Abstract. Let C ⊂ P^r be an integral and non-degenerate variety. The border rank (resp. scheme rank) of P ∈ P^r with respect to C is the minimal integer t such that P is contained in the t-secant variety of C (resp. P ∈ < Z >, where Z ⊂ X is a degree t subscheme and < > denote the linear span). In this note we study the behavior of the border rank and the scheme-rank under linear projections, when C is a linearly normal smooth curve.

Received: February 17, 2013

AMS Subject Classification: 14N05, 14H50

Key Words and Phrases: symmetric tensor rank, X-rank, cactus rank, secant variety, linearly normal curve

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