Algebraic curve

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

Historically, the study of algebraic curves begins early with the investigation of straight lines and conic sections, and reached its peak in the 19th century by August Ferdinand Möbius and Julius Plücker, and later by Bernhard Riemann.

Definition and important features

A plane algebraic curve defined over a field K by a non-constant polynomial in two variables x and y, the coefficients of which are from k. Two polynomials are identified with each other, when the one seen by multiplying a non-zero number of k from the other. The degree of the polynomial is referred to as the degree of curve.

Consider in the plane. This amount often represents an object that one would describe as vividly curve, for example,

A circle. In the definition of a constant factor does not play a role.

If the body is completed k algebraically, so you can recover after the hilbert between Nullstellensatz out of the crowd, the polynomial f, if this falls into loud several irreducible factors. In this case, therefore, need not be a sharp distinction between the defining polynomial and its zero set.

If the field k is not algebraically closed, however, it is not always a curve in the plane dar. Thus, by

In the real case the empty set or a point defined, neither one -dimensional objects. Only in the complexes generate these polynomials curves: a circle and a pair of straight lines to be cut.

It is said, therefore, a curve has a characteristic geometric, if the amount of this property has on the algebraic degree of k.

Abstract, one can define an algebraic curve as a one-dimensional -separated algebraic scheme over a field. Other conditions such as geometric minimalism or irreducibility yet are often included in the definition.

Irreducibility

Is the defining polynomial is reducible, so if it can be decomposed into two non-trivial factors, including the curve can be decomposed into two independent components. For example, the polynomial is reducible as it can be decomposed into the factors x and y. Thus defined by f curve consists of two straight lines.

When an irreducible polynomial, the curve can not be decomposed, which is then also called irreducible.

Singularities

Normally, enables any point of the algebraic curve exactly create a tangent to the curve. In this case, we call the point smooth or nonsingular. However, it may also be the case that the curve in one or more points has a self- cut or a peak. In the first case, the curve in this point has two or more tangent lines tangent to this at the second fall of multiple tangent together.

Examples of such singular points can be found in Neil 's parabola with the equation, this has a peak at zero. A double point, ie a point which is traversed twice in different directions, one finds the Cartesian leaf which is given by.

Projective curves

It is often advantageous to consider not affine algebraic curves, but in the projective plane. These can be described by the so-called homogeneous coordinates, where x, y and z can not be 0 simultaneously, and two points are the same as perceived if they arise by multiplying a different number from 0 apart. So for true. To define algebraic curves in projective, ie y is needed polynomials in three variables x, and such dignity to use any polynomials here, then big problems arise due to the fact that the representation of the points is not unique: So are the points [ 1:1:1 ] and [ 2:2:2 ] are equal, but the polynomial disappears at the first presentation, but not for the second.

This problem does not occur if we restrict ourselves to homogeneous polynomials: It is true that here also the values ​​taken by the polynomial at various illustrations are different, but the property to determine whether the polynomial has a zero, is the choice of the representation of the point independently.

In order to find an affine curve, the associated projective curve, homogenised, the defining polynomial: In each term you add a such a large z- potency that a homogeneous polynomial gives: From the equation is so.

The reverse process is referred to as Dehomogenisieren. You can place in the homogeneous polynomial of z ( or a variable, if you want dehomogenisieren by another variable ) the value 1.

Sections of two curves

Consider for example a straight line and a parabola, one expects two common points in general. Through various circumstances even less common points can occur, these cases can, however, all deal with special conditions or definitions:

• The straight line and the parabola can have no intersection, it bypasses one by one assumes that the underlying body is algebraically closed.
• The straight line can pass through the vertex of the parabola vertically upwards and thus have only one point in common with her. This does not occur if you are in the projective plane, here are straight and parabolic in this case a further intersection at infinity.
• The straight line can be a tangent to the parabola. Also in this case there is only one common point. With an appropriate definition of intersection multiplicity, however, this intersection can be counted twice.

Under the above conditions, the set of Bézout: The number of the common points of two projective plane algebraic curves of degree n and m with no common components is nm

Examples of algebraic curves

• The planar algebraic curves of degree 1 are precisely the straight line. Equations and example describing the coordinate axes, the equation, or equivalent, the first bisector.
• The planar algebraic curves of degree 2 are exactly the conic sections, including the unit circle described by and the standard parabola with the formula. The reducible curves are the degenerate conic sections.
• In grade 3 for the first time enter irreducible curves with singularities, such as the Neilsche parabola with the equation and the Cartesian leaf which is given by. The elliptic curves are also important examples of plane algebraic curves of degree 3

Dual curve

A curve can be described by their points instead of by their tangents. An important problem in this context is the question of how many tangents can be " normally" from a non lying on the curve point create a graph of order n. This number is called the class of the curve. For such a curve without singular points (such as colons or peaks ), this class is the same. Each colon reduces the class by 2 and each peak by 3 This is a major statement of the Plucker formulas that also still dealing with the number of turning points and double tangents. For this reason the body needs to be algebraically closed.

For example, a nonsingular cubic curve of 6th grade, it has a colon, is it of the fourth, and if she has a bit of the third class.

In the homogeneous case have straight tangents including an equation of the form where a, b and c may not all disappear and can be multiplied by an arbitrary number other than 0. This allows you to assign this straight the point. From the set of tangents to a given curve, one obtains a set of points in the projective plane. It turns out that this quantity itself is an algebraic curve again, the so-called dual curve.

Dual to each other the following terms:

• Curve point and curve tangent
• Colon and double tangent
• Inflection point and peak
• Regulation and class

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