Boundary and shape of Cohen-Macaulay cone

TL;DR

This paper investigates the geometric structure of the Cohen-Macaulay cone in the Grothendieck group for Cohen-Macaulay local domains, providing boundary characterizations, finiteness results, and explicit computations for specific singularities.

Contribution

It establishes boundary properties of the Cohen-Macaulay cone for domains with de Jong's alterations and computes the cone explicitly for certain hypersurface singularities.

Findings

Boundary of the Cohen-Macaulay cone characterized for domains with de Jong's alterations

Finiteness of isomorphism classes of maximal Cohen-Macaulay ideals derived

Explicit description of the cone for specific isolated hypersurface singularities

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 $\xi\eta - f(x_1, \ldots, x_n)$.