Componentwise linearity of projective varieties with almost maximal degree

Abstract

The degree of a projective subscheme has an upper bound deg(X) <= ((e+r)(e)) in terms of the codimension eand the reduction number r. It was proved in [3] that deg(X) = ((e r)(e)) if and only if Xis arithmetically Cohen-Macaulay and has an (r+ 1)-linear resolution. Moreover, if the degree of a projective variety Xsatisfies deg(X) = ((e+r)(e)) - 1, then the Betti table is described with some constraints. In this paper, we build on this work to show that most of such varieties are componentwise linear and the componentwise linearity is particularly suitable for understanding their Betti tables. As an application, the graded Betti numbers of those varieties with componentwise linear resolutions are computed. (C) 2021 Elsevier B.V. All rights reserved.