In this paper, we give an explicit presentation of the integral Chow ring of a stack of smooth plane cubics. We also determine some relations in the general case of hypersurfaces of any dimension and degree.
"The Chow Ring of the Stack of Smooth Plane Cubics." Michigan Math. J. 67 (1) 3 - 29, March 2018. https://doi.org/10.1307/mmj/1516330968