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Fix an integral projective variety, P ∈ Xreg, m > 0, L ∈ Pic(X). Let W ⊆ H 0 (X,L) be a linear subspace. Set b := dim(W(−mP)) and a := dim(W(−(m + 1)P)). Fix an integer z such that 0 ≤ z ≤ b − a. We prove the existence of a zero-dimensional scheme Z ⊂ X such that mP ⊆ Z ⊆ (m+1)P, deg(Z) = deg(mP) + z and dim(W(−Z)) = b − z.

ZERO-DIMENSIONAL SCHEMES CONTAINED BETWEEN TWO CONSECUTIVE MULTIPLE POINTS