Numerical Invariants of One Dimensional Local Rings

The purpose of this monograph is to present the main results on Hilbert functions of one-dimensional Cohen–Macaulay local rings and related numerical invariants, covering algebraic, geometric, and computational aspects. The study of Hilbert functions of local rings is a classical topic, originating in Samuel’s extension of Hilbert’s notion of characteristic functions to the local setting. We focus on the one-dimensional case, where the theory is better understood and the construction of explicit examples is more accessible. The main examples of one-dimensional Cohen–Macaulay local rings considered in these notes arise from curve singularities, with special emphasis on monomial curve singularities. From our point of view, one-dimensional local rings provide a natural and effective starting point for the study of Hilbert functions in higher dimensions. Although the main goal of this monograph is the study of Hilbert functions of one-dimensional Cohen–Macaulay local rings, in the second chapter we present a basic “toolkit” of results and techniques on Hilbert functions for higher-dimensional local rings. The results of chapter two could form the core of a graduate course on Hilbert functions, complemented by some results on Hilbert functions of one-dimensional local rings presented in other chapters of this monograph. This monograph is intended for graduate students and researchers interested in local algebra and its computational and geometric aspects.