On multisecant planes of locally non-Cohen-Macaulay surfaces

For a nondegenerate projective irreducible variety X ⊂ P^r, it is a natural problem to find an upper bound for the value of ℓ_β(X) = max{length(X ∩ L) | L = P^β ⊂ P^r, dim (X ∩ L) = 0} for each 1 ≤ β ≤ e. When X is locally Cohen-Macaulay, A. Noma in [10] proves that ℓ_β (X) is at most d − e + β where d and e are respectively the degree and the codimension of X. In this paper, we construct some surfaces S ⊂ P⁵ of degree d ∈ {7,..., 12} which satisfies the inequality (Formula Presented) This shows that Noma’s bound is no more valid for locally non-Cohen-Macaulay varieties.