Projective variety

In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. It is defined as the set of points where certain homogeneous polynomials equal zero. These polynomials help create shapes in higher-dimensional spaces that are important for studying geometry and numbers.

Projective varieties can have different dimensions. A variety with one dimension is called a projective curve, while one with two dimensions is a projective surface. When the dimension is one less than the space it lives in, it is called a projective hypersurface. These shapes help mathematicians understand complex patterns and relationships.

Projective varieties are special because they are complete, meaning no points are missing. This property makes them very useful in many areas of mathematics. By studying these varieties, mathematicians can classify shapes, understand their properties, and even connect them to other areas like Hodge theory. The study of projective varieties helps build a bridge between 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 taking all sets of 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 homogeneous polynomials, all equal zero. These equations must have the same degree for each term. This makes the shape well-behaved and allows us to study it using geometry.

Projective varieties can have different dimensions. If it has one dimension, it is called a projective curve. If it has two dimensions, 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 (a one-dimensional shape) is projective if and only 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 through a special kind of mapping. Some properties of projective varieties come from this completeness. For instance, certain functions on a projective variety behave in a simple way, much like functions on special complex shapes in another area of mathematics.

Examples and basic invariants

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

Some important types of projective varieties include products of projective spaces, Grassmannians, and flag varieties. These have been studied a lot because they help us understand deeper parts of geometry and algebra. For example, the product of two projective spaces can be neatly 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 sitting inside a bigger space. By choosing a point outside the shape, you can "project" the shape onto a smaller space. This creates a map that shows how the shape looks from that point.

This idea helps us understand complex shapes by placing them in simpler spaces. For example, a curly shape can be shown as a smoother one in a lower dimension, keeping important features intact. This method is useful in many areas 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 corresponds to a hyperplane in the original space. This creates a matching system where points in one space relate directly 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 shapes called projective schemes. Two important facts are:

1. For any number p, a certain math space linked to the shape is always a special kind of space with 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 has all its parts fitting together neatly without any rough edges or corners. For such a shape, mathematicians use a tool called the canonical sheaf, which helps describe how the shape looks and behaves.

One important idea related to these shapes is called Serre duality. It shows a balance between different ways of measuring the shape. Another key idea is the Riemann–Roch theorem, which helps us understand the relationship between certain measurements of the shape and its overall properties. These concepts are important in studying the structure and characteristics 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 understand and organize all the closed subvarieties of a projective scheme. Think of them as a special kind of space where each point represents a different subvariety.

Specifically, a Hilbert scheme focuses on subvarieties that have a particular Hilbert polynomial, which is a mathematical expression that gives us important information about the subvariety’s size and shape. When the Hilbert polynomial is of a certain form, the Hilbert scheme becomes familiar objects like the Grassmannian or the Fano scheme. These structures are important in algebraic geometry for studying shapes and their properties.

Complex projective varieties

See also: Complex projective space

Complex projective varieties are special types of shapes studied in mathematics, combining ideas from algebra and geometry. They are linked to complex numbers and have important connections with other geometric shapes called Kähler manifolds. For example, a simple shape called a compact Riemann surface is always a projective variety.

Important results help connect analytic geometry (studying shapes using complex functions) with algebraic geometry (studying shapes using equations). One key result, Chow's theorem, shows that certain analytic shapes are actually algebraic. This connection allows mathematicians to 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.