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Let X ⊂ P r be an integral and non-degenerate variety and V ⊂ P r a k-dimensional linear subspace. For any P ∈ P r the X-rank of P is the minimal cardinality of a subset of X whose linear span contains P. Here we study several integers related to the X-rank of V , e.g. the minimal X-rank of all elements of a basis of V or the minimal sum of the X-ranks of a basis of V.

RANKS OF LINEAR SUBSPACES WITH RESPECT TO AN EMBEDDED VARIETY