Graded Rings Associated with Contracted Ideals

The study of the ideals in a regular local ring (R,m) of dimension 2 has a long and important tradition dating back to the fundamental work of Zariski [ZS]. More recent contributions are due to several authors including Cutkosky, Huneke, Lipman, Sally and Tessier among others, see [C1, C2, H, HS, L, LT]. One of the main result in this setting is the unique factorization theorem for complete (i.e., integrally closed) ideals proved originally by Zariski [ZS, Thm3, Appendix 5]. It asserts that any complete ideal can be factorized as a product of simple complete ideals in a unique way (up to the order of the factors). By definition, an ideal is simple if it cannot be written as a product of two proper ideals. Another important property of a complete ideal I is that its reduction number is 1 which in turns implies that the associated graded ring grI(R) is Cohen-Macaulay and its Hilbert series is well understood; this is due to Lipman and Tessier [LT], see also [HS]. The class of contracted ideals plays an important role in the original work of Zariski as well as in the work of Huneke [H]. An m-primary ideal I of R is contracted if I = R ∩ IR[m/x] for some x ∈ m \ m2. Any complete ideal is contracted but not the other way round. The associated graded ring grI(R) of a contracted ideal I need not be CohenMacaulay and its Hilbert series can be very complicated. Our goal is to study depth, Hilbert function and defining equations of the various graded rings (Rees algebra, associated graded ring and fiber cone) of homogeneous contracted ideals in the polynomial ringR = k[x, y] over an algebraically closed field k of characteristic 0. In Section 3 we present several equivalent characterizations of contracted ideals in the graded and local case. The main result of this section is Theorem 3.11. It asserts that the depth of grI(R) is equal to the minimum of depth grI′SN (SN ), where S = R[m/x], I ′ is the transform of I and N varies in the set of maximal ideals of S containing I . An important invariant of a contracted ideal I of order (i.e., initial degree) d is the socalled characteristic form, which, in the graded setting, is nothing but GCD(Id). Here Id denotes the homogeneous component of degree d of I. The more general contracted ideals are those with a square-free (i.e., no multiple factors) characteristic form. On the other hand, the more special contracted ideals are those whose characteristic form is a power of a linear form; these ideals are exactly the so-called lex-segment ideals. The lex-segment ideals are in bijective correspondence with the Hilbert functions (in the graded sense) of graded ideals so that to specify a lex-segment ideal is equivalent to specify a Hilbert function. In the graded setting Zariski’s factorization theorem for contracted ideals [ZS, Thm1, Appendix 5] says that any contracted ideal I can be written as a product of lex-segment