Diagonal subalgebras of bigraded algebras and embeddings of blow-ups of projective spaces

Abstract

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.