Chow ring of the stack of plane nodal curves

TL;DR

The authors determine the rational Chow rings of the moduli stacks $\mathsf{PNC}_d$ (plane nodal curves) and $\mathsf{PSC}_d$ (plane smooth curves) of fixed degree, expressing them in terms of tautological generators tied to Chern classes $c_i$ of the universal bundle. They realize these stacks as quotient stacks by $\mathrm{PGL}_3$ (lift to $GL_3$), use an excision framework, and compute equivariant Chow groups via a Brauer–Severi–type analysis that extends Vial’s results to $G$-equivariant settings. A central technical achievement is the key diagram relating incidence varieties and Hessian degeneracies, which yields explicit relations and the final rings: $CH^*(\mathsf{PNC}_d)=\mathbb{Q}[c_1]/(c_1^4)$ for $d\ge4$ (with special cases for small $d$) and $CH^*(\mathsf{PSC}_d)$ collapsing to specific low-rop models ($d=1$ and $d=2$ cases) or to $\mathbb{Q}$ for $d\ge3$. The paper also connects these Chow generators to tautological classes on the moduli of curves via the map $\epsilon_d: \mathsf{PNC}_d \to \mathfrak{M}_g$, providing explicit pullback formulas for the boundary $\delta$ and Hodge $\lambda_i$ classes, and develops a broad equivariant framework for Brauer–Severi varieties that may interest future intersection-theory computations.

Abstract

We compute the rational Chow ring of the moduli stack of planar nodal curves of fixed degree and express it in terms of tautological classes. Along the way, we extend Vial's results on Chow groups of Brauer-Severi varieties to $G$-equivariant settings.

Theorems & Definitions (41)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• proof : Proof of Theorem \ref{['thm2']}