Executive Summary : | The Hilbert functions, introduced by Hilbert in 1890, have been a significant influence on mathematics, particularly in recent decades. They provide a rich source of discrete invariants of projective varieties, such as dimension, degree, and arithmetic genus. The theory of Hilbert functions offers a microscope for examining varieties and revealing their subtle geometric properties. In 1927, Macaulay characterized the possible Hilbert functions of a standard graded K-algebra, raising the question of what are the possible Hilbert functions with additional properties. This project aims to understand the Hilbert function of the standard graded Cohen-Macaulay domain, a question introduced by J. Harris in his fundamental approach to Castelnuovo theory. The goal is to determine the structure of the Hilbert function of fat points, inspired by recent work with L. Brustenga. The Hilbert function of double points was a key ingredient in a deep result by Alexander and Hirschowitz, determining the Waring rank of a generic form. The project aims to exploit these methods to study the Waring problem, especially over a field other than the field of complex numbers. They expect to characterize the Hilbert functions of points in uniform position and fat points in certain cases, aiming to find geometric interpretations of the singularities of these schemes. |