Exploring the Definition of Affine Hypersurfaces

[Ms Mrmr]

what is affine hypersurface :(

Hi all >>


please i want answer about defnition of affine hypersurface ??

thank u

 

[micromass]

If I'm not mistaken, then a hypersurface is simply the set of zero points of a polynomial. Thus if P is a polynomial, then the hypersurface defined by P is

$$\{(x_1,...,x_n)\in k^n~\vert~P(x_1,...,x_n)=0\}$$

Is this what you meant??

 

[HallsofIvy]

An "affine" hypersurface is a flat hypersurface. In one-dimension, that is line, in two-dimensions, it is a plane, in higher dimensions, a hyper-plane. That means that the polynomial, P, that micromass refers to as defining a hypersurface is linear.

 

[Hurkyl]

More context for the question would be nice.

 

[Ms Mrmr]

micromass thank u but I want the geometry definitoin for affine hypersurface


HallsofIvy thank u , ur definition is good but pleas i want More detailed about it . :shy:


Hurkyl , sorry , i tierd to explanation my question but i am not speak good english


thank u all

 

[disregardthat]

In algebraic geometry an affine hypersurface is excactly what micromass said. There are no restrictions on the polynomial in this context, which means it doesn't need to be linear.

A geometric definition of a affine hypersurface in algebraic geometry could be "a closed subset of codimension 1 of an affine space". (an affine space is normally k^n, where k is algebraically closed field in the zariski-topology, or some subset of this if you want more generality) In other words it is a closed subset of an affine space with dimension one less than the affine space itself. Intuitively you can imagine a 2-dimensional surface (such as a plane, a sphere, a plane intersecting a sphere etc.. in euclidean 3-space). This means it is generated by a single polynomial. Sometimes a hypersurface refers to such an irreducible set (which is what I've seen, but I will not insist on this), which means that the generating polynomial needs to be irreducible.

Hypersurfaces generated by a linear polynomial are generally called hyperplanes (and specifically lines if the dimension is 1).

 

[Ms Mrmr]

Jarle thank you very much