Algebraic curve

An algebraic curve is a one-dimensional algebraic variety , so it can be described by a polynomial equation . An important special case are the plane algebraic curves, that is, algebraic curves that run in the affine or projective plane .

Historically, the preoccupation with algebraic curves began in antiquity with the investigation of straight lines and conic sections . In the 17th century they became the subject of analysis within the framework of analytic geometry, and Isaac Newton systematically dealt with cubics. The preoccupation with them reached a high point in the 19th century through the treatment in the context of projective geometry (among others August Ferdinand Möbius , Julius Plücker ). The point in infinity is systematically taken into account. The natural approach is based on the fundamental theorem of algebra over complex numbers, and the classical theory was placed on a new basis through the connection to Riemann surfaces - which are in complex curves - discovered by Bernhard Riemann . In number theory (arithmetic geometry) curves over bodies K other than real and complex numbers and over rings are also considered.

Algebraic curves are among the simplest objects of algebraic geometry , in which they are treated with purely algebraic methods and not with methods of analysis. Higher-dimensional varieties of algebraic geometry are, for example, algebraic surfaces . You can also examine algebraic curves in the context of complex analysis.

In the following, the terms used are explained in the simplest case of plane algebraic curves. You can define algebraic curves as intersection curves of algebraic surfaces in more than two dimensions. Their classification in three dimensions according to grade d and gender g was the subject of two large papers on the Steiner Prize in the 1880s by Max Noether and Georges Henri Halphen , the evidence and work of which, however, was still incomplete. The object of the classification is to determine which pairs (d, g) exist. Algebraic curves can always be embedded in the three-dimensional projective space, so that the consideration of two and three spatial dimensions is sufficient.

Definition and important properties

A plane algebraic curve over a body is defined by a non-constant polynomial in two variables and whose coefficients come from . Two polynomials are identified with one another if one is derived from the other by multiplying by a number other than zero . The degree of the polynomial is called the degree of the curve. ${\ displaystyle K}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle K}$ ${\ displaystyle K}$

This definition is based on the following motivation: If there is such a polynomial, the set of zeros can be used ${\ displaystyle f}$

{\ displaystyle V (f) = \ {(x, y) \ in K ^ {2} | f (x, y) = 0 \}}

look at the plane . This set often represents an object that one would clearly describe as a curve, for example ${\ displaystyle K ^ {2}}$

{\ displaystyle \ {(x, y) \ in \ mathbb {R} ^ {2} | x ^ {2} + y ^ {2} -1 = 0 \}}

a circle. A constant factor does not play a role in the definition of either. ${\ displaystyle V (f)}$

If the field is algebraically closed , the polynomial can be recovered from the set using Hilbert's zero theorem , if it breaks down into a number of different irreducible factors. In this case there is no need to make a strict distinction between the defining polynomial and its set of zeros. ${\ displaystyle K}$ ${\ displaystyle V (f)}$ ${\ displaystyle f}$

If, on the other hand, the body is not algebraically closed, it does not always represent a curve in the plane ${\ displaystyle K}$ ${\ displaystyle V (f)}$

{\ displaystyle \ {(x, y) \ in \ mathbb {R} ^ {2} | x ^ {2} + y ^ {2} + 1 = 0 \}}

and

{\ displaystyle \ {(x, y) \ in \ mathbb {R} ^ {2} | x ^ {2} + y ^ {2} = 0 \}}

in reality the empty set or a point is defined, neither of which is one-dimensional objects. These polynomials only generate curves when they are complex : a circle and an intersecting pair of lines.

One therefore says that a curve has a property geometric , if the set has this property over the algebraic closure of . ${\ displaystyle V (f)}$ ${\ displaystyle K}$

In a more abstract way, an algebraic curve can also be defined as a one-dimensional separated algebraic scheme over a body. Often other requirements such as geometric simplicity or irreducibility are included in the definition.

Irreducibility

If the defining polynomial is reducible , i.e. if it can be broken down into two nontrivial factors, then the curve can also be broken down into two independent components. For example, the polynomial is reducible because it can be broken down into the factors and . The curve defined by therefore consists of two straight lines. ${\ displaystyle f (x, y) = xy}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle f}$

In the case of an irreducible polynomial, the curve cannot be broken down, which is then also called irreducible.

Singularities

Neil's parabola with apex in the zero point

Cartesian sheet with a single colon in the zero point

Normally, exactly one tangent can be drawn to the curve at each point of the algebraic curve. In this case the point is called smooth or non-singular . However, it can also happen that the curve has a self-intersection or a peak at one or more points. In the first case the curve has two or more tangents at this point, in the second these tangents coincide to form a multiple tangent.

Examples of such singular points can be found in Neil's parabola with the equation , this has a tip at the zero point. A colon , i.e. a point that is traversed twice in different directions, can be found in the Cartesian leaf , which is given by . ${\ displaystyle y ^ {2} = x ^ {3}}$ ${\ displaystyle x ^ {3} + y ^ {3} -3xy = 0}$

Projective curves

It is often advantageous to consider algebraic curves not in the affine but in the projective plane . This can be described by so-called homogeneous coordinates , whereby and are not allowed to be at the same time and two points are considered to be the same if they emerge from one another by multiplying them by one of different numbers. For so true . In order to define algebraic curves in the projective, one needs polynomials in three variables and . If you were to use any polynomials here, big problems would arise due to the fact that the representation of the points is not unambiguous: The points and are the same, but the polynomial disappears in the first representation, but not in the second. ${\ displaystyle [x: y: z]}$ ${\ displaystyle x, y}$ ${\ displaystyle z}$ ${\ displaystyle 0}$ ${\ displaystyle 0}$ ${\ displaystyle \ lambda \ neq 0}$ ${\ displaystyle [x: y: z] = [\ lambda \ cdot x: \ lambda \ cdot y: \ lambda \ cdot z]}$ ${\ displaystyle x, y}$ ${\ displaystyle z}$ ${\ displaystyle \ left [1: 1: 1 \ right]}$ ${\ displaystyle \ left [2: 2: 2 \ right]}$ ${\ displaystyle f (x, y, z) = x ^ {2} -y}$

This problem does not arise if you limit yourself to homogeneous polynomials : Although the values that the polynomial assumes in different representations can also differ here, the property of whether the polynomial has a zero depends on the choice of representation of the Point independent .

In order to find the corresponding projective curve for an affine curve, one homogenizes the defining polynomial: In each term one inserts such a large -potency that a homogeneous polynomial results: The equation becomes . ${\ displaystyle z}$ ${\ displaystyle x ^ {2} -y + 1 = 0}$ ${\ displaystyle x ^ {2} -yz + z ^ {2} = 0}$

The reverse process is called dehomogenization. Here you insert the value into the homogeneous polynomial for (or a variable, if you want to dehomogenize according to another variable) . ${\ displaystyle z}$ ${\ displaystyle 1}$

Intersections of two curves

For example, if you consider a straight line and a parabola, you generally expect two points in common. Due to various circumstances, less common points can arise, but these cases can all be avoided by special requirements or definitions:

The straight line and the parabola cannot have any point of intersection; this can be avoided by assuming that the body on which it is based is algebraically closed.
The straight line can run vertically upwards through the vertex of the parabola and thus only have one point in common with it. This does not occur when you are in the projective plane, here the straight line and the parabola have another point of intersection at infinity in this case.
The straight line can be a tangent to the parabola. In this case, too, there is only one common point. However, with a suitable definition of intersection multiplicity , this intersection can be counted twice.

Under the above assumptions, Bézout's theorem applies : The number of common points of two projective plane algebraic curves of degree n and m without common components is nm .

Examples of algebraic curves

Curves sorted by degree

The plane algebraic curves of degree 1 are exactly the straight lines. The equations and describe, for example, the coordinate axes, the equation or, equivalently, the first bisector.

{\ displaystyle x = 0}

{\ displaystyle y = 0}

{\ displaystyle x = y}

{\ displaystyle xy = 0}

The plane algebraic curves of degree 2 are exactly the conic sections, including the unit circle described by and the normal parabola with the formula . The reducible curves are the degenerate conic sections. ${\ displaystyle x ^ {2} + y ^ {2} = 1}$ ${\ displaystyle y = x ^ {2}}$

At degree 3, irreducible curves with singularities appear for the first time, for example Neil's parabola with the equation and the Cartesian leaf given by . The elliptic curves are also important examples of plane algebraic curves of degree 3. ${\ displaystyle y ^ {3} = x ^ {2}}$ ${\ displaystyle x ^ {3} + y ^ {3} -3xy = 0}$

A spiral curve is an algebraic curve of degree 4. Special cases are the Cassinian curve , Bernoulli's lemniscate and Booth's lemniscate .

Curves sorted by gender

Gender 0 curves are rational curves .

Gender 1 curves are elliptic curves .

The curves of gender at least 2 include hyperelliptic curves , the Klein quartic, and the Fermat curve . ${\ displaystyle x ^ {3} y + y ^ {3} z + z ^ {3} x = 0}$ ${\ displaystyle x ^ {n} + y ^ {n} -z ^ {n} = 0}$

Dual curve

A curve can also be described by its tangents instead of its points . An important problem in this context is the question of how many tangents can “usually” be placed from a point not lying on the curve to a curve of the nth order. This number is called the class of the curve. This class is the same for such a curve without singular points (such as colons or peaks) . Each colon reduces the class by 2 and each point by 3. This is a main statement of Plücker's formulas , which also deal with the number of turning points and double tangents. For this, the basic field must be algebraically closed. ${\ displaystyle n (n-1)}$

For example, a third-order curve free of singularities is of the 6th class, if it has a colon it is of the fourth, and if it has a peak, of the third class.

In the homogeneous case, straight lines, including tangents, have an equation of the form , whereby and not all may vanish and may be multiplied by any number other than 0. This allows you to assign the point to this straight line . A set of points in the projective plane is thus obtained from the set of tangents to a given curve. It turns out that this set is itself an algebraic curve, the so-called dual curve . ${\ displaystyle ax + by + cz = 0}$ ${\ displaystyle a, b}$ ${\ displaystyle c}$ ${\ displaystyle [a: b: c]}$

The following terms are dual to each other:

Curve point and curve tangent
Colon and double tangent
Turning point and tip
Order and class

The dual curve of the dual curve is again the original curve.

literature

Egbert Brieskorn , Horst Knörrer : level algebraic curves, Birkhäuser 1981
Gerd Fischer : Plane algebraic curves, Vieweg 1994
William Fulton : Algebraic Curves, an introduction to algebraic geometry, Benjamin 1969, online edition 2008, pdf
Klaus Hulek : Elementary Algebraic Geometry. Vieweg, Braunschweig / Wiesbaden 2000

Web links

Wikiversity: A Lecture on Algebraic Curves - Course Materials

Barth, level algebraic curves, script, University of Marburg, pdf

Individual evidence

↑ for example presented in: Griffiths, Harris, Principles of Algebraic Geometry, Wiley 1978
↑ Hartshorne, Algebraic Geometry, Springer 1977, p. 349ff
↑ Hartshorne, Algebraic Geometry, Springer, p. 307

[1] r example presented in: Griffiths, Harris, Principles of Algebraic Geometry, Wiley 1978

[2] Hartshorne, Algebraic Geometry, Springer 1977, p. 349ff

[3] Hartshorne, Algebraic Geometry, Springer, p. 307