The possible dimensions of the higher secant varieties

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Abstract
Let X ⊂ P be an irreducible closed subvariety of projective space. The kth higher secant variety of X, denoted Xk, is the closure of the union of all linear spaces spanned by k points of X. Let ak(X) = dim (Xk) -dim (Xk-1). It is known that the sequence ak(X) is nonincreasing, and that there may be at most one 1 in such a sequence. We prove that these two conditions are also sufficient conditions for an arbitrary sequence λk of nonnegative integers to be the ak(X)'s for an irreducible variety X. To this end we exhibit the ideals of the higher secant varities of rational normal scrolls as determinantal ideals. This allows us to compute their dimensions, showing that there is in fact always a rational normal scroll X such that ak(X) = λk for any sequence λk satisfying the two conditions above.

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