A nonspecial divisor in the moduli space of cubic fourfolds via 10-nodal plane sextics

In the moduli space $\mathcal{C}$ of complex cubic hypersurfaces $X\subset {\mathbb{P}}^5$, we study the condition that $X$ admits a net of polar quadrics whose discriminant locus is a $10$-nodal irreducible plane sextic curve. Our main result is that such a condition defines an irreducible divisor in $\mathcal{C}$ which is not of Noether-Lefschetz type.