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

Abstract

Let [Formula: see text] be an [Formula: see text]-dimensional nondegenerate irreducible projective variety of degree [Formula: see text] and codimension [Formula: see text]. For [Formula: see text] and a [Formula: see text]-dimensional linear subspace [Formula: see text] satisfying dim([Formula: see text], [Formula: see text] is defined as the possibly maximal length of the scheme theoretic intersection [Formula: see text]. Then it is well known that [Formula: see text] if [Formula: see text] 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 [Formula: see text] when [Formula: see text] is locally Cohen–Macaulary. On the other hand, the possible values of [Formula: see text] are unknown if [Formula: see text] is not locally Cohen–Macaulay. In this paper, we construct surfaces [Formula: see text] of maximal sectional regularity (which are not locally Cohen–Macaulay) and of degree [Formula: see text] for every [Formula: see text] such that [Formula: see text] for all [Formula: see text].