Intersection of Curves and Bezout's Theorem
Date of Graduation
Master of Science in Mathematics
This thesis is a study of three topics in the theory of plane algebraic curves, the theory of intersection multiplicities, the famous Theorem of Bezout, and an interesting property of cubic curves. An algebraic curve, roughly speaking, is the graph of f (x, y)=0, where f ∈k[x, y] is a plynomial in two variables with coefficients in a field k. We will study projective spaces in general, including, briefly, the real projective plane, P² (ℝ), and then the complex projective plane, P² (ℂ). Next, we will discuss the concept of intersection multiplicity of two algebraic curves, which is necessary to a proof of Bezout's Theorem. We define intersection numbers, prove some of their interesting properties and give examples. We then prove Bezout's Theorem, including examples. Bezout's Theorem states that if f is a curve of degree m and g is a curve of degree n, f and g intersect mn times. In order for this to be true, we need to count the intersections properly using the intersection multiplicity concept. We also need the field k to be algebraically closed and we need to take into account points at infinity, which necessitates the previous introduction of the complex projective plane. Finally, we will specialize our study of algebraic curves to the interesting class of cubic curves, f (x, y)=0, where f ∈k[x, y] is a polynomial of degree 3. For these curves, a group structure can be put on the points (x, y) satisfying the equation of the curve, with associativity being a result of Bezout's Theorem. Group law on the cubic curve was the author's original incentive to pursue these topics.
© Bryan Chapman
Chapman, Bryan, "Intersection of Curves and Bezout's Theorem" (2001). MSU Graduate Theses. 2432.