Osculating Varieties of Veronese Varieties and Their Higher Secant Varieties

We consider the k-osculating varieties O(k,n,d) to the (Veronese) d-uple embeddings of P^n. We study the dimension of their higher secant varieties via inverse systems (apolarity). By associating certain 0-dimensional schemes of P-n to O^s(k,n,d) and by studying their Hilbert functions, we are able, in several cases, to determine whether those secant varieties are defective or not