  We introduce the notion of a relative marked basis over quasi-stable ideals,
together with constructive methods and a functorial interpretation, developing
computational methods for the study of Hilbert schemes over quotients of
polynomial rings. Then we focus on two applications.
  The first has a theoretical flavour and produces an explicit open cover of
the Hilbert scheme when the quotient ring is Cohen-Macaulay on quasi-stable
ideals. Together with relative marked bases, we use suitable general changes of
variables which preserve the structure of the quasi-stable ideal, against the
expectations.
  The second application has a computational flavour. When the quotient rings
are Macaulay-Lex on quasi-stable ideals, we investigate the lex-point of the
Hilbert schemes and find examples of both smooth and singular lex-points.
