The Geometry of Border Bases

TL;DR

This paper explores the structure of border bases by constructing explicit flat families, covering the punctual Hilbert scheme, and connecting radical points through flat families, advancing understanding of border basis schemes.

Contribution

It provides explicit constructions of flat families of border bases, covers the punctual Hilbert scheme with border basis schemes, and connects radical points via flat families.

Findings

Constructed explicit flat families of border bases.

Covered the punctual Hilbert scheme with border basis schemes.

Connected radical points through flat families on the principal component.

Abstract

The main topic of the paper is the construction of various explicit flat families of border bases. To begin with, we cover the punctual Hilbert scheme Hilb^\mu(A^n) by border basis schemes and work out the base changes. This enables us to control flat families obtained by linear changes of coordinates. Next we provide an explicit construction of the principal component of the border basis scheme, and we use it to find flat families of maximal dimension at each radical point. Finally, we connect radical points to each other and to the monomial point via explicit flat families on the principal component.