In this paper we consider a large class of coordinate rings of certain unions of projective scrolls. This class appears in several recent works studying properties of the special fibre of an ideal I in a local ring. We prove that the reduction number of these algebras is always equal to one. We also prove that these reduced algebras are Cohen-Macaulay of minimal degree, so that the above assertion also follows from a theorem by Eisenbud and Goto. Our methods, however, are constructive, and we can explicitly describe a Noether subalgebra. This can be used to find explicit reductions for the ideal I. As an applica- tion we describe the special fibre when I is the defining ideal of a projective monomial variety of codimension 2, and prove a conjecture by Gimenez: we show that the Rees algebra of I is defined by relations of degree two at most.
On certain algebras of reduction number one
BARILE, Margherita;
1998-01-01
Abstract
In this paper we consider a large class of coordinate rings of certain unions of projective scrolls. This class appears in several recent works studying properties of the special fibre of an ideal I in a local ring. We prove that the reduction number of these algebras is always equal to one. We also prove that these reduced algebras are Cohen-Macaulay of minimal degree, so that the above assertion also follows from a theorem by Eisenbud and Goto. Our methods, however, are constructive, and we can explicitly describe a Noether subalgebra. This can be used to find explicit reductions for the ideal I. As an applica- tion we describe the special fibre when I is the defining ideal of a projective monomial variety of codimension 2, and prove a conjecture by Gimenez: we show that the Rees algebra of I is defined by relations of degree two at most.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
