R. Hartshorne and A. Hirschowitz proved that a generic collection of lines on ℙn, n≥3, has bipolynomial Hilbert function. We extend this result to a specialization of the collection of generic lines, by considering a union of lines and 3-dimensional sundials (i.e., a union of schemes obtained by degenerating pairs of skew lines).
We consider the k-osculating varietiesO to the Veronese d?uple embeddings of P. By studying the Hilbert function of certain zero-dimensional schemes Y ⊂ P, we find the dimension of O , the (s?1) secant varieties of O, for 3 ≤ s ≤ 6 and s = 9, and we determine whether those secant varieties are defective or not.
In this paper we compute the dimension of all the s higher secant varieties of the Segre-Veronese embeddings Y of the product P × P × P in the projective space P via divisors of multidegree d = (a,b,c) (N = (a+1)(b+1)(c+1) - 1). We find that Y has no deficient higher secant varieties, unless d = (2,2,2) and s = 7, or d = (2h,1,1) and s = 2h + 1, with defect 1 in both cases.
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