Hypersurface

A hypersurface that is a smooth manifold is called a smooth hypersurface.

In Rⁿ, a smooth hypersurface is orientable.[2] 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.[3]

An algebraic hypersurface is an algebraic variety that may be defined by a single implicit equation of the form

${\displaystyle p(x_{1},\ldots ,x_{n})=0,}$

where p is a multivariate polynomial. Generally the polynomial is supposed to be irreducible. When this is not the case, the hypersurface is not an algebraic variety, but only an algebraic set. It may depend on the authors or the context whether a reducible polynomial defines a hypersurface. For avoiding ambiguity, the term irreducible hypersurface is often used.

As for algebraic varieties, the coefficients of the defining polynomial may belong to any fixed field k, and the points of the hypersurface are the zeros of p in the affine space ${\displaystyle K^{n},}$ where K is an algebraically closed extension of k.

A hypersurface may have singularities, which are the common zeros, if any, of the defining polynomial and its partial derivatives. In particular, a real algebraic hypersurface is not necessarily a manifold.

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 ${\displaystyle P(x_{0},x_{1},\ldots ,x_{n})}$ in n + 1 indeterminates. As usual, homogeneous polynomial means that all monomials of P have the same degree, or, equivalently that ${\displaystyle P(cx_{0},cx_{1},\ldots ,cx_{n})=c^{d}P(x_{0},x_{1},\ldots ,x_{n})}$ 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 ${\displaystyle x_{0}=0}$ 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 ${\displaystyle P(1,x_{1},\ldots ,x_{n})=0.}$ Conversely, given an affine hypersurface of equation ${\displaystyle p(x_{1},\ldots ,x_{n})=0,}$ 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 ${\displaystyle P(x_{0},x_{1},\ldots ,x_{n})=0,}$ with

${\displaystyle P(x_{0},x_{1},\ldots ,x_{n})=x_{0}^{d}p(x_{1}/x_{0},\ldots ,x_{n}/x_{0}),}$

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

${\displaystyle x^{2}+y^{2}-1=0}$

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

• Affine sphere
• Coble hypersurface
• Dwork family
• Null hypersurface
• Polar hypersurface

• "Hypersurface", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Shoshichi Kobayashi and Katsumi Nomizu (1969), Foundations of Differential Geometry Vol II, Wiley Interscience
• P.A. Simionescu & D. Beal (2004) Visualization of hypersurfaces and multivariable (objective) functions by partial globalization, The Visual Computer 20(10):665–81.