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

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.

💡 Research Summary

The paper introduces a novel computational framework based on marked bases—a concept closely related to, yet distinct from, Gröbner bases—to detect and construct homogeneous polynomial ideals that are Cohen‑Macaulay, Gorenstein, or strict complete intersections. A marked basis is defined over a quasi‑stable monomial ideal J: each generator is a polynomial together with a distinguished head term belonging to the Pommaret basis of J, while all other terms lie in the complement N(J). This structure yields a confluent, Noetherian rewriting system, guaranteeing a unique normal form for any polynomial with respect to the ideal generated by the marked set.

The authors first formalize the “marked functor” Mf_J, which assigns to any Noetherian K‑algebra A the set of J‑marked bases in A