Some Formulae Arising in Projective-Differential Geometry

We set up a framework in which some projective-differential intuitive concepts can be very easily formalized. Then we prove some general formulas. Among them, two "inversion formulas" about deformations of families of nonreduced subschemes are particularly remarkable. As applications, we give convenient definitions about some classical topics, such as higher order foci; finally we propose a new proof of the classic structure theorem about the first Gauss map.