Algebraic curve

In mathematics, an algebraic curve is a special shape made from equations.

The simplest type is called an affine algebraic plane curve. This is the set of points where a polynomial (a special kind of math expression) in two variables equals zero.

Another type is the projective algebraic plane curve. This uses a special kind of polynomial in three variables and exists in a projective plane. These two types are closely related.

If the polynomial that defines the curve cannot be broken down into simpler parts, it is called an irreducible plane algebraic curve. If it can be broken down, the curve is made up of several simpler curves.

More generally, an algebraic curve is a one-dimensional object in algebraic geometry. Most of the time, studying these curves is the same as studying plane curves. These curves can also be found in spaces beyond flat planes and are sometimes called space curves or skew curves.

In Euclidean geometry

An algebraic curve in the Euclidean plane is the set of points whose coordinates solve a special kind of math problem called a bivariate polynomial equation p(x, y) = 0. This is often called the implicit equation of the curve.

When working with such a curve, the first tasks are to understand its shape and how to draw it. These tasks can be tricky because the equation is more complex than simple function graphs. However, since the equation is a polynomial, the curve has special patterns.

Every algebraic curve can be broken down into smooth parts called arcs, sometimes linked by special points, and may also include isolated points called acnodes. These arcs can either go on forever or end at certain points. For instance, the Tschirnhausen cubic has arcs that meet at the origin and other special points. Knowing these special points helps in drawing the curve clearly.

Plane projective curves

Sometimes, it is helpful to look at curves in projective space. A plane projective curve is a set of points in a projective plane. These points make a special kind of equation, called a homogeneous polynomial in three variables, equal to zero.

Any curve that we see in our normal plane (called an affine curve) can be changed into a projective curve. This is done by changing the curve's equation. This helps us understand the curve better by including special points called "points at infinity." These points help mathematicians study the curve more easily.

Remarkable points of a plane curve

Analytic structure

Studying the area around a special point on a math shape called an algebraic curve helps us understand its shape. Near a smooth part of the curve, we can describe one direction using a simple math rule. But near a special point, things get trickier and need more detailed math rules to explain.

We can move the special point to the start point to make it easier to study. The curve is described by a math rule called a polynomial. By using special math series, we can break down the curve into simpler parts that help us see its shape near the special point.

Non-plane algebraic curves

An algebraic curve is a special kind of shape in mathematics that has one dimension. We use equations made from polynomials to describe these curves.

When we study curves that are not flat, we use a special method. We start with two variables, like x₁ and x₂, and use polynomials to create rules for these variables. This helps us understand the shape and properties of the curve, even if it is in a space with more than two dimensions.

Algebraic function fields

Studying algebraic curves is the same as studying special kinds of math fields called algebraic function fields. These fields help us understand curves.

For example, using complex numbers, we can create a special field from an equation like y² = x³ − x − 1. This shows how algebraic curves and function fields are connected.

Complex curves and real surfaces

A complex projective algebraic curve lives in a space called CPⁿ. This space has a real shape that can be touched and measured. It is whole, linked together, and can be turned without tearing.

The shape of this surface, called its topological genus, tells us how many "handles" or "donut holes" it has. We can find this number using math rules. For a simple curve, the genus can be figured out by a special math formula.

A Riemann surface is a special kind of surface. It helps us connect different areas of math, letting us use tools from one area to solve problems in another. This helps experts in different math fields work together.

Singularities

Points on a special math shape called an algebraic curve can be smooth or singular. We can tell them apart by looking at something called the tangent space. If some special numbers, called partial derivatives, all become zero at a point, that point is singular. Otherwise, it is smooth.

Singular points are where the curve crosses itself or forms a point like a sharp tip. These points help mathematicians learn more about the curve.

Examples of curves

Rational curves

A rational curve is a curve that matches with a straight line. We can describe the curve's functions as ratios of simple functions. If the main number system is closed, this is the same as a curve with a special property called "genus" equal to zero.

In simple terms, a rational curve can be described using equations with one changing value. An example is the rational normal curve, where all these equations are simple powers.

Any shape made by cutting a circle or ellipse can be a rational curve. By drawing a line through a special point on the shape, we can find other points on the curve using simple math.

Elliptic curves

An elliptic curve is a special type of curve with genus one and a known point. A common way to show it is using a certain type of equation. These curves have a group structure, meaning we can add points on the curve together in a specific way.

Curves of genus greater than one

Curves with genus greater than one are different from rational and elliptic curves. According to a mathematical rule, these curves can only have a limited number of known points if they use rational numbers. Examples include hyperelliptic curves and special shapes like the Fermat curve.

Projective plane curves

Plane curves of a certain level can be made by finding where a special math expression equals zero. The level of these curves can be figured out using a math tool. For example, the curve shown by the equation x⁴ + y⁴ + z⁴ is a smooth curve with a level of 3.

Curves in product of projective lines

Curves in a mix of two projective lines can also be studied. For certain values, these curves have levels that can be calculated. This helps in building curves of any level using this method.