2010/12/16 by Weronika Buczyńska, Buczyńska, Weronika, Jarosław Buczyński +1 · 2 citations
Mathematics · #14M12 (Primary) 13H10 #14M17 (Secondary) #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Mathematics #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1012.3563
openalex publication_date 2010/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the secant varieties of the Veronese varieties and of Veronese\nreembeddings of a smooth projective variety. We give some conditions, under\nwhich these secant varieties are set-theoretically cut out by determinantal\nequations. More precisely, they are given by minors of a catalecticant matrix.\nThese conditions include the case when the dimension of the projective variety\nis at most 3 and the degree of reembedding is sufficiently high. This gives a\npositive answer to a set-theoretic version of a question of Eisenbud in\ndimension at most 3. For dimension four and higher we produce plenty of\nexamples when the catalecticant minors are not enough to set-theoretically\ndefine the secant varieties to high degree Veronese varieties. This is done by\nrelating the problem to smoothability of certain zero-dimensional Gorenstein\nschemes.\n