On zero-dimensional linear sections of surfaces of maximal sectional regularity

Let X ℙr be an n-dimensional nondegenerate irreducible projective variety of degree d and codimension e. For 1 ≤ β ≤ e and a β-dimensional linear subspace L r satisfying dim(X L) = 0, β(X) is defined as the possibly maximal length of the scheme theoretic intersection X L. Then it is well known that 1(X) ≤ d-e + 1 if X is a curve. Also it was generalized by Noma [Multisecant lines to projective varieties, Projective Varieties with Unexpected Properties (Walter de Gruyter, GmbH and KG, Berlin, 2005), pp. 349-359] that β(X) ≤ d-e + βfor all 1 ≤ β ≤ e, when X is locally Cohen-Macaulary. On the other hand, the possible values of β(X) are unknown if X is not locally Cohen-Macaulay. In this paper, we construct surfaces S 5 of maximal sectional regularity (which are not locally Cohen-Macaulay) and of degree d for every d ≥ 7 such that β(S) ≥ d-3 + β + (β-1) d 2-1-2, for all β {2, 3}.