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

We study the secant varieties of the Veronese varieties and of Veronese reembeddings of a smooth projective variety. We give some conditions, under which these secant varieties are set-theoretically cut out by determinantal equations. More precisely, they are given by minors of a catalecticant matrix. These conditions include the case when the dimension of the projective variety is at most 3 and the degree of reembedding is sufficiently high. This gives a positive answer to a set-theoretic version of a question of Eisenbud in dimension at most 3. For dimension four and higher we produce plenty of examples when the catalecticant minors are not enough to set-theoretically define the secant varieties to high degree Veronese varieties. This is done by relating the problem to smoothability of certain zero-dimensional Gorenstein schemes. e-mail addresses: wkrych@mimuw.edu.pl and jabu@mimuw.edu.pl address: Institut Fourier, 100 rue des Maths, BP 74, 38402 St Martin d’Hères Cedex, France keywords: secant variety, catalecticant minors, Veronese variety, Veronese reembeddings, cactus variety, smoothable zero-dimensional schemes. AMS Mathematical Subject Classification 2010: Primary: 14M12; Secondary: 13H10, 14M17.

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