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We study the secant varieties of the Veronese varieties and of Veronesereembeddings of a smooth projective variety. We give some conditions, underwhich these secant varieties are set-theoretically cut out by determinantalequations. More precisely, they are given by minors of a catalecticant matrix.These conditions include the case when the dimension of the projective varietyis at most 3 and the degree of reembedding is sufficiently high. This gives apositive answer to a set-theoretic version of a question of Eisenbud indimension at most 3. For dimension four and higher we produce plenty ofexamples when the catalecticant minors are not enough to set-theoreticallydefine the secant varieties to high degree Veronese varieties. This is done byrelating the problem to smoothability of certain zero-dimensional Gorensteinschemes.

Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes