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

초록

Let X subset of P-r be an n-dimensional nondegenerate irreducible projective variety of degree d and codimension e. For 1 <= beta <= e and a beta-dimensional linear subspace L subset of P-r satisfying dim(X boolean AND L) = 0, l(beta)(X) is defined as the possibly maximal length of the scheme theoretic intersection X boolean AND L. Then it is well known that l(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 l(beta) (x) <= d - e + beta for all 1 <= beta <= e, when X is locally Cohen-Macaulary. On the other hand, the possible values of l(beta) (X) are unknown if X is not locally Cohen-Macaulay. In this paper, we construct surfaces S subset of P-5 of maximal sectional regularity (which are not locally Cohen-Macaulay) and of degree d for every d >= 7 such that l(beta) (S) >= d - 3 + beta + (beta - 1) (left perpendicular d/2 right perpendicular - 1) -2, for all beta is an element of {2, 3}.

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