Graded rings associated with contracted ideals

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.