Integral Chow rings of modular compactifications of $\mathcal{M}_{1,n\leq 6}$

Abstract

For $n\leq 6$, we compute the integral Chow ring of every modular compactification of $\mathcal{M}_{1,n}$ parametrising only Gorenstein curves with smooth, distinct markings. These include the Deligne--Mumford, Schubert, and Smyth compactifications, and many more. They can all be excised from the stack of log-canonically polarised Gorenstein curves. The Chow ring of the latter admits a simple, combinatorial description, which we compute by patching along a natural stratification by core level. We further deduce that all these modular compactifications satisfy the Chow-K\"{u}nneth generation property, that the cycle class map is an isomorphism, and for $n=4$ we study whether the Getzler's relation hold integrally.