  In this paper we introduce an effective method to construct rational
deformations between couples of Borel-fixed ideals. These deformations are
governed by flat families, so that they correspond to rational curves on the
Hilbert scheme. Looking globally at all the deformations among Borel-fixed
ideals defining points on the same Hilbert scheme, we are able to give a new
proof of the connectedness of the Hilbert scheme and to introduce a new
criterion to establish whenever a set of points defined by Borel ideals lies on
a common component of the Hilbert scheme. The paper contains a detailed
algorithmic description of the technique and all the algorithms are made
available.
