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arXiv · 2203.11770

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

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

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Cristina Bertone, Francesca Cioffi, Matthias Orth, Werner M. Seiler. 2022-03-22. Open Covers and Lex Points of Hilbert schemes over quotient rings via relative marked bases. https://doi.org/10.1080/10586458.2024.2309517

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