A Discussion of Gröbner Bases and the Hilbert Scheme

Abstract

In this expository thesis, we want to present an introduction to computational algebraic geometry through the lens of Gröbner bases and describe the construction and some pathological examples on the Hilbert scheme. In section 2, we introduce some algebra preliminaries and define Gröbner bases, while in section 3 we focus on discussing Buchberger’s algorithm for computing the Gröbner basis of a given ideal. In section 4, we give two definitions for the Hilbert polynomial associated to a subscheme, and use Gröbner bases to prove the existence of this polynomial and to show a way of computing it. In section 5, we present an extended example and compute the Hilbert polynomial associated to smooth algebraic curves, for which we introduce Weil divisors. The Hillbert polynomial will help us introduce the Hilbert scheme in section 6, which parametrizes subschemes of the projective space with a fixed Hilbert polynomial. We also describe the construction of the Hilbert scheme in this section. Then, in sections 7 and 8, we discuss the Hilbert scheme of twisted cubics, for which we introduce the notion of extraneous component, and Mumford’s example.