n+1 өлшемді кеңістіктегі n өлшемді манифольдтар мен алгебралық сорттар

Manifold or algebraic variety of dimension n in a space of dimension n+1
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension n − 1, which is embedded in an ambient space of dimension n, generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation

defines an algebraic hypersurface of dimension n − 1 in the Euclidean space of dimension n. This hypersurface is also a smooth manifold, and is called a hypersphere or an (n – 1) sphere.

A hypersurface that is a smooth manifold is called a smooth hypersurface. In 'R'^(n), a smooth hypersurface is orientable. Every connected compact smooth hypersurface is a level set, and separates Rn into two connected components; this is related to the Jordan–Brouwer separation theorem.

Hypersurfaces have some specific properties that are not shared with other algebraic varieties. One of the main such properties is Hilbert's Nullstellensatz, which asserts that a hypersurface contains a given algebraic set if and only if the defining polynomial of the hypersurface has a power that belongs to the ideal generated by the defining polynomials of the algebraic set. A corollary of this theorem is that, if two irreducible polynomials (or more generally two square free polynomials) define the same hypersurface, then one is the product of the other by a nonzero constant. Hypersurfaces are exactly the subvarieties of dimension n – 1 of an affine space of dimension of n. This is the geometric interpretation of the fact that, in a polynomial ring over a field, the height of an ideal is 1 if and only if the ideal is a principal ideal. In the case of possibly reducible hypersurfaces, this result may be restated as follows: hypersurfaces are exactly the algebraic sets whose all irreducible components have dimension n – 1.

A projective (algebraic) hypersurface of dimension n – 1 in a projective space of dimension n over a field k is defined by a homogeneous polynomial in n + 1 indeterminates. As usual, homogeneous polynomial means that all monomials of P have the same degree, or, equivalently that for every constant c, where d is the degree of the polynomial. The points of the hypersurface are the points of the projective space whose projective coordinates are zeros of P.

If one chooses the hyperplane of equation as hyperplane at infinity, the complement of this hyperplane is an affine space, and the points of the projective hypersurface that belong to this affine space form an affine hypersurface of equation Conversely, given an affine hypersurface of equation it defines a projective hypersurface, called its projective completion, whose equation is obtained by homogenizing p. That is, the equation of the projective completion is with

where d is the degree of P.

These two processes projective completion and restriction to an affine subspace are inverse one to the other. Therefore, an affine hypersurface and its projective completion have essentially the same properties, and are often considered as two points of view for the same hypersurface. However, it may occur that an affine hypersurface is nonsingular, while its projective completion has singular points. In this case, one says that the affine surface is singular at infinity. For example, the circular cylinder of equation

in the affine space of dimension three has a unique singular point, which is at infinity, in the direction 1=x = 0, y = 0.