On the Cohen-Macaulayness of diagonal subalgebras of the Rees algebra

Abstract

We consider the blowing up of ℙ k/n−1 along a closed subscheme defined by a homogeneous idealI ∪A=k[X 1, …,X n ] generated by forms of degree ≤d, and its projective embeddings by the linear systems corresponding to (I e) c , forc≥de+1. The homogeneous coordinate rings of these embeddings arek[(I e) c ]. One wants to study the Cohen-Macaulay property of these rings. We will prove that if the Rees algebraR A (I) is Cohen-Macaulay, thenk[(I e) c ] are Cohen-Macaulay forc>>e>0, thus proving a conjecture stated by A. Conca, J. Herzog, N.V. Trung and G. Valla.