The diagonal subalgebra of a blow-up algebra

Abstract

Given a bigraded k-algebra S = ⊕ ( u,ν) S ( u,ν) , ( u,ν) εℕ×ℕ, ( k a field), one attaches to it the so-called diagonal subalgebra S δ = ⊕ ( u,u) S ( u,u) . This notion generalizes the concept of Segre product of graded algebras. The classical situation has S = k[S (1,0), S (0,1)], whereby taking generators of S (0,1) and S (0,1) yields a closed embedding Proj (S) ↪ℙ k n − 1 ×ℙ k r − 1 , for suitable n,r; the resulting generators of S (1,1) make S δ isomorphic to the homogeneous coordinate ring of the image of Proj ( S) under the Segre mapℙ k n − 1 ×ℙ k r − 1 →ℙ k nr − 1 . The main results of this paper deal with the situation where S is the Rees algebra of a homogeneous ideal generated by polynomials in a fixed degree. In this framework, S δ is a standard graded algebra which, in some case, can be seen as the homogeneous coordinate ring of certain rational varieties embedded in projective space. This includes some examples of rational surfaces in p k 5 and toric varieties inℙ k n . The main concern is then with the normality and the Cohen-Macaulayness of s δ. One can describe the integral closure of s δ explicitly in terms of the given ideal and show that normality carries from S to S δ. In contrast to normality, Cohen-Macaulayness fails to behave similarly, even in the case of the Segre product of Cohen-Macaulay graded algebras. The problem is rather puzzling, but one is able to treat a few interesting classes of ideals under which the corresponding Rees algebras yield Cohen-Macaulay diagonal subalgebras. These classes include complete intersections and determinantal ideals generated by the maximal minors of a generic matrix.