On Fiber Cones of 𝑚-Primary Ideals

Abstract

Abstract Two formulas for the multiplicity of the fiber cone of an 𝑚-primary ideal of a d -dimensional Cohen–Macaulay local ring ( R , 𝑚) are derived in terms of the mixed multiplicity e d –1 (𝑚| I ), the multiplicity e ( I ), and superficial elements. As a consequence, the Cohen–Macaulay property of F ( I ) when I has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of I and lengths of certain ideals. We also characterize the Cohen–Macaulay and Gorenstein properties of fiber cones of 𝑚–primary ideals with a d –generated minimal reduction J satisfying ℓ( I 2 / JI ) = 1 or ℓ( I 𝑚/ J 𝑚) = 1.