The integral Chow ring of $\mathcal{R}_2$

TL;DR

This work determines the integral Chow ring of the moduli stack $\mathcal{R}_2$ of genus-2 Prym pairs by constructing an explicit quotient presentation $\mathcal{R}_2\cong [\mathrm{Sym}^4(V^\vee)\otimes\det(V)\otimes\Gamma\setminus\Delta]/G$ with $G=(\mathbb{G}_m\times\mathbb{G}_m)\rtimes \mathbb{Z}/2\mathbb{Z}$. Using equivariant intersection theory, the authors build a $G$-equivariant envelope for the delta locus and develop detailed pushforward formulas for multiplication maps to compute $\mathrm{CH}^*(\mathcal{R}_2)$, obtaining the torsion relations $(2\lambda_1,2\gamma,8\lambda_2,\gamma^2+\lambda_1\gamma,\lambda_1^2+\lambda_1\gamma)$. They identify the generators with geometric classes: $\lambda_i=(-1)^i\lambda_i$ are the Chern classes of the Hodge bundle, and $\gamma$ comes from the pushforward along the degree-2 Prym cover, giving a clear tautological interpretation. The final result is $\mathrm{CH}^*(\mathcal{R}_2)=\mathbb{Z}[\lambda_1,\lambda_2,\gamma]/(2\lambda_1,2\gamma,8\lambda_2,\gamma^2+\lambda_1\gamma,\lambda_1^2+\lambda_1\gamma)$ over any algebraically closed field with char not equal to $2$ or $3$, providing a foundational step for integral Chow theories of Prym moduli and paving the way for extensions to hyperelliptic Prym loci.

Abstract

In this paper we compute the integral Chow ring of the moduli stack $\mathcal{R}_2$ of Prym pairs of genus 2 with integral coefficients.

Theorems & Definitions (62)

• Definition 1
• Theorem 2
• Lemma 3
• Theorem 4
• Definition 5
• Definition 6
• Definition 7
• Remark 8
• Definition 9
• Remark 10