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.
Open Covers and Lex Points of Hilbert Schemes Over Quotient Rings via Relative Marked Bases / Bertone, C.; Cioffi, F.; Orth, M.; Seiler, W. M.. - In: EXPERIMENTAL MATHEMATICS. - ISSN 1058-6458. - 00:0(2024), pp. 1-15. [10.1080/10586458.2024.2309517]
Open Covers and Lex Points of Hilbert Schemes Over Quotient Rings via Relative Marked Bases
Cioffi F.;
2024
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.