The diagonal subalgebra of a blow-up algebra Original Research Article

Given a bigradedk-algebraS = circled plus(u,ν) S(u,ν), (u,ν) εthe semi-ring of natural numbers×the semi-ring of natural numbers, (k a field), one attaches to it the so-called diagonal subalgebraSδ = circled plus(u,u) S(u,u). This notion generalizes the concept of Segre product of graded algebras. The classical situation hasS = k[S(1,0), S(0,1)], whereby taking generators ofS(0,1) andS(0,1) yields a closed embedding Proj(S) right arrow-hookedopen face Pkn − 1 ×open face Pkr − 1, for suitablen,r; the resulting generators ofS(1,1) makeSδ isomorphic to the homogeneous coordinate ring of the image of Proj (S) under the Segre mapopen face Pkn − 1 ×open face Pkr − 1 →open face Pknr − 1. The main results of this paper deal with the situation whereS is the Rees algebra of a homogeneous ideal generated by polynomials in a fixed degree. In this framework,Sδ 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 inpk5 and toric varieties inopen face Pkn. The main concern is then with the normality and the Cohen-Macaulayness ofsδ. One can describe the integral closure ofsδ explicitly in terms of the given ideal and show that normality carries fromS toSδ. 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.