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PLANE ALGEBRAIC CURVES An algebraic curve defined over a field F may be considered as the locus of points in Fn determined by n-1 independent polynomial functions in n variables with coefficients in F, gi(x1, ..., xn), where the curve is definied by setting each gi=0. Using the Resultant , we can eliminate all but two of the varibles and reduce the curve to a Birationally Equivalent plane curve, f(x,y) = 0, still with coefficients in F, but usually of higher degree, and often possessing additional singularities. For example, eliminating z between the two equations x2+y2-z2=0 and x+2y+3z-1=0, which defines an intersection of a cone and a plane in three dimensions, we obtain the Conic Section 8x2 + 5y2-4xy+2x+4y-1=0, which in this case is an ellipse. If we eliminate z between 4x2+y2-z2 = 1 and z=x2, we obtain y2=x4 - 4x2 +1, which is the equation of an Elliptic Curve . PROJECTIVE CURVES It is often desirable to consider that curves are a locus of points in Projective Space . In the set of equations gi=0, we can replace each xk with xk/x0, and multiply by x0n, where n is the degree of gi. In this way we obtain n-1 Homogeneous polynomial functions, which define the corresponding curve in projective space. For a plane algebraic curve we have a single equation f(z,y,z)=0, where f is homogeneous; for example, the Fermat Curve xn+yn+zn=0 is a projective curve. ALGEBRAIC FUNCTION FIELDS The study of algebraic curves can be reduced to the study of Irreducible algbebraic curves. Up to Birational equivalence, these are Categorically Equivalent to Algebraic Function Field s. An algebraic function field is a field of algebraic functions in one variable K defined over a given field F. This means there exists an element x of K which is transcendental over F, and such that K is a finite algebraic extension of F(x), which is the field of rational functions in the indeterminate x over F. For example, consider the field C of complex numbers, over which we may define the field C(x) of rational functions in C. If y2 = x3-x-1, then the field C(x,y) is an Elliptic Function Field . The element x is not uniquely determined; the field can also be regarded, for instance, as an extension of C(y). The algebraic curve corresponding to the function field is simply the set of points (x,y) in C2 satisfying y2 = x3-x-1. If the field F is not Algebraically Closed , the point of view of function fields is a little more general than that of considering the locus of points, since we include, for instance, "curves" with no points on them. If the base field F is the field R of real numbers, then x2+y2=-1 defines an algebraic extension field of R(x), but the corresponding curve considered as a locus has no points in R. However, it does have points defined over the algebraic closure '''C''' of R. RATIONAL CURVES A rational curve, also called a unicursal curve, is any curve which is Birationally Equivalent to a line, which we may take to be a projective line and identify with the field of rational functions in one indeterminate F(x). If F is algebraically closed, this is equivalent to a curve of Genus zero; however the field R(x,y) with x2+y2=-1 is a field of genus zero which is not a rational function field. Concretely, a rational curve of dimension n over F can be parametrized (except for isolated exceptional points) by means of n rational functions defined in terms of a single parameter t; by clearing denominators we can turn this into n+1 polynomial functions in projective space. An example would be the Rational Normal Curve . Any Conic Section defined over F with a rational point in F is a rational curve. It can be parametrized by drawing a line with slope t through the rational point, and intersection with the plane quadradic curve; this gives a polynomial with F-rational coefficients and one F-rational root, hence the other root is F-rational (ie, belongs to F) also. For example, consider the ellipse x2 + xy + y2 = 1, where (-1, 0) is a rational point. Drawing a line with slope t from (-1,0), y=t(x+1), substituting it in the equation of the ellipse, factoring, and solving for x, we obtain : We then have that the equation for y is : which defines a rational parametrization of the ellipse and hence shows the ellipse is a a rational curve. All points of the ellipse are given, except for (-1,1), which corresponds to ; the entire curve is parametrized therefore by the real projective line. Viewing rational parametrizations with rational coefficients projectively, we can view them as giving number theoretical information about homogeneous equations defined over the integers. For example from the above, we obtain : for which : is true for integer ''X'', ''Y'' and ''Z'' if ''t'' is an integer. Hence we obtain triangles with integer length sides, such as sides of length 3, 7, and 8, where one of the angles is 60 degrees, from relationships such as . SINGULARITIES Using the intrinsic concept of Tangent Space , points P on an algebraic curve ''C'' are classified as ''smooth'' or ''non-singular'', or else '' Singular ''. Given n-1 homogeneous polynomial functions in n+1 variables, we may find the Jacobian Matrix as the (n-1)x(n+1) matrix of partial derivatives. If the Rank of this matrix at a point P on the curve has the maximal value of n-1, then the point is a smooth point. In particular, if the curve is a plane projective algebraic curve, defined by a single homogeneous polynomial equation f(x, y, z)=0, then the singular points are precisely the points P where the rank of the 1 by n+1 matrix is zero, that is, where :. Since f is a polynomial, this definition is purely algebraic and makes no assumption about the nature of the field F, which in particular need not be the real or complex numbers. It should of course be recalled that (0,0,0) is not a point of the curve and hence not a singular point. The singularities of a curve are not birational invariants. However, locating and classifying the singularities of a curve is one way of computing the genus, which is a birational invariant. For this to work, we should consider the curve projectively and require F to be algebraically closed, so that all the singularities which belong to the curve are considered. Singular points include crossings over itself, and also types of ''cusp'', for example that shown by the curve with equation x3 = y2 at (0,0). A curve ''C'' has at most a finite number of singular points. If it has none, it can be called ''non-singular''. For this definition to be correct, we must use an Algebraically Closed field and a curve ''C'' in Projective Space (i.e. ''complete'' in the sense of algebraic geometry). If for example we simply look at a curve in the real affine plane there might be singular points 'at infinity', or that needed complex number co-ordinates for their expression. COMPACT RIEMANN SURFACES The theory of non-singular algebraic curves over the complex numbers coincides with that of the Compact Riemann Surface s. Every algebraic curve has a Genus defined. In the Riemann surface case that is the same as the topologist's idea of genus of a 2- Manifold . The genus enters into the statement of the Riemann-Roch Theorem and can be characterized as the only integer that makes this theorem correct. This can serve as a definition of the genus for curves over other fields. EXAMPLES OF CURVES The case of genus one - Elliptic Curve s - has in itself a large number of deep and interesting features. For higher genus ''g'' some of those carry over to the Jacobian Variety , an Abelian Variety of dimension ''g''. See also List Of Curves . REFERENCES
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