$ is a symmetric $Δ$-complex in the sense of Chan, Galatius, and Payne, and is identified with the dual intersection complex of the boundary divisor in the Deligne-Mumford-Knudsen moduli space $\overline{\mathcal{M}}_{g, n}$ of stable curves. After the work of Massarenti, who has shown that $\mathrm{Aut}(\overline{\mathcal{M}}_g)$ is trivial for $g \geq 2$ while $\mathrm{Aut}(\overline{\mathcal{M}}_{g, n}) \cong S_n$ when $n \geq 1$ and $2g - 2 + n \geq 3$, our result implies that the tropical moduli space $Δ_{g, n}$ faithfully reflects the symmetries of the algebraic moduli space for general $g$ and $n$.

math.CO↗

arXiv2021

Automorphisms of tropical Hassett spaces

Given an integer $g \geq 0$ and a weight vector $w \in \mathbb{Q}^n \cap (0, 1]^n$ satisfying $2g - 2 + \sum w_i > 0$, let $Δ_{g, w}$ denote the moduli space of $n$-marked, $w$-stable tropical curves of genus $g$ and volume one. We calculate the automorphism group $\mathrm{Aut}(Δ_{g, w})$ for $g \geq 1$ and arbitrary $w$, and we calculate the group $\mathrm{Aut}(Δ_{0, w})$ when $w$ is heavy/light. In both of these cases, we show that $\mathrm{Aut}(Δ_{g, w}) \cong \mathrm{Aut}(K_w)$, where $K_w$ is the abstract simplicial complex on $\{1, \ldots, n\}$ whose faces are subsets with $w$-weight at most $1$. We show that these groups are precisely the finite direct products of symmetric groups. The space $Δ_{g, w}$ may also be identified with the dual complex of the divisor of singular curves in the algebraic Hassett space $\overline{\mathcal{M}}_{g, w}$. Following the work of Massarenti and Mella on the biregular automorphism group $\mathrm{Aut}(\overline{\mathcal{M}}_{g, w})$, we show that $\mathrm{Aut}(Δ_{g, w})$ is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves.

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arXiv2020

Chow Rings of Heavy/Light Hassett Spaces via Tropical Geometry

We compute the Chow ring of an arbitrary heavy/light Hassett space $\bar{M}_{0, w}$. These spaces are moduli spaces of weighted pointed stable rational curves, where the associated weight vector $w$ consists of only heavy and light weights. Work of Cavalieri et al. exhibits these spaces as tropical compactifications of hyperplane arrangement complements. The computation of the Chow ring then reduces to intersection theory on the toric variety of the Bergman fan of a graphic matroid. Keel has calculated the Chow ring $A^*(\bar{M}_{0, n})$ of the moduli space $\bar{M}_{0, n}$ of stable nodal $n$-marked rational curves; his presentation is in terms of divisor classes of stable trees of $\mathbb{P}^1$'s having one nodal singularity. Our presentation of the ideal of relations for the Chow ring $A^*(\bar{M}_{0, w})$ is analogous. We show that pulling back under Hassett's birational reduction morphism $ρ_w: \bar{M}_{0, n} \to \bar{M}_{0, w}$ identifies the Chow ring $A^*(\bar{M}_{0, w})$ with the subring of $A^*(\bar{M}_{0, n})$ generated by divisors of $w$-stable trees, which are those trees which remain stable in $\bar{M}_{0, w}$.

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