Abstract
Hilbert schemes of zerodimensional ideals in a polynomial ring can be covered with suitable affine open subschemeswhose construction is achieved using border bases. Moreover, border bases have proved to be an excellent tool for describing zero-dimensional ideals when the coefficients are inexact. and in this situation they show a clear advantage with respect to Grobner bases which, nevertheless, can also be used in the study of Hilbert schemes, since they provide tools for constructing suitable stratifications.
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