Plane sextics and a nonspecial divisor in the moduli space of cubic fourfolds

We show that the property of admitting a net of polar quadrics with discriminant locus a plane sextic curve with at least 10 nodes is nongeneral for a cubic fourfold. In particular, we study cubic fourfolds having a polar net with an irreducible 10-nodal discriminant curve. They form a divisor in the moduli space of cubic fourfolds which we prove is not a Noether-Lefschetz divisor.