The geometry of border bases (Q534027)

Border bases schemes have attracted a growing interest in recent years and there are still a large number of open questions. In this paper the authors provide some important contributions: First the authors detail a number of flat families of border bases. This is important since it can happen that there are flat families whose general fiber has a border basis but whose special fiber does not. Secondly the authors provide an explicit construction of the principle component of the border basis scheme. This construction differs from previous ones (e.g. Grothendieck's approach via Grassmannians). Further this construction allows them to find flat families of maximal dimension at each radical point. Finally the authors are also able to connect two radical points with a given order basis via explicit flat families. Further they connect any radical point to a monomial point via one explicit flat family. Thus the authors are able to connect an arbitrary radical point to a fixed monomial point. The paper is thorough and concludes with some illuminating examples. It should be of interest to anyone studying border bases.