Is there any literature dealing with cubic hypersurfaces in full generality (over $\mathbb{C}$)? Couldn't find any.
We know everything about hyperplanes. We also know a lot of things about quadric hypersurfaces. The main tool is the relation with the associated bilinear form: this allows to describe a lot of interesting properties of the quadric, such as
- its singular locus, which is the projectivization of the kernel of the bilinear form;
- its linear subspaces, namely the linear subspaces of the projective space on which the bilinear form is identically zero;
- its structure as a cone through a plane over a smooth quadric in smaller dimension.
All of this is very standard and can be found f.e. in the preliminaries of the sixth section of Griffiths and Harris' "Principles of algebraic geometry".
Do you know something along this line about cubic hypersurfaces? I feel like we should rely on a generalization of the bilinear form, but I'm not able to find any literature about it. Is it a much more difficult problem?
P.S.: I don't know if "cubic equations" is an appropriate tag for this...