  The stratum St(J,<) (the homogeneous stratum Sth(J,<) respectively) of a
monomial ideal J in a polynomial ring R is the family of all (homogeneous)
ideals of R whose initial ideal with respect to the term order < is J. St(J,<)
and Sth(J,<) have a natural structure of affine schemes. Moreover they are
homogeneous w.r.t. a non-standard grading called level. This property allows us
to draw consequences that are interesting from both a theoretical and a
computational point of view. For instance a smooth stratum is always isomorphic
to an affine space (Corollary 3.6). As applications, in Sec. 5 we prove that
strata and homogeneous strata w.r.t. any term ordering < of every saturated
Lex-segment ideal J are smooth. For Sth(J,Lex) we also give a formula for the
dimension. In the same way in Sec. 6 we consider any ideal R in k[x0,..., xn]
generated by a saturated RevLex-segment ideal in k[x,y,z]. We also prove that
Sth(R,RevLex) is smooth and give a formula for its dimension.
