Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes
By Weronika Buczy'nska, Jaroslaw Buczy'nski. 2010
136 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Computational Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/739999451df5b54f9ca94bdff7bdadbd9232c656
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In Algebra and Number Theory
- M. Catalisano
- Roberto Notari
- Alessandro Gimigliano
- Alessandra Bernardi
- Enrico Carlini
- G. Casnati
- Joachim Jelisiejew
In Computational Mathematics
- Alessandra Bernardi
- A. Oneto
- Alessandro Gimigliano
- Enrico Carlini
- M. Catalisano
- Roberto Notari
- G. Casnati
- Joachim Jelisiejew
In Computational Theory and Mathematics
- Alessandra Bernardi
- M. Catalisano
- A. Oneto
- Alessandro Gimigliano
- Roberto Notari
- Enrico Carlini
- G. Casnati
- Joachim Jelisiejew
Citations in the corpus
- Irreducibility of the Gorenstein loci of Hilbert schemes via ray families (2014, methodology): “it is open, but in general not dense, inside HilbdPN. Recently, interesting interactions between HilbG d P N and the geometry of secant varieties and general topology have been found (see for example [BB14], [BJJM]). Some results about HilbG d P N are known. The irreducibility and smoothness of Hilb”
- The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, background): “Another very related notion of rank is the following; see [169, 82].”