Chow and cohomology rings of moduli stacks of plane quartics

TL;DR

This paper computes the Chow and cohomology rings of the moduli stack of plane quartics, revealing its structure and relations through explicit calculations and stack-theoretic constructions.

Contribution

It constructs a smooth proper stack resolving wall crossings and determines the generators, relations, and Poincaré polynomial of its Chow ring.

Findings

Computed the Poincaré polynomial of the moduli stack.

Proved the cycle class map is an isomorphism with rational coefficients.

Determined generators and relations for the Chow ring.

Abstract

This paper studies the Chow and cohomology rings of the Hacking moduli stack $\mathcal{P}^{\mathrm{H}}$ of plane quartics. We construct a smooth proper Deligne--Mumford stack resolving the Calabi--Yau wall crossing between the KSBA and K-moduli compactifications for plane quartics via stack-theoretic weighted blowups. Its coarse moduli space is, up to normalization, the fiber product of the natural diagram relating the KSBA, K-moduli, and boundary polarized Calabi--Yau compactifications. From this, we compute the Poincar\'e polynomial of $\mathcal{P}^{\mathrm{H}}$, show that the cycle class map is an isomorphism with rational coefficients, and determine generators and relations for its Chow ring in terms of tautological classes. Analogous results are established for the GIT and K-moduli stacks.