Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Open Covers and Lex Points of Hilbert schemes over quotient rings via relative marked bases

Cristina Bertone, Francesca Cioffi, Matthias Orth, Werner M. Seiler

Abstract

We introduce the notion of a relative marked basis over quasi-stable ideals, together with constructive methods and a functorial interpretation, developing computational methods for the study of Hilbert schemes over quotients of polynomial rings. Then we focus on two applications. The first has a theoretical flavour and produces an explicit open cover of the Hilbert scheme when the quotient ring is Cohen-Macaulay on quasi-stable ideals. Together with relative marked bases, we use suitable general changes of variables which preserve the structure of the quasi-stable ideal, against the expectations. The second application has a computational flavour. When the quotient rings are Macaulay-Lex on quasi-stable ideals, we investigate the lex-point of the Hilbert schemes and find examples of both smooth and singular lex-points.

Open Covers and Lex Points of Hilbert schemes over quotient rings via relative marked bases

Abstract

We introduce the notion of a relative marked basis over quasi-stable ideals, together with constructive methods and a functorial interpretation, developing computational methods for the study of Hilbert schemes over quotients of polynomial rings. Then we focus on two applications. The first has a theoretical flavour and produces an explicit open cover of the Hilbert scheme when the quotient ring is Cohen-Macaulay on quasi-stable ideals. Together with relative marked bases, we use suitable general changes of variables which preserve the structure of the quasi-stable ideal, against the expectations. The second application has a computational flavour. When the quotient rings are Macaulay-Lex on quasi-stable ideals, we investigate the lex-point of the Hilbert schemes and find examples of both smooth and singular lex-points.
Paper Structure (12 sections, 21 theorems, 30 equations, 1 algorithm)

This paper contains 12 sections, 21 theorems, 30 equations, 1 algorithm.

Table of Contents

  1. Preliminaries
  2. Marked bases and marked functors
  3. Marked bases
  4. Marked functors
  5. Parametrisation of saturated ideals in quotient rings
  6. Relative marked bases and reduction relations
  7. Relative marked functor
  8. Case $I=\tilde{I}$.
  9. An open cover of a Hilbert scheme over a Cohen-Macaulay quotient ring on a quasi-stable ideal
  10. Lex-point in Hilbert schemes over Macaulay-Lex quotients on quasi-stable ideals
  11. Generalities
  12. Examples of smooth and singular lex-points

Key Result

Proposition 1.2

If $\tilde{I}$ is a quasi-stable ideal, then $\mathcal{B}_{\tilde{I}}\subseteq \mathcal{P}_{\tilde{I}}$ holds and the regularity $\mathrm{reg}(\tilde{I})$ coincides with the maximum degree of a term in $\mathcal{P}_{\tilde{I}}$.

Theorems & Definitions (67)

  • Definition 1.1
  • Proposition 1.2: Seiler2009II, Seiler:InvolutionBook
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 57 more