Algebras of Minors

Abstract

Let I be an ideal of a Noetherian ring R. The Rees algebra of I is the graded R-algebra ⊕k=0Ik. The study of the properties of Rees algebras (and of the other blow-up algebras) has attracted the attention of many researchers in the last three decades. For a detailed account of the theories that have been developed and of the results that have been proved, the reader should consult the monograph of Vasconcelos [V]. In this paper we treat a special and interesting case: the Rees algebras of determinantal ideals and their special fibers. Let X = xij be a generic matrix of size m × n over a field K and let S be the polynomial ring K xij . Let It be the ideal of S generated by the minors of size t of X. Finally, let t be the Rees algebra of It , and let At be the subalgebra of S generated by the t-minors of X. In the case of maximal minors, i.e., t = min m n , the algebra At is nothing but the homogeneous coordinate ring of the Grassmann variety which is known to be a