Morphism of algebraic varieties

In algebraic geometry, a morphism between algebraic varieties is a special kind of function. This function is described using polynomials, which are expressions made from variables and operations like addition and multiplication. Such a function is also called a regular map.

When a regular map has another regular map as its inverse, it is called biregular. Biregular maps are the same as isomorphisms of algebraic varieties, meaning they perfectly match one variety to another.

Because regular maps can be very limited—especially for projective varieties—mathematicians also use rational and birational maps. These are like pieces of regular maps, defined using rational fractions instead of just polynomials. An algebraic variety also has a structure called a locally ringed space, and a morphism between these varieties matches these structures exactly.

Definition

A morphism between algebraic varieties is a special kind of function. Think of it as a way to map one shape to another using equations made from polynomials. These equations help us understand how points on one shape relate to points on another.

When we talk about a morphism from one variety to a simple line, we call this a regular function. This idea helps us study the shapes and their properties in algebraic geometry.

Regular functions

When we talk about special kinds of maps in math, we call them regular functions. These are like smooth paths that connect points in shapes we study.

A regular function is very simple: it connects a shape to a straight line. This helps us understand the shape better, just like how smooth lines help us study curves in other parts of math.

Comparison with a morphism of schemes

When we talk about special shapes called affine schemes, there is a way to connect them using special rules. If we have two of these shapes, we can use a special kind of math rule to create a path between them. This path is made by looking at certain important points on the shapes.

When we look at simpler shapes called affine varieties, the same idea works. These shapes have special points, and the paths between them follow the same rules as with the more complex shapes. This shows that the study of these simpler shapes fits nicely inside the study of the more complex ones.

Examples

See also: Morphism of schemes § Examples

In algebraic geometry, there are special ways to connect shapes called "morphisms." These are like rules that help us understand how one shape relates to another.

For example, imagine a simple curve where one coordinate is the square of the other. We can create a rule that maps points on this curve to a straight line, and this rule follows the rules of algebraic geometry.

Another example involves more complex curves and mappings between them. These mappings help us study the properties of the shapes and how they connect.

Properties

A morphism between varieties is a special kind of function that connects two shapes in a smooth way. It keeps the shapes' basic properties while moving from one to the other.

When we look at these shapes up close, the morphism still works well, even if the overall shape isn't simple. Also, if a morphism fully covers a part of the second shape, it helps us understand how these shapes relate to each other in a deeper way.

Morphisms to a projective space

When we talk about special kinds of shapes in math, we can describe how one shape changes into another using rules called morphisms. One important case is when a shape called a projective variety changes into a space called a projective space.

For a point on the shape, some parts of its description are not zero. We can focus on areas where these parts stay away from zero, turning the shape into a simpler space called an affine space. In these areas, the morphism can be described using special math rules called regular functions. These rules help us understand how points move and change in a clear, organized way.

Fibers of a morphism

This section talks about special rules for shapes in a branch of math called algebraic geometry.

It explains that when you connect two shapes using a certain kind of math rule, there are important patterns about how their parts relate in size. The rules help mathematicians understand these connections better.

The section mentions a few math ideas used in the proof, like Noether's normalization lemma and generic freeness, but these details are for older students who study this topic more deeply.

Degree of a finite morphism

When we talk about a special kind of map between shapes in algebra, we call it a finite surjective morphism. This map has a "degree," which tells us something important about how the shapes relate to each other.

If the map is special enough (called étale) and the shapes are complete, then the degree helps us understand how certain features of one shape match up with the other. This idea is useful in studying how these shapes behave and connect.