dorsal/arxiv
View SchemaVirtual Hodge numbers of $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$: stability and calculations
| Authors | Siddarth Kannan, Terry Dekun Song |
|---|---|
| Categories | |
| ArXiv ID | 2601.07981vv1 |
| URL | https://arxiv.org/abs/2601.07981 |
| License | http://creativecommons.org/licenses/by-nc-sa/4.0/ |
Abstract
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. Since Getzler has calculated the Serre characteristic of $\mathcal{M}_{1, n}$, our formula in particular determines the Serre characteristic of $\mathcal{M}_{1, n}(\mathbb{P}^r, d)$ for arbitrary $n$, $r$, and $d$. Our formula also calculates the Serre characteristic of $\mathcal{M}_{2, n}(\mathbb{P}^r, d)$ whenever those of $\mathcal{M}_{2, n +k}$ are known for $k \leq d$.
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"abstract": "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. Since Getzler has calculated the Serre characteristic of $\\mathcal{M}_{1, n}$, our formula in particular determines the Serre characteristic of $\\mathcal{M}_{1, n}(\\mathbb{P}^r, d)$ for arbitrary $n$, $r$, and $d$. Our formula also calculates the Serre characteristic of $\\mathcal{M}_{2, n}(\\mathbb{P}^r, d)$ whenever those of $\\mathcal{M}_{2, n +k}$ are known for $k \\leq d$.",
"arxiv_id": "2601.07981",
"authors": [
"Siddarth Kannan",
"Terry Dekun Song"
],
"categories": [
"math.AG",
"math.CO"
],
"license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
"title": "Virtual Hodge numbers of $\\mathcal{M}_{g, n}(\\mathbb{P}^r, d)$: stability and calculations",
"url": "https://arxiv.org/abs/2601.07981",
"version": "v1"
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