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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Induction for secant varieties of Segre varieties


Authors: Hirotachi Abo, Giorgio Ottaviani and Chris Peterson
Journal: Trans. Amer. Math. Soc. 361 (2009), 767-792
MSC (2000): Primary 15A69, 15A72, 14Q99, 14M12, 14M99
Published electronically: September 29, 2008
MathSciNet review: 2452824
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Abstract: This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective $ p$-secant varieties to Segre varieties for $ p\leq 6$. We generalize a theorem of Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers of $ \mathbb{P}{n}$. We determine the set of $ p$ for which unbalanced Segre varieties have defective $ p$-secant varieties. In addition, we completely describe the dimensions of the secant varieties to the deficient Segre varieties $ \mathbb{P}{1}\times\mathbb{P}{1} \times\mathbb{P}{n} \times \mathbb{P}{n}$ and $ \mathbb{P}{2}\times\mathbb{P}{3} \times \mathbb{P}{3}$. In the final section we propose a series of conjectures about defective Segre varieties.


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Additional Information

Hirotachi Abo
Affiliation: Department of Mathematics, University of Idaho, Moscow, Idaho 83844
Email: abo@uidaho.edu

Giorgio Ottaviani
Affiliation: Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy
Email: ottavian@math.unifi.it

Chris Peterson
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80525
Email: peterson@math.colostate.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04725-9
PII: S 0002-9947(08)04725-9
Keywords: Secant varieties, partial secant variety, joins, Segre varieties, tensor algebra, hypermatrices, tensor rank, border rank, computational algebraic geometry
Received by editor(s): February 23, 2007
Published electronically: September 29, 2008
Additional Notes: The authors would like to thank the Department of Mathematics of the Università di Firenze, the Department of Mathematics of Colorado State University, the National Science Foundation and GNSAGA of Italian INDAM and the FY 2008 Seed Grant Program of the University of Idaho Research Office.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.