  By definition, an $\m$-primary ideal $I$ in a 2-dimensional regular local
ring $(R, \m)$ is contracted if $I=R \cap IR[\m/x]$ for some $x \in \m
\setminus \m^2$. Contracted ideals have been introduced by Zariski and used for
proving the unique factorization theorem for complete (i.e. integrally closed)
ideals. Any complete ideal is contracted but not the other way round. While the
associated graded rings to complete ideals are always Cohen-Macaulay, this is
not the case for contracted ideals. Our goal is to study depth, Hilbert
function and defining equations of the graded rings of homogeneous contracted
ideals. We show, by using quadratic transform, that the depth of the associated
graded ring to a contracted ideal $I$ is determined by depth of the associated
graded rings to a certain family of monomial ideals (indeed lex-segments) which
are naturally attached to $I$. For certain classes of contracted ideals we show
that the associated graded ring is Cohen-Macaulay or at least has positive
depth.
