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

TL;DR

This paper introduces new computational techniques based on marked bases to detect and construct special classes of polynomial ideals, and proves the openness of certain loci in Hilbert schemes.

Contribution

It develops novel methods using marked bases for identifying Cohen-Macaulay, Gorenstein, and complete intersection ideals, with a new proof of openness in Hilbert schemes.

Findings

New computational methods for ideal detection and construction.

Elementary proof of openness of specific loci in Hilbert schemes.

Applicable to non-constant Hilbert polynomials.

Abstract

Using techniques coming from the theory of marked bases, we develop new computational methods for detection and construction of Cohen-Macaulay, Gorenstein and complete intersection homogeneous polynomial ideals. Thanks to the functorial properties of marked bases, an elementary and effective proof of the openness of arithmetically Cohen-Macaulay, arithmetically Gorenstein and strict complete intersection loci in a Hilbert scheme follows, for a non-constant Hilbert polynomial.