Subsets of a projective variety X⊂PnX⊂Pn spanning a given P∈PnP∈Pn

Fix an integral variety X⊂PnX⊂Pn, P∈PnP∈Pn, and an integer k > 0. Let S(X,P,k) be the set of all subsets S ⊂ X such that ♯(S) = k, P ∈ 〈S〉 and P ∉ 〈S′〉 for any S′ ⫋ S  . Here we study S(X,P,k) (non-emptiness and dimension) in the extremal case k = n − dim(X) + 1.