Mathematics – Algebraic Geometry
Scientific paper
2010-12-16
Mathematics
Algebraic Geometry
28 pages, 10 figures; v2: improved presentation in accordance with referee's suggestions. The corrections were made while the
Scientific paper
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.
Buczynska Weronika
Buczyński Jarosław
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