On a Lemma of Bompiani

We reprove in modern terms and extend to arbitrary dimension a classical result of Enrico Bompiani on algebraic surfaces X ⊂ P having very degenerate osculating spaces. Let X ⊂ P be an integral nondegenerate projective variety of dimension n defined over the field C. In order to introduce the osculating space of order m to a smooth point p ∈ X, fix a lifting U ⊆ C −→ C \ {0} t 7−→ p(t) of a local parametrization of X centered in p and define T (m, p,X) to be the linear span of the points [pI(0)] ∈ P , where I is a multi-index such that |I| ≤ m. The starting point of our research was the following result, which we read in a recent paper by Luca Chiantini and Ciro Ciliberto (see [3], Proposition 2.3): Proposition 1. Let X ⊂ P be a smooth variety and let p ∈ X be a general point. Assume that dimT (m, p,X) = dimT (m+1, p,X) = h. Then X ⊆ P. Our natural question was: can one go a little bit further? Namely, if the dimension of the osculating space at a general point does not increase too much while passing from order m to order m + 1, what can one say about the projective geometry of X? Following a suggestion of Ciro Ciliberto, to whom we are grateful, we looked for an answer among the works of Bompiani (see [2]). Enrico Bompiani (1889–1975) was a student of Guido Castelnuovo and in more than three hundreds papers intensively studied the differential geometry of projective varieties; in particular, he deeply investigated the ∗This research is part of the T.A.S.C.A. project of I.N.d.A.M., supported by P.A.T. (Trento) and M.I.U.R. (Italy).