On the locus of genus $3$ curves that admit meromorphic differentials with a zero of order $6$ and a pole of order $2$

The main goal of this article is to compute the class of the divisor of ${\bar{\mathcal{M}}}_3$ obtained by taking the closure of the image of $\mathrm{\Omega }{\mathcal{M}}_3(6;-2)$ by the forgetful map. This is done using Porteous formula and the theory of test curves. For this purpose, we study the locus of meromorphic differentials of the second kind, computing the dimension of the map of these loci to ${\mathcal{M}}_g$ and solving some enumerative problems involving such differentials in low genus. A key tool of the proof is the compactification of strata recently introduced by Bainbridge-Chen-Gendron-Grushevsky-M"oller.