Bézout's theorem, at least the original version, concerns the number of intersection points of two curves in projective plane. The main purpose of this thesis, apart from proving the classical version of Bézout's theorem, is to give multiple generalizations for it.
The first proper chapter, Chapter 2, is devoted to the proof of classical Bézout's theorem. In the first two sections of the chapter we define projective and affine plane curves, and show some of their basic properties. In the third section we define the resultant of two polynomials, and use the newly acquired tool to prove the upper bound version of Bézout's theorem. The fourth section discusses the multiplicity of a point of intersection. This multiplicity, dependent of algebraic data associated to the intersection, is needed for stating the equality version of the classical Bézout's theorem. In the fifth section we prove this using properties of the intersection multiplicity proved in the fourth section.
The third chapter extends the classical Bézout's theorem beyond its original scope. In Section 3.1 we define a crucial tool, Hilbert polynomial, which allows us to keep track of algebraic information associated to the projective scheme cut out by a homogeneous ideal. This polynomial is not an invariant of the scheme itself; rather it should be thought as containing information concerning both the intrinsic structure of the subscheme, and about how the subscheme is located in the ambient projective space.
The second section of the third chapter quickly summarizes the parts of modern algebraic geometry that are of most use later. Section 3.3 gives the first proper generalization of Bézout's theorem. This generalization is more a quantitative than a qualitative one, as it deals with intersections of projective hyperplanes. The fourth chapter gives a generalization of the upper bound version of the Bézout's theorem to a very general case. We define the geometric multiplicity of a closed subscheme of a projective space, and show that it behaves well under intersections. The geometric multiplicity gives an upper bound for the number of components, hence the generalization of inequality version of Bézout. In the final section, 3.5, we define Serre's multiplicity of a component of intersection, and show that the multiplicities given by this formula satisfy the equality version of Bézout's theorem in proper intersections of equidimensional subchemes.
