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 flavor 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 flavor. 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.

Keywords:

• marked basis
• Hilbert scheme
• Cohen-Macaulay ring
• Macaulay-Lex ring
• open cover
• lex-point

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13P10
• 14C05
• 14M05
• 13H10
• 13F55

Additional information

Funding

The first and second authors are members of GNSAGA (INdAM, Italy).