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

Buczyńska, Weronika; Buczyński, Jarosław

doi:10.1090/S1056-3911-2013-00595-0

Authors: Weronika Buczyńska and Jarosław Buczyński
Journal: J. Algebraic Geom. 23 (2014), 63-90
DOI: https://doi.org/10.1090/S1056-3911-2013-00595-0
Published electronically: September 19, 2013
MathSciNet review: 3121848
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Abstract | References | Additional Information

Abstract: 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.

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Additional Information

Weronika Buczyńska
Affiliation: Institut Mittag-Leffler, Auravägen 17, SE-182 60 Djursholm, Sweden
Address at time of publication: Institut of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland
Email: wkrych@mimuw.edu.pl

Jarosław Buczyński
Affiliation: Institut Fourier, 100 rue des Maths, BP 74, 38402 St Martin d’Hères Cedex, France
Address at time of publication: Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland
Email: jabu@mimuw.edu.pl

Received by editor(s): December 16, 2010
Received by editor(s) in revised form: March 3, 2011
Published electronically: September 19, 2013
Additional Notes: The first author gratefully acknowledges the support from the AXA Mittag-Leffler Fellowship Project, sponsored by the AXA Research Fund. The second author was supported by Marie Curie International Outgoing Fellowship “Contact Manifolds”.
Article copyright: © Copyright 2013 University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.