Tensor ranks on tangent developable of Segre varieties

Abstract

We describe the stratification by tensor rank of the points belonging to the tangent developable of any Segre variety. We give algorithms to compute the rank and a decomposition of a tensor belonging to the secant variety of lines of any Segre variety. We prove Comon's conjecture on the rank of symmetric tensors for those tensors belonging to tangential varieties to Veronese varieties.

Keywords:

• secant varieties
• tensor rank
• tangent developable
• Segre varieties
• Comon's conjecture

AMS Subject Classifications::

• 14N05
• 14Q05

Acknowledgements

The first author was partially supported by MIUR and GNSAGA of INdAM (Italy). The second author was partially supported by CIRM-FBK (TN-Italy), Marie-Curie FP7-PEOPLE-2009-IEF, INRIA Sophia Antipolis Mediterranée Project Galaad (France) and Mittag-Leffler Institut (Sweden).

We would like to thank B. Sturmfels for asking one of us this question at the Mittag-Leffler Institut during the Spring semester 2011 ‘Algebraic Geometry with a view towards applications’. We also thank the Mittag-Leffler Institut (Djursholm, Stokholm, Sweden), for its hospitality and opportunities.