Polar geometry of cubic hypersurfaces and a nonspecial divisor in the moduli space of cubic fourfolds

The study of cubic hypersurfaces in $\mathbb{P}^{n+1}$ is still one of the focal points of Algebraic Geometry. In fact, classically, these have represented the first obstacle in the study of rationality. In particular, the first cubic about which we still cannot say anything in general about the problem of rationality is the cubic fourfold and this is the original reason for choosing the topic of this thesis. Specifically, on the rationality of cubic fourfolds, there is a conjecture due to Kuznetsov locating the rational ones in the union of special divisors of the moduli space of such varieties. It is therefore interesting to study the locus of cubic fourfolds outside of special divisors: in this perspective, the construction and the description of nonspecial divisors with special geometric properties is a quite natural issue. Also, only three nonspecial divisors are known and described in the literature. In this work we construct a divisor in the moduli space of cubic fourfolds, that we call Severi divisor, which we prove to be nonspecial: to do so, we use the classical theory of polar hypersurfaces and the ancient result known as Dixon's lemma, now generalized to stable curves. We then generalize this construction to the moduli space of cubics of any dimension $n$. The powerfulness of these divisors is still unexplored with respect to the study of cubic hypersurfaces: in general and with respect to the rationality problem. It would be interesting to study the intersection locus of the Severi divisor of cubic fourfolds with the special divisors or even repeat a similar construction to obtain other nonspecial divisors. It would be even more interesting to be able to use the very explicit properties of the cubic fourfolds parametrized by the points of this divisor to say something about rationality.