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

Let X r be an n-dimensional nondegenerate irreducible projective variety of degree d and codimension e. For 1≤β≤e and a β-dimensional linear subspace r satisfying dim(XL)=0, β(X) is defined as the possibly maximal length of the scheme theoretic intersection XL. 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 S5 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)d2-1-2, for all β {2,3}.