# 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 .

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#### References

 [a1] M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish (1975) pp. 1–5 MR0394453 MR0394452 MR0372756 Zbl 0306.53003 Zbl 0306.53002 Zbl 0306.53001

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

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 MR0605336 Zbl 0479.14007 [a2] R. Donagi, "Generic Torelli for projective hypersurfaces" Compos. Math. , 50 (1983) pp. 325–353 MR0720291 Zbl 0598.14007 [a3] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381 MR0463157 Zbl 0367.14001

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