HILBERT POLYNOMIAL AND ITS RELATION TO THE RIEMANN ROCH THEOREM

Abstract

This M.Sc. project delves into the intricate relationship between Hilbert polynomials and the cornerstone Riemann-Roch theorem, unveiling the geometric properties of curves. We begin by establishing the foundation in algebraic geometry with Zariski topology on affine and projective spaces. Building upon this, we explore the powerful tools of category theory, sheaves, and schemes, with a particular emphasis on projective schemes and constructions like Proj. The project then delves into the construction of Hilbert functions and polynomials for finitely generated graded modules, culminating in their application to projective varieties. Intriguingly, we investigate how the coefficients of these polynomials encode valuable geometric information about the variety, including its degree and dimension. In the last part of the project we state the Riemann Roch Theorem for plane curves and try to understand the interlink between the coefficient of Hilbert polynomial and the Riemann Roch Theorem.