Algebraic curve

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.

The tangent at a point (a, b) of the curve is the line of equation ${\displaystyle (x-a)p'_{x}(a,b)+(y-b)p'_{y}(a,b)=0}$, 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:

$${\displaystyle xp'_{x}(a,b)+yp'_{y}(a,b)+p'_{\infty }(a,b)=0,}$$

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

If ${\displaystyle p'_{x}(a,b)=p'_{y}(a,b)=0,}$ 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

$${\displaystyle xP'_{x}(a,b,c)+yP'_{y}(a,b,c)+zP'_{z}(a,b,c)=0,}$$

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

$${\displaystyle P'_{x}(a,b,c)=P'_{y}(a,b,c)=P'_{z}(a,b,c)=0.}$$

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

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

$${\displaystyle P={^{h}p}=p_{d}+zp_{d-1}+\cdots +z^{d}p_{0}}$$

and

$${\displaystyle P'_{z}(a,b,0)=p_{d-1}(a,b).}$$

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

$${\displaystyle xq'_{x}(a,b)+yq'_{y}(a,b)+p_{d-1}(a,b)=0.}$$

If ${\displaystyle q'_{x}(a,b)=q'_{y}(a,b)=0}$ and ${\displaystyle p_{d-1}(a,b)\neq 0,}$ 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

$${\displaystyle q'_{x}(a,b)=q'_{y}(a,b)=p_{d-1}(a,b)=0,}$$

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.

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:

$${\displaystyle p'_{x}(x,y)=p'_{y}(x,y)=p(x,y)=0.}$$

In characteristic zero, this system is equivalent to

$${\displaystyle p'_{x}(x,y)=p'_{y}(x,y)=p'_{\infty }(x,y)=0,}$$

where, with the notation of the preceding section, ${\displaystyle p'_{\infty }(x,y)=P'_{z}(x,y,1).}$ 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

$${\displaystyle P'_{x}(x,y,z)=P'_{y}(x,y,z)=P'_{z}(x,y,z)=0}$$

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.

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.

x³ = y²

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 ${\displaystyle {\widetilde {{\mathcal {O}}_{P}}}/{\mathcal {O}}_{P}}$ where ${\displaystyle {\mathcal {O}}_{P}}$ is the local ring at P and ${\displaystyle {\widetilde {{\mathcal {O}}_{P}}}}$ is its integral closure.[1]

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,

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].

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.

x² + xy + y² = 1

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

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

Eliminating T and U between these equations we get again the projective equation of the ellipse

$${\displaystyle X^{2}+X\,Y+Y^{2}=Z^{2},}$$

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 are rational curves embedded into ${\displaystyle \mathbb {P} ^{2}}$. Given generic sections ${\displaystyle s_{1},s_{2},s_{3}\in \Gamma (\mathbb {P} ^{1},{\mathcal {O}}(d))}$ 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 ${\displaystyle d}$.[2] 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 ${\displaystyle \Gamma (\mathbb {P} ^{1},{\mathcal {O}}(d))}$ 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.[3] It is given by the recursive relation

$${\displaystyle N_{d}=\sum _{d_{A}+d_{B}=d}N_{d_{A}}N_{d_{B}}d_{A}^{2}d_{B}\left(d_{B}{\binom {3d-4}{3d_{A}-2}}-d_{A}{\binom {3d-4}{3d_{A}-1}}\right)}$$

where ${\displaystyle N_{1}=N_{2}=1}$.

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

$${\displaystyle y^{2}z+a_{1}xyz+a_{3}yz^{2}=x^{3}+a_{2}x^{2}z+a_{4}xz^{2}+a_{6}z^{3}.}$$

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 differ markedly from both rational and elliptic curves. Such curves defined over the rational numbers, by Faltings's 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.

Plane curves ${\displaystyle C\subset \mathbb {P} ^{2}}$ of degree ${\displaystyle k}$, which can be constructed as the vanishing locus of a generic section ${\displaystyle s\in \Gamma (\mathbb {P} ^{2},{\mathcal {O}}(k))}$, has 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 Bezouts theorem, should intersect at most ${\displaystyle 2}$ points, 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 ${\displaystyle [0:1:{\sqrt {-1}}]}$.

Curve ${\displaystyle C\subset \mathbb {P} ^{1}\times \mathbb {P} ^{1}}$ given by the vanishing locus of ${\displaystyle s\in \Gamma (\mathbb {P} ^{1}\times \mathbb {P} ^{1},{\mathcal {O}}(a,b))}$, for ${\displaystyle a,b\geq 2}$, 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

bidegree	${\displaystyle (2,2)}$	${\displaystyle (2,3)}$	${\displaystyle (2,4)}$	${\displaystyle (2,5)}$
genus	1	2	3	4

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

bidegree	${\displaystyle (3,2)}$	${\displaystyle (3,3)}$	${\displaystyle (3,4)}$	${\displaystyle (3,5)}$
genus	2	4	6	8