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 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 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 |
|---|---|
| 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.”