Algebraic curve

Plane projective curves

It is often desirable to consider curves in the projective space. An algebraic curve in the projective plane or plane projective curve is the set of the points in a projective plane whose projective coordinates are zeros of a homogeneous polynomial in three variables P(x, y, z).

Every affine algebraic curve of equation p(x, y) = 0 may be completed into the projective curve of equation ^hp(x, y, z) = 0, where $${\displaystyle {}^{h}p(x,y,z)=z^{\mathrm{deg}(p)}p\left(\frac{x}{z},\frac{y}{z}\right)}$$ is the result of the homogenization of p. Conversely, if P(x, y, z) = 0 is the homogeneous equation of a projective curve, then P(x, y, 1) = 0 is the equation of an affine curve, which consists of the points of the projective curve whose third projective coordinate is not zero. These two operations are reciprocal one to the other, as $${\displaystyle {}^{h}p(x,y,1)=p(x,y)}$$ and, if p is defined by p(x, y) = P(x, y, 1), then $${\displaystyle {}^{h}p(x,y,z)=P(x,y,z)}$$ as soon as the homogeneous polynomial P is not divisible by z.

For example, the projective curve of equation $${\displaystyle x^2+y^2-z^2}$$ is the projective completion of the unit circle of equation $${\displaystyle x^2+y^2-1=0.}$$

This implies that an affine curve and its projective completion are the same curves, or, more precisely that the affine curve is a part of the projective curve that is large enough to well define the "complete" curve. This point of view is commonly expressed by calling "points at infinity" of the affine curve the points (in finite number) of the projective completion that do not belong to the affine part.

Projective curves are frequently studied for themselves. They are also useful for the study of affine curves. For example, if p(x, y) is the polynomial defining an affine curve, beside the partial derivatives p'_x and p'_y, it is useful to consider the derivative at infinity $${\displaystyle p_{\mathrm{\infty }}^{\prime }(x,y)={}^h{p}_z^{\prime }(x,y,1).}$$

For example, the equation of the tangent of the affine curve of equation p(x, y) = 0 at a point (a, b) is $${\displaystyle x{p}_x^{\prime }(a,b)+y{p}_y^{\prime }(a,b)+p_{\mathrm{\infty }}^{\prime }(a,b)=0.}$$

Remarkable points of a plane curve

In this section, we consider a plane algebraic curve defined by a bivariate polynomial p(x, y) and its projective completion, defined by the homogenization ${\displaystyle P(x,y,z)={}^{h}p(x,y,z)}$ of p.

Intersection with a line

Knowing the points of intersection of a curve with a given line is frequently useful. The intersection with the axes of coordinates and the asymptotes are useful to draw the curve. Intersecting with a line parallel to the axes allows one to find at least a point in each branch of the curve. If an efficient root-finding algorithm is available, this allows to draw the curve by plotting the intersection point with all the lines parallel to the y-axis and passing through each pixel on the x-axis.

If the polynomial defining the curve has a degree d, any line cuts the curve in at most d points. Bézout's theorem asserts that this number is exactly d, if the points are searched in the projective plane over an algebraically closed field (for example the complex numbers), and counted with their multiplicity. The method of computation that follows proves again this theorem, in this simple case.

To compute the intersection of the curve defined by the polynomial p with the line of equation ax+by+c = 0, one solves the equation of the line for x (or for y if a = 0). Substituting the result in p, one gets a univariate equation q(y) = 0 (or q(x) = 0, if the equation of the line has been solved in y), each of whose roots is one coordinate of an intersection point. The other coordinate is deduced from the equation of the line. The multiplicity of an intersection point is the multiplicity of the corresponding root. There is an intersection point at infinity if the degree of q is lower than the degree of p; the multiplicity of such an intersection point at infinity is the difference of the degrees of p and q.

Tangent at a point

The tangent at a point (a, b) of the curve is the line of equation ${\displaystyle (x-a)p_x^{\prime }(a,b)+(y-b)p_y^{\prime }(a,b)=0}$, like for every differentiable curve defined by an implicit equation. In the case of polynomials, another formula for the tangent has a simpler constant term and is more symmetric:

$${\displaystyle x{p}_x^{\prime }(a,b)+y{p}_y^{\prime }(a,b)+p_{\mathrm{\infty }}^{\prime }(a,b)=0,}$$

where ${\displaystyle p_{\mathrm{\infty }}^{\prime }(x,y)=P_z^{\prime }(x,y,1)}$ is the derivative at infinity. The equivalence of the two equations results from Euler's homogeneous function theorem applied to P.

If ${\displaystyle p_x^{\prime }(a,b)=p_y^{\prime }(a,b)=0,}$ the tangent is not defined and the point is a singular point.

This extends immediately to the projective case: The equation of the tangent of at the point of projective coordinates (a:b:c) of the projective curve of equation P(x, y, z) = 0 is

$${\displaystyle x{P}_x^{\prime }(a,b,c)+y{P}_y^{\prime }(a,b,c)+z{P}_z^{\prime }(a,b,c)=0,}$$

and the points of the curves that are singular are the points such that

$${\displaystyle P_x^{\prime }(a,b,c)=P_y^{\prime }(a,b,c)=P_z^{\prime }(a,b,c)=0.}$$

(The condition P(a, b, c) = 0 is implied by these conditions, by Euler's homogeneous function theorem.)

Asymptotes

Every infinite branch of an algebraic curve corresponds to a point at infinity on the curve, that is a point of the projective completion of the curve that does not belong to its affine part. The corresponding asymptote is the tangent of the curve at that point. The general formula for a tangent to a projective curve may apply, but it is worth to make it explicit in this case.

Let ${\displaystyle p=p_d+\cdots +p_0}$ be the decomposition of the polynomial defining the curve into its homogeneous parts, where p_i is the sum of the monomials of p of degree i. It follows that $${\displaystyle P={}^{h}p=p_d+z{p}_{d-1}+\cdots +z^d{p}_0}$$ and $${\displaystyle P_z^{\prime }(a,b,0)=p_{d-1}(a,b).}$$

A point at infinity of the curve is a zero of p of the form (a, b, 0). Equivalently, (a, b) is a zero of p_d. The fundamental theorem of algebra implies that, over an algebraically closed field (typically, the field of complex numbers), p_d factors into a product of linear factors. Each factor defines a point at infinity on the curve: if bx − ay is such a factor, then it defines the point at infinity (a, b, 0). Over the reals, p_d factors into linear and quadratic factors. The irreducible quadratic factors define non-real points at infinity, and the real points are given by the linear factors. If (a, b, 0) is a point at infinity of the curve, one says that (a, b) is an asymptotic direction. Setting q = p_d the equation of the corresponding asymptote is $${\displaystyle x{q}_x^{\prime }(a,b)+y{q}_y^{\prime }(a,b)+p_{d-1}(a,b)=0.}$$

If ${\displaystyle q_x^{\prime }(a,b)=q_y^{\prime }(a,b)=0}$ and ${\displaystyle p_{d-1}(a,b)\ne 0,}$ the asymptote is the line at infinity, and, in the real case, the curve has a branch that looks like a parabola. In this case one says that the curve has a parabolic branch. If $${\displaystyle q_x^{\prime }(a,b)=q_y^{\prime }(a,b)=p_{d-1}(a,b)=0,}$$ the curve has a singular point at infinity and may have several asymptotes. They may be computed by the method of computing the tangent cone of a singular point.

Singular points

The singular points of a curve of degree d defined by a polynomial p(x,y) of degree d are the solutions of the system of equations: $${\displaystyle p_x^{\prime }(x,y)=p_y^{\prime }(x,y)=p(x,y)=0.}$$ In characteristic zero, this system is equivalent to $${\displaystyle p_x^{\prime }(x,y)=p_y^{\prime }(x,y)=p_{\mathrm{\infty }}^{\prime }(x,y)=0,}$$ where, with the notation of the preceding section, ${\displaystyle p_{\mathrm{\infty }}^{\prime }(x,y)=P_z^{\prime }(x,y,1).}$ The systems are equivalent because of Euler's homogeneous function theorem. The latter system has the advantage of having its third polynomial of degree d-1 instead of d.

Similarly, for a projective curve defined by a homogeneous polynomial P(x,y,z) of degree d, the singular points have the solutions of the system $${\displaystyle P_x^{\prime }(x,y,z)=P_y^{\prime }(x,y,z)=P_z^{\prime }(x,y,z)=0}$$ as homogeneous coordinates. (In positive characteristic, the equation ${\displaystyle P(x,y,z)}$ has to be added to the system.)

This implies that the number of singular points is finite as long as p(x,y) or P(x,y,z) is square free. Bézout's theorem implies thus that the number of singular points is at most (d − 1)², but this bound is not sharp because the system of equations is overdetermined. If reducible polynomials are allowed, the sharp bound is d(d − 1)/2, this value is reached when the polynomial factors in linear factors, that is if the curve is the union of d lines. For irreducible curves and polynomials, the number of singular points is at most (d − 1)(d − 2)/2, because of the formula expressing the genus in term of the singularities (see below). The maximum is reached by the curves of genus zero whose all singularities have multiplicity two and distinct tangents (see below).

The equation of the tangents at a singular point is given by the nonzero homogeneous part of the lowest degree in the Taylor series of the polynomial at the singular point. When one changes the coordinates to put the singular point at the origin, the equation of the tangents at the singular point is thus the nonzero homogeneous part of the lowest degree of the polynomial, and the multiplicity of the singular point is the degree of this homogeneous part.

Analytic structure

The study of the analytic structure of an algebraic curve in the neighborhood of a singular point provides accurate information of the topology of singularities. In fact, near a singular point, a real algebraic curve is the union of a finite number of branches that intersect only at the singular point and look either as a cusp or as a smooth curve.

Near a regular point, one of the coordinates of the curve may be expressed as an analytic function of the other coordinate. This is a corollary of the analytic implicit function theorem, and implies that the curve is smooth near the point. Near a singular point, the situation is more complicated and involves Puiseux series, which provide analytic parametric equations of the branches.

For describing a singularity, it is worth to translate the curve for having the singularity at the origin. This consists of a change of variable of the form ${\displaystyle X=x-a,Y=y-b,}$ where ${\displaystyle a,b}$ are the coordinates of the singular point. In the following, the singular point under consideration is always supposed to be at the origin.

The equation of an algebraic curve is ${\displaystyle f(x,y)=0,}$ where f is a polynomial in x and y. This polynomial may be considered as a polynomial in y, with coefficients in the algebraically closed field of the Puiseux series in x. Thus f may be factored in factors of the form ${\displaystyle y-P(x),}$ where P is a Puiseux series. These factors are all different if f is an irreducible polynomial, because this implies that f is square-free, a property which is independent of the field of coefficients.

The Puiseux series that occur here have the form $${\displaystyle P(x)=\sum _{n=n_0}^{\mathrm{\infty }}{a}_n{x}^{n/d},}$$ where d is a positive integer, and ⁠${\displaystyle n_0}$⁠ is an integer that may also be supposed to be positive, because we consider only the branches of the curve that pass through the origin. Without loss of generality, we may suppose that d is coprime with the greatest common divisor of the n such that ⁠${\displaystyle a_n\ne 0}$⁠ (otherwise, one could choose a smaller common denominator for the exponents).

Let ⁠${\displaystyle {\omega }_d}$⁠ be a primitive dth root of unity. If the above Puiseux series occurs in the factorization of ⁠${\displaystyle f(x,y)=0}$⁠, then the d series $${\displaystyle P_i(x)=\sum _{n=n_0}^{\mathrm{\infty }}{a}_n{\omega }_d^i{x}^{n/d}}$$ occur also in the factorization (a consequence of Galois theory). These d series are said conjugate, and are considered as a single branch of the curve, of ramification index d.

In the case of a real curve, that is a curve defined by a polynomial with real coefficients, three cases may occur. If none ⁠${\displaystyle P_i(x)}$⁠ has real coefficients, then one has a non-real branch. If some ⁠${\displaystyle P_i(x)}$⁠ has real coefficients, then one may choose it as ⁠${\displaystyle P_0(x)}$⁠. If d is odd, then every real value of x provides a real value of ⁠${\displaystyle P_0(x)}$⁠, and one has a real branch that looks regular, although it is singular if d > 1. If d is even, then ⁠${\displaystyle P_0(x)}$⁠ and ⁠${\displaystyle P_{d/2}(x)}$⁠ have real values, but only for x ≥ 0. In this case, the real branch looks as a cusp (or is a cusp, depending on the definition of a cusp that is used).

For example, the ordinary cusp has only one branch. If it is defined by the equation ${\displaystyle y^2-x^3=0,}$ then the factorization is ${\displaystyle (y-x^{3/2})(y+x^{3/2});}$ the ramification index is 2, and the two factors are real and define each a half branch. If the cusp is rotated, it equation becomes ${\displaystyle y^3-x^2=0,}$ and the factorization is ${\displaystyle (y-x^{2/3})(y-j^2{x}^{2/3})(y-(j^2{)}^2{x}^{2/3}),}$ with ${\displaystyle j=(1+\sqrt{-3})/2}$ (the coefficient ⁠${\displaystyle (j^2{)}^2}$⁠ has not been simplified to j for showing how the above definition of ⁠${\displaystyle P_i(x)}$⁠ is specialized). Here the ramification index is 3, and only one factor is real; this shows that, in the first case, the two factors must be considered as defining the same branch.

Non-plane algebraic curves

An algebraic curve is an algebraic variety of dimension one. This implies that an affine curve in an affine space of dimension n is defined by, at least, n − 1 polynomials in n variables. To define a curve, these polynomials must generate a prime ideal of Krull dimension 1. This condition is not easy to test in practice. Therefore, the following way to represent non-plane curves may be preferred.

Let ${\displaystyle f,g_0,g_3,\dots ,g_n}$ be n polynomials in two variables x₁ and x₂ such that f is irreducible. The points in the affine space of dimension n such whose coordinates satisfy the equations and inequations

$${\displaystyle \begin{array}{cc}& f(x_1,x_2)=0\\ & g_0(x_1,x_2)\ne 0\\ x_3& =\frac{g_3(x_1,x_2)}{g_0(x_1,x_2)}\\ & ⋮\\ x_n& =\frac{g_n(x_1,x_2)}{g_0(x_1,x_2)}\end{array}}$$

are all the points of an algebraic curve in which a finite number of points have been removed. This curve is defined by a system of generators of the ideal of the polynomials h such that it exists an integer k such ${\displaystyle g_0^{k}h}$ belongs to the ideal generated by ${\displaystyle f,x_3{g}_0-g_3,\dots ,x_n{g}_0-g_n}$. This representation is a birational equivalence between the curve and the plane curve defined by f. Every algebraic curve may be represented in this way. However, a linear change of variables may be needed in order to make almost always injective the projection on the two first variables. When a change of variables is needed, almost every change is convenient, as soon as it is defined over an infinite field.

This representation allows us to deduce easily any property of a non-plane algebraic curve, including its graphical representation, from the corresponding property of its plane projection.

For a curve defined by its implicit equations, above representation of the curve may easily deduced from a Gröbner basis for a block ordering such that the block of the smaller variables is (x₁, x₂). The polynomial f is the unique polynomial in the base that depends only of x₁ and x₂. The fractions g_i/g₀ are obtained by choosing, for i = 3, ..., n, a polynomial in the basis that is linear in x_i and depends only on x₁, x₂ and x_i. If these choices are not possible, this means either that the equations define an algebraic set that is not a variety, or that the variety is not of dimension one, or that one must change of coordinates. The latter case occurs when f exists and is unique, and, for i = 3, ..., n, there exist polynomials whose leading monomial depends only on x₁, x₂ and x_i.

Algebraic function fields

The study of algebraic curves can be reduced to the study of irreducible algebraic curves: those curves that cannot be written as the union of two smaller curves. Up to birational equivalence, the irreducible curves over a field F are categorically equivalent to algebraic function fields in one variable over F. Such an algebraic function field is a field extension K of F that contains an element x 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. For example, 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 subset of R² has no points. The equation x² + y² = −1 does define an irreducible algebraic curve over R in the scheme sense (an integral, separated one-dimensional schemes of finite type over R). In this sense, the one-to-one correspondence between irreducible algebraic curves over F (up to birational equivalence) and algebraic function fields in one variable over F holds in general.

Two curves can be birationally equivalent (i.e. have isomorphic function fields) without being isomorphic as curves. The situation becomes easier when dealing with nonsingular curves, i.e. those that lack any singularities. Two nonsingular projective curves over a field are isomorphic if and only if their function fields are isomorphic.

Tsen's theorem is about the function field of an algebraic curve over an algebraically closed field.

Complex curves and real surfaces

A complex projective algebraic curve resides in n-dimensional complex projective space CPⁿ. This has complex dimension n, but topological dimension, as a real manifold, 2n, and is compact, connected, and orientable. An algebraic curve over C likewise has topological dimension two; in other words, it is a surface.

The topological genus of this surface, that is the number of handles or donut holes, is equal to the geometric genus of the algebraic curve that may be computed by algebraic means. In short, if one consider a plane projection of a nonsingular curve that has degree d and only ordinary singularities (singularities of multiplicity two with distinct tangents), then the genus is (d − 1)(d − 2)/2 − k, where k is the number of these singularities.

Compact Riemann surfaces

A Riemann surface is a connected complex analytic manifold of one complex dimension, which makes it a connected real manifold of two dimensions. It is compact if it is compact as a topological space.

There is a triple equivalence of categories between the category of smooth irreducible projective algebraic curves over C (with non-constant regular maps as morphisms), the category of compact Riemann surfaces (with non-constant holomorphic maps as morphisms), and the opposite of the category of algebraic function fields in one variable over C (with field homomorphisms that fix C as morphisms). This means that in studying these three subjects we are in a sense studying one and the same thing. It allows complex analytic methods to be used in algebraic geometry, and algebraic-geometric methods in complex analysis and field-theoretic methods to be used in both. This is characteristic of a much wider class of problems in algebraic geometry.

See also algebraic geometry and analytic geometry for a more general theory.

Singularities

Using the intrinsic concept of tangent space, points P on an algebraic curve C are classified as smooth (synonymous: non-singular), or else singular. Given n − 1 homogeneous polynomials in n + 1 variables, we may find the Jacobian matrix as the (n − 1)×(n + 1) matrix of the partial derivatives. If the rank of this matrix is n − 1, then the polynomials define an algebraic curve (otherwise they define an algebraic variety of higher dimension). If the rank remains n − 1 when the Jacobian matrix is evaluated at a point P on the curve, then the point is a smooth or regular point; otherwise it is a singular 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×(n + 1) matrix is zero, that is, where $${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }x}(P)=\frac{\mathrm{\partial }f}{\mathrm{\partial }y}(P)=\frac{\mathrm{\partial }f}{\mathrm{\partial }z}(P)=0.}$$

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.

Similarly, for an affine algebraic curve defined by a single polynomial equation f(x,y) = 0, then the singular points are precisely the points P of the curve where the rank of the 1×n Jacobian matrix is zero, that is, where

$${\displaystyle f(P)=\frac{\mathrm{\partial }f}{\mathrm{\partial }x}(P)=\frac{\mathrm{\partial }f}{\mathrm{\partial }y}(P)=0.}$$

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.

Classification of singularities

Singular points include multiple points where the curve crosses over itself, and also various 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 smooth or non-singular. Commonly, this definition is understood over an algebraically closed field and for a curve C in a projective space (i.e., complete in the sense of algebraic geometry). For example, the plane curve of equation ${\displaystyle y-x^3=0}$ is considered as singular, as having a singular point (a cusp) at infinity.

In the remainder of this section, one considers a plane curve C defined as the zero set of a bivariate polynomial f(x, y). Some of the results, but not all, may be generalized to non-plane curves.

The singular points are classified by means of several invariants. The multiplicity m is defined as the maximum integer such that the derivatives of f to all orders up to m – 1 vanish (also the minimal intersection number between the curve and a straight line at P). Intuitively, a singular point has delta invariant δ if it concentrates δ ordinary double points at P. To make this precise, the blow up process produces so-called infinitely near points, and summing m(m − 1)/2 over the infinitely near points, where m is their multiplicity, produces δ. For an irreducible and reduced curve and a point P we can define δ algebraically as the length of ${\displaystyle {\tilde{\mathcal{O}}}_P/{\mathcal{O}}_P}$ where ${\displaystyle {\mathcal{O}}_P}$ is the local ring at P and ${\displaystyle {\tilde{\mathcal{O}}}_P}$ is its integral closure.

The Milnor number μ of a singularity is the degree of the mapping ⁠grad f(x,y)/|grad f(x,y)|⁠ on the small sphere of radius ε, in the sense of the topological degree of a continuous mapping, where grad f is the (complex) gradient vector field of f. It is related to δ and r by the Milnor–Jung formula,

μ = 2δ − r + 1.

Here, the branching number r of P is the number of locally irreducible branches at P. For example, r = 1 at an ordinary cusp, and r = 2 at an ordinary double point. The multiplicity m is at least r, and that P is singular if and only if m is at least 2. Moreover, δ is at least m(m-1)/2.

Computing the delta invariants of all of the singularities allows the genus g of the curve to be determined; if d is the degree, then

$${\displaystyle g=\frac{1}{2}(d-1)(d-2)-\sum _P{\delta }_P,}$$

where the sum is taken over all singular points P of the complex projective plane curve. It is called the genus formula.

Assign the invariants [m, δ, r] to a singularity, where m is the multiplicity, δ is the delta-invariant, and r is the branching number. Then an ordinary cusp is a point with invariants [2,1,1] and an ordinary double point is a point with invariants [2,1,2], and an ordinary m-multiple point is a point with invariants [m, m(m − 1)/2, m].

Examples of curves

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; accordingly, we may identify the function field of the curve 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 of all real algebraic functions defined on the real algebraic variety x² + y² = −1 is a field of genus zero which is not a rational function field.

Concretely, a rational curve embedded in an affine space of dimension n over F can be parameterized (except for isolated exceptional points) by means of n rational functions of a single parameter t; by reducing these rational functions to the same denominator, the n+1 resulting polynomials define a polynomial parametrization of the projective completion of the curve in the projective space. An example is the rational normal curve, where all these polynomials are monomials.

Any conic section defined over F with a rational point in F is a rational curve. It can be parameterized by drawing a line with slope t through the rational point, and an intersection with the plane quadratic curve; this gives a polynomial with F-rational coefficients and one F-rational root, hence the other root is F-rational (i.e., 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

$${\displaystyle x=\frac{1-t^2}{1+t+t^2}.}$$

Then the equation for y is

$${\displaystyle y=t(x+1)=\frac{t(t+2)}{1+t+t^2},}$$

which defines a rational parameterization of the ellipse and hence shows the ellipse is a rational curve. All points of the ellipse are given, except for (−1,1), which corresponds to t = ∞; the entire curve is parameterized therefore by the real projective line.

Such a rational parameterization may be considered in the projective space by equating the first projective coordinates to the numerators of the parameterization and the last one to the common denominator. As the parameter is defined in a projective line, the polynomials in the parameter should be homogenized. For example, the projective parameterization of the above ellipse is

$${\displaystyle X=U^2-T^2,Y=T(T+2U),Z=T^2+TU+U^2.}$$

Eliminating T and U between these equations we get again the projective equation of the ellipse $${\displaystyle X^2+XY+Y^2=Z^2,}$$ which may be easily obtained directly by homogenizing the above equation.

Many of the curves on Wikipedia's list of curves are rational and hence have similar rational parameterizations.

Rational plane curves

Rational plane curves are rational curves embedded into ${\displaystyle {\mathbb{P}}^2}$. Given generic sections ${\displaystyle s_1,s_2,s_3\in \mathrm{\Gamma }({\mathbb{P}}^1,\mathcal{O}(d))}$ of degree ${\displaystyle d}$ homogeneous polynomials in two coordinates, ${\displaystyle x,y}$, there is a map$${\displaystyle s:{\mathbb{P}}^1\to {\mathbb{P}}^2}$$ given by $${\displaystyle s([x:y])=[s_1([x:y]):s_2([x:y]):s_3([x:y])]}$$defining a rational plane curve of degree ${\displaystyle d}$. There is an associated moduli space ${\displaystyle \mathcal{M}={\overline{\mathcal{M}}}_{0,0}({\mathbb{P}}^2,d\cdot [H])}$ (where ${\displaystyle [H]}$ is the hyperplane class) parametrizing all such stable curves. A dimension count can be made to determine the moduli spaces dimension: There are ${\displaystyle d+1}$ parameters in ${\displaystyle \mathrm{\Gamma }({\mathbb{P}}^1,\mathcal{O}(d))}$ giving ${\displaystyle 3d+3}$ parameters total for each of the sections. Then, since they are considered up to a projective quotient in ${\displaystyle {\mathbb{P}}^2}$ there is ${\displaystyle 1}$ less parameter in ${\displaystyle \mathcal{M}}$. Furthermore, there is a three dimensional group of automorphisms of ${\displaystyle {\mathbb{P}}^1}$, hence ${\displaystyle \mathcal{M}}$ has dimension ${\displaystyle 3d+3-1-3=3d-1}$. This moduli space can be used to count the number ${\displaystyle N_d}$ of degree ${\displaystyle d}$ rational plane curves intersecting ${\displaystyle 3d-1}$ points using Gromov–Witten theory. It is given by the recursive relation$${\displaystyle N_d=\sum _{d_A+d_B=d}{N}_{d_A}{N}_{d_B}{d}_A^2{d}_B\left(d_B(\genfrac{}{}{0ex}{}{3d-4}{3d_A-2})-d_A(\genfrac{}{}{0ex}{}{3d-4}{3d_A-1})\right)}$$where ${\displaystyle N_1=N_2=1}$.

Elliptic curves

An elliptic curve may be defined as any curve of genus one with a rational point: a common model is a nonsingular cubic curve, which suffices to model any genus one curve. In this model the distinguished point is commonly taken to be an inflection point at infinity; this amounts to requiring that the curve can be written in Tate-Weierstrass form, which in its projective version is $${\displaystyle y^{2}z+a_{1}xyz+a_{3}y{z}^2=x^3+a_2{x}^{2}z+a_{4}x{z}^2+a_6{z}^3.}$$

If the characteristic of the field is different from 2 and 3, then a linear change of coordinates allows putting ${\displaystyle a_1=a_2=a_3=0,}$ which gives the classical Weierstrass form $${\displaystyle y^2=x^3+px+q.}$$

Elliptic curves carry the structure of an abelian group with the distinguished point as the identity of the group law. In a plane cubic model three points sum to zero in the group if and only if they are collinear. For an elliptic curve defined over the complex numbers the group is isomorphic to the additive group of the complex plane modulo the period lattice of the corresponding elliptic functions.

The intersection of two quadric surfaces is, in general, a nonsingular curve of genus one and degree four, and thus an elliptic curve, if it has a rational point. In special cases, the intersection either may be a rational singular quartic or is decomposed in curves of smaller degrees which are not always distinct (either a cubic curve and a line, or two conics, or a conic and two lines, or four lines).

Curves of genus greater than one

Curves of genus greater than one differ markedly from both rational and elliptic curves. Such curves defined over the rational numbers, by Faltings' theorem, can have only a finite number of rational points, and they may be viewed as having a hyperbolic geometry structure. Examples are the hyperelliptic curves, the Klein quartic curve, and the Fermat curve xⁿ + yⁿ = zⁿ when n is greater than three. Also projective plane curves in ${\displaystyle {\mathbb{P}}^2}$ and curves in ${\displaystyle {\mathbb{P}}^1\times {\mathbb{P}}^1}$ provide many useful examples.

Projective plane curves

Plane curves ${\displaystyle C\subset {\mathbb{P}}^2}$ of degree ${\displaystyle k}$, which can be constructed as the vanishing locus of a generic section ${\displaystyle s\in \mathrm{\Gamma }({\mathbb{P}}^2,\mathcal{O}(k))}$, have genus $${\displaystyle \frac{(k-1)(k-2)}{2}}$$ which can be computed using coherent sheaf cohomology. Here's a brief summary of the curves' genera relative to their degree

For example, the curve ${\displaystyle x^4+y^4+z^4}$ defines a curve of genus ${\displaystyle 3}$ which is smooth since the differentials ${\displaystyle 4x^3,4y^3,4z^3}$ have no common zeros with the curve. A non-example of a generic section is the curve ${\displaystyle x(x^2+y^2+z^2)}$ which, by Bezout's theorem, should intersect at most ${\displaystyle 2}$ points; it is the union of two rational curves ${\displaystyle C_1\cup C_2}$ intersecting at two points. Note ${\displaystyle C_1}$ is given by the vanishing locus of ${\displaystyle x}$ and ${\displaystyle C_2}$ is given by the vanishing locus of ${\displaystyle x^2+y^2+z^2}$. These can be found explicitly: a point lies in both if ${\displaystyle x=0}$. So the two solutions are the points ${\displaystyle [0:y:z]}$ such that ${\displaystyle y^2+z^2=0}$, which are ${\displaystyle [0:1:-\sqrt{-1}]}$ and ${\displaystyle [0:1:\sqrt{-1}]}$.

Curves in product of projective lines

Curve ${\displaystyle C\subset {\mathbb{P}}^1\times {\mathbb{P}}^1}$ given by the vanishing locus of ${\displaystyle s\in \mathrm{\Gamma }({\mathbb{P}}^1\times {\mathbb{P}}^1,\mathcal{O}(a,b))}$, for ${\displaystyle a,b\ge 2}$, give curves of genus$${\displaystyle ab-a-b+1}$$which can be checked using coherent sheaf cohomology. If ${\displaystyle a=2}$, then they define curves of genus ${\displaystyle 2b-2-b+1=b-1}$, hence a curve of any genus can be constructed as a curve in ${\displaystyle {\mathbb{P}}^1\times {\mathbb{P}}^1}$. Their genera can be summarized in the table

and for ${\displaystyle a=3}$, this is