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

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[X₁, . . . , 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.