# Classification of algebraic varieties

## What is a quasi affine variety?

A quasi-affine variety is

**an open subvariety of an affine algebraic variety**. (As an open subspace of a Noetherian space it is automatically compact.) An example of a quasi-affine variety that is not affine is C2∖{(0,0)}.## Is Z an affine variety?

In particular,

**Z is not an affine variety**. 5. Let V ⊂ kn and W ⊂ km be affine varieties.## What is a projective scheme?

**A scheme X → S is called projective over S if it factors as a closed immersion**.

**followed by the projection to S**. A line bundle (or invertible sheaf) on a scheme X over S is said to be very ample relative to S if there is an immersion (i.e., an open immersion followed by a closed immersion)

## What is a variety in algebraic geometry?

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as

**the set of solutions of a system of polynomial equations over the real or complex numbers**.## What is zariski problem?

The Zariski Cancellation Problem for Affine Spaces

**asks whether the affine space A k n is cancellative**, i.e., if is an affine k-variety such that V × A k 1 ≅ A k n + 1 , does it follow that V ≅ A k n ? Equivalently, if A is an affine k-algebra such that A [ 1 ] = k [ n + 1 ] , does it follow that A = k [ n ] ?## What is a graded algebra?

A graded algebra is an associative algebra which is with a labelling on its elements by elements of some monoid or group, and such that the multiplication in the algebra is reflected in the multiplication in the labelling group.

## Is projective space an affine variety?

The theorem that

**projective spaces are not affine varieties**is a theorem over the complex numbers. As you note, your construction fails over the complex numbers, so there is no contradiction.## Is projective space Compact?

**A (finite dimensional) projective space is compact**. For every point P of S, the restriction of π to a neighborhood of P is a homeomorphism onto its image, provided that the neighborhood is small enough for not containing any pair of antipodal points. This shows that a projective space is a manifold.

## What is Fano geometry?

In finite geometry, the Fano plane (after Gino Fano) is

**a finite projective plane with the smallest possible number of points and lines**: 7 points and 7 lines, with 3 points on every line and 3 lines through every point.## Why is projective geometry important?

In general, by ignoring geometric measurements such as distances and angles, projective geometry

**enables a clearer understanding of some more generic properties of geometric objects**. Such insights have since been incorporated in many more advanced areas of mathematics.## What is rp2 topology?

Alternatively, each line in through the origin in R

^{3}– 0 meets the unit sphere S^{2}in a pair of antipodal points. Thus RP^{2}is**the space we get from the sphere by identifying antipodal points**. As a topological space, RP^{2}is non-orientable.## What is a projective plane topology?

But, more generally, the notion “projective plane” refers to

**any topological space homeomorphic to**. It can be proved that a surface is a projective plane iff it is a one-sided (with one face) connected compact surface of genus 1 (can be cut without being split into two pieces).## What is a homogeneous vector?

## What are the properties of a projective plane?

Definition. A projective plane consists of a set of lines, a set of points, and a relation between points and lines called incidence, having the following properties:

**Given any two distinct points, there is exactly one line incident with both of them**.## What is an example of a finite plane?

i.e. not intersect in any point. . Another kind of finite plane is an affine plane, which can be obtained from a projective plane by removing one line (and all the points on it).

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Example.

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Example.

Title | finite plane |
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Defines | Fano plane |

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22 mar 2013

## What is a finite plane?

A finite plane of order n is

**one such that each line has n points (for an affine plane), or such that each line has n + 1 points (for a projective plane)**. One major open question in finite geometry is: Is the order of a finite plane always a prime power? This is conjectured to be true.## What are the homogeneous coordinates of a point?

Any point in the projective plane is represented by

**a triple (X, Y, Z)**, called homogeneous coordinates or projective coordinates of the point, where X, Y and Z are not all 0. The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor.## What is the difference between projective and homogeneous coordinates?

Projective geometry has an extra dimension, called W, in addition to the X, Y, and Z dimensions. This four-dimensional space is called “projective space,” and

**coordinates in projective space are called “homogeneous coordinates.”**