Cohen-Macaulay, Gorenstein and complete intersection conditions by marked bases

Abstract

Using techniques from the theory of marked bases, we develop new effective methods for detecting and constructing Cohen-Macaulay, Gorenstein and complete intersection homogeneous polynomial ideals over a field $K$. Due to the functorial properties of marked bases, an elementary proof follows for the openness of the arithmetically Cohen-Macaulay, arithmetically Gorenstein and strict complete intersection $K$-rational points loci in a Hilbert scheme with a non-constant Hilbert polynomial.