Algebraic curve

In mathematics, an algebraic curve is a special shape made from equations. The simplest type is called an affine algebraic plane curve, which 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, which uses a special kind of polynomial in three variables and exists in a projective plane. These two types are closely related, and you can change one into the other using a process called homogenizing.

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 called its components.

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, but some properties, like how smooth the curve is or its degree, need to be studied even when the curve is not flat. 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, unlike other curves where y can be directly found from x.

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 that can help solve these problems.

Every algebraic curve can be broken down into smooth, steady 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, their directions, and how the arcs connect 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 where the points' coordinates 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 expanded into a projective curve. This is done by changing the curve's equation into a homogeneous polynomial. This changed equation helps us understand the curve better by including special points called "points at infinity." These points complete the curve and help mathematicians study it 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. To describe these curves, we often use equations made from polynomials.

When we want to study curves that are not flat, we can use a special method. We start with two variables, like x₁ and x₂, and use polynomials to create rules that these variables must follow. This helps us understand the shape and properties of the curve, even if it lives 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 that cannot be broken into smaller parts.

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. Even when a curve seems to have no points, like x² + y² = −1 using real numbers, it still fits into this special math structure.

Complex curves and real surfaces

A complex projective algebraic curve lives in a space called CPⁿ, which has a real shape that can be touched and measured. This space 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 of a certain level of difficulty, the genus can be figured out by a special math formula.

A Riemann surface is a special kind of surface that helps us connect different areas of math, letting us use tools from one area to solve problems in another. This connection lets experts in different math fields work together and understand each other better.

Singularities

Points on a special type of math shape called an algebraic curve can be either smooth or singular. To tell them apart, we look at something called the tangent space. If certain special numbers, called partial derivatives, all equal zero at a point, that point is singular. Otherwise, it is smooth.

Singular points are where the curve crosses itself or forms a cusp, like the point at the origin for the curve defined by x³ = y². Curves can have many such points, but only a limited number. If there are none, the curve is smooth. These singular points help mathematicians understand more about the curve.

Examples of curves

Rational curves

A rational curve is any curve that can be matched with a straight line. We can think of the curve's functions as ratios of single-variable 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.