The Geometry of Border Bases

In this paper we continue the study of the border basis scheme we started in another paper. The main topic is the construction of various explicit flat families of border bases. To begin with, we cover the punctual Hilbert scheme $\Hilb^\mu(\mathbb{A}^n)$ 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 th