Projective-differential properties of point correspondences between three hypersurfaces

We study the point correspondences between three hypersurfaces in projective spaces with the help of G. F. Laptev invariant methods. We establish main equations and geometrical objects of correspondences. We construct invariant normalizations of hypersurfaces and main tensors of correspondences, and describe a connection of the studied correspondences with multidimensional $3$-webs.