Stratification of the fourth secant variety of Veronese variety via the symmetric rank

Details Stratification of the fourth secant variety of Veronese variety via the symmetric rank Stratification of the fourth secant variety of Veronese variety via the symmetric rank

2010-05-19

arxiv.org/abs/1005.3465v3

Mathematics

Algebraic Geometry

Scientific paper

If $X\subset \mathbb{P}^n$ is a projective non degenerate variety, the $X$-rank of a point $P\in \mathbb{P}^n$ is defined to be the minimum integer $r$ such that $P$ belongs to the span of $r$ points of $X$. We describe the complete stratification of the fourth secant variety of any Veronese variety $X$ via the $X$-rank. This result has an equivalent translation in terms both of symmetric tensors and homogeneous polynomials. It allows to classify all the possible integers $r$ that can occur in the minimal decomposition of either a symmetric tensor or a homogeneous polynomial of $X$-border rank 4 (i.e. contained in the fourth secant variety) as a linear combination of either completely decomposable tensors or powers of linear forms respectively.

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