Border Basis Schemes

The basic idea of border basis theory is to describe a zero-dimensional ring P/I by an order ideal of terms whose residue classes form a K-vector space basis of P/I. The O-border basis scheme is a scheme that parametrizes all zero-dimensional ideals that have an O-border basis. In general, the O-border basis scheme is not an affine space. Subsequently, in [Huib09] it is proved that if an order ideal with "d" elements is defined in a two-dimensional polynomial ring and it is of some special shapes, then the O-border basis scheme is isomorphic to the affine space of dimension 2d. This thesis is dedicated to find a more general condition for an O-border basis scheme to be isomorphic to an affine space of dimension "nd" that is independent of the shape of the order ideal with "d" elements and "n" is the dimension of the polynomial ring that the order ideal is defined in. We accomplish this in 6 Chapters. In Chapters 2 and 3 we develop the concepts and properties of border basis schemes. In Chapter 4 we transfer the smoothness criterion (see [Huib05]) for the point (0,...,0) in a Hilbert scheme of points to the monomial point of the border basis scheme by employing the tools from border basis theory. In Chapter 5 we explain trace and Jacobi identity syzygies of the defining equations of a O-border basis scheme and characterize them by the arrow grading. In Chapter 6 we give a criterion for the isomorphism between 2d dimensional affine space and O-border basis scheme by using the results from Chapters 3 and Chapter 4. The techniques from other chapters are applied in Chapter 6.1 to segment border basis schemes and in Chapter 6.2 to O-border basis schemes for which O is of the sawtooth form.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Passau

Year

2017

Author dc:creator

• Sipal, Bilge

Contributors dc:contributor

• Kreuzer, Martin
• Sezer, Mufit

• Border Bases, Border Basis Scheme, Monomial point, Cotangent Space, Hilbert Schemes
• Kommutative Algebra, Randbasen, Randbasen Schema

Statement dc:rights

• CC by: Creative Commons - Namensnennung

Repository record source_url

https://opus4.kobv.de/opus4-uni-passau/frontdoor/index/index/docId/470

OAI identifier oai:identifier

oai:kobv.de-opus4-uni-passau:470