Hironaka, in his paper [H1] on desingularization of algebraic varieties over a field of characteristic 0, to deal with singular points develops the algebraic apparatus of the associated graded ring, introducing standard bases of ideals, numerical characters ν* and τ* etc. Such a point of view involves a deep investigation of the ideal b* generated by the initial forms of the elements of an ideal A of a local ring, with respect to a certain ideal a.

A graded K-algebra R is said to be Koszul if the minimal R-free graded resolution of K is linear. In this paper we study the Koszul property of the homogeneous coordinate ring R of a set of s points in the complex projective space P n . Kempf proved that R is Koszul if s ≤ 2n and the points are in general linear position. If the coordinates of the points are algebraically independent over Q, then we prove that R is Koszul if and only if s ≤ 1 + n + n 2 /4. If s ≤ 2n and the points are in linear general position, then we show that there exists a system of coordinates x 0 , . . . , x n of P n such that all the ideals (x 0 , x 1 , . . . , x i ) with 0 ≤ i ≤ n have a linear R-free resolution.

Let V be closed subscheme of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] defined by a homogeneous ideal I ⊆ A = K [ X 1 , . . . , X n ], and let X be the ( n - 1)-fold obtained by blowing-up [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] along V . If one embeds X in some projective space, one is led to consider the subalgebra K [( I e ) c ] of A for some positive integers c and e . The aim of this paper is to study ring-theoretic properties of K [( I e ) c ]; this is achieved by developing a theory which enables us to describe the local cohomology of certain modules over generalized Segre products of bigraded algebras. These results are applied to the study of the Cohen-Macaulay property of the homogeneous coordinate ring of the blow-up of the projective space along a complete intersection. We also study the Koszul property of diagonal subalgebras of bigraded standard k -algebras.

No abstract

An Artinian ideal I of k[x, y] has many Hilbert-Burch matrices. We show that there is a canonical choice. As an application, we determine the dimension of certain affine Gröbner cells and their Betti strata recovering results of Ellingsrud and Strømme, Göttsche and Iarrobino.