Boundary and shape of Cohen-Macaulay cone

Abstract

Let $R$ be a Cohen-Macaulay local domain. In this paper we study the cone of Cohen-Macaulay modules inside the Grothendieck group of finitely generated $R$-modules modulo numerical equivalences, introduced in \cite{CK}. We prove a result about the boundary of this cone for Cohen-Macaulay domain admitting de Jong's alterations, and use it to derive some corollaries on finiteness of isomorphism classes of maximal Cohen-Macaulay ideals. Finally, we explicitly compute the Cohen-Macaulay cone for certain isolated hypersurface singularities defined by $\xi \eta -f(x_1,\dots ,x_n)$.