On the X-ranks associated to a linear subspace

Abstract

Let X ⊂ ℙ^N be an integral non-degenerate variety. For any q ∈ ℙ^N the X-rank r_X(q) of q is the minimal cardinality of a set S ⊂ X whose linear span contains q. For any r-dimensional linear subspace V ⊂ ℙ^N the X-rank r_X(V) of V is the minimal cardinality of a set S ⊂ X whose linear span contains V. We define several invariants related to the X-rank, e.g. the open rank of V, r+1 integers r_X (q_i), 0 ≤ i ≤ r, with r_X(q_i) ≤ r_X(q_j) for all i < j and q₀, ... , q_r a basis of V and minimal η_X(V) := r_X(q₀) + ... + r_X(q_r). We always have η_X(V) ≥ r_X(V) and we give an upper bound for η_X(V), which ensures that we may compute r_X(V) with knowledge of q₀, ... , q_r and in particular check if η_X(V) = r_X(V) (the general case, but not always). The integer r_X(q_r) is the minimal integer t such that V is spanned by points of X-rank ≤ t. Many of these invariants of V give lower bounds for the invariants of any linear subspace of V.