## 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 R3 – 0 meets the unit sphere S2 in a pair of antipodal points. Thus RP2 is the space we get from the sphere by identifying antipodal points. As a topological space, RP2 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 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).

Example.
Titlefinite plane
DefinesFano plane
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.”