Projective variety

In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. It is the set of points where some homogeneous polynomials are zero. These polynomials help make shapes in spaces with many dimensions. These shapes are important for studying geometry and numbers.

Projective varieties can have different sizes. A variety that is one-dimensional is called a projective curve. One that is two-dimensional is a projective surface. When the size is one less than the space it is in, it is called a projective hypersurface. These shapes help mathematicians see complex patterns and connections.

Projective varieties are special because they are complete, which means no points are missing. This makes them useful in many parts of mathematics. By studying these varieties, mathematicians can group shapes, learn about their features, and link them to other areas like Hodge theory. Learning about projective varieties helps connect algebra, geometry, and analysis.

Variety and scheme structure

Projective space is a way to describe all the lines that go through the center of a space. We write this space as Pⁿ. It is made by grouping together points that are multiples of each other.

A projective variety is a special shape inside this projective space. It is formed where several equations, made from balanced polynomials, all equal zero. These equations must have the same degree for each term. This makes the shape easy to study using geometry.

Projective varieties can have different sizes. If it has one size, it is called a projective curve. If it has two sizes, it is called a projective surface. These shapes help us understand more complex geometric ideas.

Relation to complete varieties

Projective varieties are special types of shapes in mathematics that are always "complete," meaning they don’t have any missing points. For example, a smooth curve is projective if it is complete.

There is a connection between complete and projective varieties. Chow's lemma tells us that any complete variety can be closely related to a projective variety. Some properties of projective varieties come from this completeness.

Examples and basic invariants

Projective varieties are special shapes in mathematics. They live in a space called projective space. These shapes are defined by equations where all terms have the same degree. This makes them interesting to study because they have nice properties and appear in many areas of math.

Some important types of projective varieties include products of projective spaces, Grassmannians, and flag varieties. These shapes are studied a lot because they help us understand deeper parts of geometry and algebra. For example, the product of two projective spaces can be placed inside a larger projective space using something called the Segre embedding.

Projections

A projection is a way to "flatten" shapes in space. Imagine you have a shape inside a bigger space. By picking a point outside the shape, you can "project" the shape onto a smaller space. This makes a map that shows what the shape looks like from that point.

This idea helps us understand tricky shapes by putting them in simpler spaces. For example, a curly shape can look smoother in a lower dimension, but it still keeps its important features. This method is useful in many parts of geometry.

Main article: Noether's normalization lemma

Duality and linear system

The dual of a projective space helps us understand hyperplanes, which are like flat slices of the space. For a projective space, each point in its dual matches a hyperplane in the original space. This creates a matching system where points in one space relate to flat slices in another.

A special set of hyperplanes called a "pencil" moves together in a way that can be described using another projective space. When we look at spaces made from collections of sections of line bundles, we can map points to these hyperplanes, forming structures that help us study the original space.

Cohomology of coherent sheaves

Main article: coherent sheaf

This section talks about special math ideas called cohomology of coherent sheaves. These ideas help mathematicians understand certain shapes.

Two important facts are:

1. For any number p, a certain math space linked to the shape always has a fixed size.
2. There is a number n₀ so that, for all numbers bigger than n₀, another math space linked to the shape becomes zero.

These facts help mathematicians study and describe these shapes better.

Smooth projective varieties

Imagine a special kind of shape in math called a smooth projective variety. This shape fits together neatly, with no rough edges or corners.

Mathematicians use a tool called the canonical sheaf to describe how the shape looks and behaves.

Two important ideas help us understand these shapes:

• Serre duality shows a balance between different ways of measuring the shape.
• The Riemann–Roch theorem helps us see the relationship between certain measurements of the shape and its overall properties.

These ideas are important for studying the structure and features of these special mathematical shapes.

Main article: Serre duality

Main articles: Hirzebruch–Riemann–Roch theorem, Grothendieck–Riemann–Roch theorem

Hilbert schemes

Hilbert schemes help us organize and understand closed subvarieties of a projective scheme. Think of them as a special space where each point stands for a different subvariety.

A Hilbert scheme looks at subvarieties with a specific Hilbert polynomial. This polynomial is a math expression that tells us useful facts about the size and shape of the subvariety. When the Hilbert polynomial has a certain form, the Hilbert scheme turns into well-known objects like the Grassmannian or the Fano scheme. These are important in algebraic geometry for studying shapes and their properties.

Complex projective varieties

See also: Complex projective space

Complex projective varieties are special shapes studied in mathematics. They mix ideas from algebra and geometry. These shapes are linked to complex numbers and relate to other geometric shapes called Kähler manifolds. For example, a simple shape called a compact Riemann surface is always a projective variety.

Important results connect analytic geometry (studying shapes using complex functions) with algebraic geometry (studying shapes using equations). One key result, Chow's theorem, shows that some analytic shapes are actually algebraic. This helps mathematicians use tools from both areas to understand these varieties better.

Related notions

Some related ideas in algebraic geometry include the multi-projective variety. There is also something called a weighted projective variety, which is a closed subvariety of a weighted projective space. These concepts help mathematicians study shapes and spaces in more complex ways.