This paper continues the theme of equivariant enumerative geometry. This time, we compute what can be thought of as the "fundamental class of a general point in the stack of cubic surfaces." More descriptively, we establish a universal expression in characteristic classes which counts the number of times a given, general, cubic surface arises in any 4-dimensional family of cubic surfaces. For instance, Ranestad and Sturmfels, in their "Twenty Seven problems on Cubic surfaces" ask how many times a general isomorphism class of cubic surface arises in a general 4-dimensional linear system of surfaces. When we plug this problem into our formula, it outputs 96120. This number was previously discovered (though not proven) by Brustenga i Moncusi, Timme, and Weinstein using numerical algebraic geometry software.

The paper can be (somewhat unfairly) summarized as an exercise in the technique of undetermined coefficients. But this overall technique might offer an avenue of attack for analogous enumerative questions for other hypersurfaces. This is very much an ongoing area of investigation. Furthermore, by refining the overall technique it might just barely be possible to establish the analogous universal expression for every smooth cubic surface. Again, these further explorations are ongoing.