Macaulay-like marked bases

Abstract

We define marked sets and bases over a quasi-stable ideal $\mathfrak{j}$ in a polynomial ring on a Noetherian $K$-algebra, with $K$ a field of any characteristic. The involved polynomials may be non-homogeneous, but their degree is bounded from above by the maximum among the degrees of the terms in the Pommaret basis of $\mathfrak{j}$ and a given integer $m$. Due to the combinatorial properties of quasi-stable ideals, these bases behave well with respect to homogenization, similarly to Macaulay bases. We prove that the family of marked bases over a given quasi-stable ideal has an affine scheme structure, is flat and, for large enough $m$, is an open subset of a Hilbert scheme. Our main results lead to algorithms that explicitly construct such a family. We compare our method with similar ones and give some complexity results.

30 pages. Final version. In the present version Section 6 about flatness is improved, and new subsections concerning comparison with other existing computational methods (Section 7.1) and some complexity results (Section 7.2) were added

References in corpus (1)

• Flat families by strongly stable ideals and a generalization of Groebner bases

Cited by in corpus (5)

• Strongly stable ideals and Hilbert polynomials
• The close relation between border and Pommaret marked bases
• Open Covers and Lex Points of Hilbert schemes over quotient rings via relative marked bases
• Functors of Liftings of Projective Schemes
• Cohen-Macaulay, Gorenstein and complete intersection conditions by marked bases