This paper is a continuation of the theme of equivariant enumerative geometry found in this paper and this paper. . This time, we compute the "fundamental class of a point in the moduli stack of quartic plane curves." This paper is actually an equivariant homage to the remarkable work of Aluffi and Faber on the geometry of GL(3)-orbit closures in the complete linear system of plane curves. To appreciate the power of one of our formulas, consider the puzzle: How many times does a general genus 3 curve arise as a 2-plane slice of a fixed, general quartic threefold? Plugging this problem into our general formula gives: 510720, obviously.

Our point of view in this paper is slightly different from our other work in this area: We want to know how equivariant classes change under popular specializations. The idea here being that such an investigation might then provide a network of relationships which, when back-traced, might lead to a computation of the equivariant fundamental class of an arbitrary point in the stack of smooth plane curves. The implementation of this strategy is the subject of ongoing work.