## Induction for secant varieties of Segre varieties

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- by Hirotachi Abo, Giorgio Ottaviani and Chris Peterson PDF
- Trans. Amer. Math. Soc.
**361**(2009), 767-792 Request permission

## 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.## References

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## Additional Information

**Hirotachi Abo**- Affiliation: Department of Mathematics, University of Idaho, Moscow, Idaho 83844
- MR Author ID: 614361
- Email: abo@uidaho.edu
**Giorgio Ottaviani**- Affiliation: Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy
- MR Author ID: 134700
- Email: ottavian@math.unifi.it
**Chris Peterson**- Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80525
- MR Author ID: 359254
- Email: peterson@math.colostate.edu
- 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.
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**361**(2009), 767-792 - MSC (2000): Primary 15A69, 15A72, 14Q99, 14M12, 14M99
- DOI: https://doi.org/10.1090/S0002-9947-08-04725-9
- MathSciNet review: 2452824