Remarks on osculating linear spaces to projective varieties

Let X ⊂ PN be an integral n-dimensional variety and m(X, P, i) (resp. m(X, i)), 1 ≤ i ≤ N - n + 1, the Hermite invariants of X measuring the osculating behaviour of X at P (resp. at its general point). Here we prove m(X, x) + m(X, y) ≤ m(X, x + y) and m(X, P, x) + m(X, y) ≤ m(X, P, x + y) for all integers x, y such that x + y ≤ N - n + 1, the case n = 1 being known (M. Homma, A. Garcia and E. Esteves).