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PLANE ALGEBRAIC CURVES Contents 1. Introduction 1 2 PLANE ALGEBRAIC CURVES ANDREW DING Abstract. We go over some of the basics of plane algebraic curves, which are planar curves described as the set of solutions of a polynomial in two variables. We study many basic notions, such as projective space, parametrization, and the intersection of two curves. We end with the group law on the cubic and search for torsion points. Contents 1. Introduction 1 2. Parametrization of a Conic 2 3. Projective Space 3 4. Lines and Conics 5 5. Parametrization of Singular Cubics 8 6. Cayley-Bacharach Theorem 9 7. Group Law on a Cubic 12 8. Weierstrass Form 14 9. Torsion Points 14 9.1. 2-Torsion Points 15 9.2. 3-Torsion Points 15 9.3. More on Torsion Points 16 Acknowledgments 16 References 16 1. Introduction Solving systems of polynomial equations in several variables is of interest in algebraic geometry. It is a generalization of both solving for roots of polynomials (algebra) and solving systems of linear equations (linear algebra). The main insight of algebraic geometry is to view solutions to a system of polynomial equations geometrically. For example, the set of solutions to a polynomial equation in two variables forms an algebraic curve. We can study the topological and geometrical properties of such a curve. In addition, geometric pictures provide valuable insight in solving problems in algebraic geometry. Unlike topology, which focuses on continuous functions, and differential geome- try, which focuses on smooth functions, the main focus in algebraic geometry are polynomial functions. The main advantage of this fact is that a majority of results work over an arbitrary field. Occasionally, an algebraically closed field is needed. Date: 08/26/2012. 1 2 ANDREW DING Another advantage of polynomials is a sense of finiteness. For example, a polyno- mial over one variable has finitely many roots, and Sd, the space of all homogeneous polynomials of degree d, is a finite-dimensional vector space. Some curves can be parametrized, which allows us to completely understand these curves. Notably, parametrizing a curve over any field gives all rational solu- tions, which is of interest in number theory. Another object of study is algebraic subsets in projective space. In the projective plane, every pair of distinct lines intersect at exactly one point. In the affine plane, two distinct lines generally intersect at one point, with the exception of parallel lines, which are a special case. Projective geometry eliminates this special case, and allows the statement of Bezout's Theorem, which fully describes the number of intersection of two algebraic curves. In addition, there is a handful of theorems that are stated in the context of projective space, such as the Cayley-Bacharach Theorem and Pascal's Theorem. Given certain conditions, Bezout's Theorem states that a line and a cubic will intersect at three points. Given this fact and that two points determine a line, if the cubic is 'nice enough', then we can define a group over the set of points on the cubic. These types of cubics are called elliptic curves. The theory of elliptic curves is extremely rich and remains an active area of research. 2. Parametrization of a Conic Let's start with an example that hopefully both gives motivation to study alge- braic geometry and demonstrates the interplay between algebra and geometry. Example 2.1. Find all integer solutions to the equation X2 + Y 2 = Z2. Proof. At first, this seems like a purely algebraic question. In addition, we can find several solutions, such as (1; 0; 1) and (3; 4; 5), but it may seem daunting to find all solutions. We can start by narrowing our search to all primitive integer solutions, that is, solutions such that gcd(X; Y; Z) = 1, because all solutions come from a primitive one. For example, (12; 16; 20) comes from (3; 4; 5). Substitute x = X=Z; y = Y=Z. We obtain x2 + y2 = 1. Since (X; Y; Z) is a primitive solution, we have a bijection from the rational points on the circle C to 3 4 primitive integer solutions. For example, (3; 4; 5) would correspond to ( 5 ; 5 ). To find all the rational points, we use a technique called projection. First of all, we need one rational point, so we will use (−1; 0). Next, note that every line through (−1; 0) (except one) intersects the circle through another point. Each of these lines has a different slope λ and hits the circle at different points Pλ. We have 2λ λ2−1 a bijection from R to Cn(−1; 0). Simplifying, we obtain Pλ = ( λ2+1 ; λ2+1 ). 2 2 Note if λ 2 Q, then Pλ 2 Q . Conversely, if Pλ 2 Q , then by calculating the slope of the line through (−1; 0) and P , we get λ 2 Q. This gives a bijection from Q to rational points on the circle. m m We are almost done. Let λ = n 2 Q, where m; n are coprime. Substitute n for λ. Clearing denominators, we get X = 2mn; Y = m2 − n2;Z = m2 + n2. If l; m are both odd, then divide X; Y; Z by 2. This gives us all primitive Pythagorean triples. We can make a few important observations. PLANE ALGEBRAIC CURVES 3 (1) We were working with the equation X2 + Y 2 = Z2, which is a homoge- neous equation. This process does not work for polynomials in general. Homogeneous polynomials are convenient for many reasons, one of which is highlighted in Section 3. (2) We started projection with the rational point (−1; 0). Projection does not help us find this first rational point. In fact, some conics will have no rational points, such as 2x2 + y2 = 5. (To verify this, work with the polynomial 2X2 + Y 2 = 5Z2 modulo 5). (3) The vertical line through (−1; 0) should be mentioned. It intersects the circle at (−1; 0) with multiplicity 2, so the vertical line corresponds to the point (−1; 0). We did not miss the associated primitive solution; by allowing m = −1; n = 0 in the last step, we get the point (−1; 0; 1). (4) The fact that we got a bijection from the set of rational points on the conic to the set of rational numbers is not a coincidence. In fact, this will hold for conics in general. This will be explained in more detail in Section 4. 3. Projective Space We take a break from conics to develop the notion of projective space. It is often convenient to work in projective space when discussing the number of intersection points of two algebraic curves. For example, in the affine plane, two distinct lines intersect at one point, unless they are parallel. In the projective plane, any two distinct lines will always intersect at exactly one point. We will begin with the projective line and the projective plane. 2 1 Definition 3.1. Let k be a field, O = (0; 0) 2 k . The projective line Pk is defined as the geometry with flines through O 2 k2g as the set of points. Let us try to imagine what the projective line looks like. In the affine plane, a line is determined by a point on the line and its slope. Since all lines go through O, the only difference between two distinct lines is their slopes. Hence, we have the following injection: 1 ' : k ! Pk;'(λ) = line with slope λ through O: However, this misses the vertical line through O. Let's say it has slope 1. Adding it to the domain, we have: 1 ∼ Pk = k [ f1g: Now, let us develop a system of coordinates to describe these points. Suppose 1 2 we have a point P 2 Pk. The point P is an affine line l through O 2 k . The line l must pass through another point, say (A; B). We give P the homogeneous coordinates [A : B]. If λ 6= 0, then [λA : λB] represents P as well. This is analogous A λA to the fact that, if B 6= 0, B and λB represent the same number. In mathematical language, 1 2 Pk = k − f(0; 0)g= ∼; where (a; b) ∼ (c; d) if c = λa, d = λb for some λ 6= 0: In other words, we define a point on the projective line as an equivalence class of points on the affine plane, where two points are related if they lie on the same veritable line through the origin. 4 ANDREW DING The advantage of using homogeneous coordinates is that it describes projective points without the use of infinity, making them easier to use in algebraic manipu- lation. Homogeneous coordinates treat all points equally, which is also true for the original definition of the projective line. Now we will move to a more interesting concept, the projective plane. As the projective line is an 'extension' of the affine line, the projective plane is also an 'extension' of the affine plane. The process is similar. 3 2 Definition 3.2. Let k be a field, O = (0; 0; 0) 2 k . The projective plane Pk is defined as the geometry with flines through O 2 k3g as the set of points and fplanes through O 2 k3g as the set of lines. 2 Let us try to imagine what the projective plane looks like. Let P 2 Pk. Then P is a line l through O 2 k3. Let (A; B; C) be a nonzero point on l. There are two cases to consider. 3 A B If C 6= 0, then l intersects the plane Z = 1 in k at the unique point ( C ; C ; 1).
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