Hypersurface

A generalization of the concept of an ordinary surface in three-dimensional space to the case of an -dimensional space. The dimension of a hypersurface is one less than that of its ambient space.

If and are differentiable manifolds, , and if an immersion has been defined, then is a hypersurface in . Here is a differentiable mapping whose differential at any point is an injective mapping of the tangent space to at into the tangent space to at .

References

[a1]	M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish (1975) pp. 1–5

An algebraic hypersurface is a subvariety of an algebraic variety that is locally defined by one equation. An algebraic hypersurface in the affine space over a field is globally defined by one equation

An algebraic hypersurface in a projective space is defined by an equation

where is a homogeneous form in variables. The degree of this form is said to be the degree (order) of the hypersurface. A closed subscheme of a scheme is said to be a hypersurface if the corresponding sheaf of ideals is a sheaf of principal ideals. For a connected non-singular algebraic variety this condition means that the codimension of in is one. For each non-singular algebraic hypersurface of order (often denoted by ) the following holds:

a) the canonical class is equal to where is the class of a hyperplane section of ;

b) the cohomology groups for , and

c) if , the fundamental group (algebraic or topological if ) ;

d) if , the Picard group and is generated by the class of a hyperplane section.

I.V. Dolgachev

Comments

The cohomology ring of a smooth complex projective hypersurface can be expressed completely in terms of rational differential forms on the ambient projective space, [a1]. In most cases, the period mapping for these hypersurfaces has been shown to be of degree one [a2].

References

[a1]	J. Carlson, P. Griffiths, "Infinitesimal variations of Hodge structure and the global Torelli problem" A. Beauville (ed.) , Algebraic geometry (Angers, 1979) , Sijthoff & Noordhoff (1980) pp. 51–76
[a2]	R. Donagi, "Generic Torelli for projective hypersurfaces" Compos. Math. , 50 (1983) pp. 325–353
[a3]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381

An analytic hypersurface is a set in a complex Euclidean space that, in a neighbourhood of each of its points , is defined by an equation , where the function is continuous with respect to the parameter , , and, for each fixed , is holomorphic in in a neighbourhood which is independent of ; moreover, for all . In other words, an analytic hypersurface is a set in that is locally the union of a continuous one-parameter family of complex-analytic surfaces of complex codimension one. For instance, if a function is holomorphic in a domain and in , then the sets , , etc., are analytic hypersurfaces.

A twice-differentiable hypersurface in is an analytic hypersurface if and only if its Levi form vanishes identically on or if is locally pseudo-convex on both sides.

E.M. Chirka

Comments

Sometimes the phrase "analytic hypersurface" is also used for an analytic set of complex codimension 1, analogously to 3) above, cf. [a1]. An analytic hypersurface as in 4) is also called a foliation by analytic varieties of codimension 1. The result concerning a twice-differentiable , mentioned above, can be found in [a2].

References

[a1]	H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1977) (Translated from German)
[a2]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[a3]	L. Kaup, B. Kaup, "Holomorphic functions of several variables" , de Gruyter (1983) (Translated from German)