Hypersurface

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

If $ M $ and $ N $ are differentiable manifolds, $ \mathop{\rm dim} N - \mathop{\rm dim} M = 1 $, and if an immersion $ f: M \rightarrow N $ has been defined, then $ f( M) $ is a hypersurface in $ N $. Here $ f $ is a differentiable mapping whose differential $ df $ at any point $ x \in M $ is an injective mapping of the tangent space $ M _ {x} $ to $ M $ at $ x $ into the tangent space $ N _ {f(} x) $ to $ N $ at $ f( x) $.

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 $ A _ {k} ^ {n} $ over a field $ k $ is globally defined by one equation

$$ f ( x _ {1} \dots x _ {n} ) = 0. $$

An algebraic hypersurface $ W $ in a projective space $ P ^ {n} $ is defined by an equation

$$ F ( x _ {0} \dots x _ {n} ) = 0, $$

where $ F $ is a homogeneous form in $ n + 1 $ variables. The degree $ m $ of this form is said to be the degree (order) of the hypersurface. A closed subscheme $ W $ of a scheme $ V $ is said to be a hypersurface if the corresponding sheaf of ideals $ I _ {W} \subset {\mathcal O} _ {V} $ is a sheaf of principal ideals. For a connected non-singular algebraic variety this condition means that the codimension of $ W $ in $ V $ is one. For each non-singular algebraic hypersurface $ W \subset P _ {k} ^ {n} $ of order $ m $( often denoted by $ V _ {n} ^ {m} $) the following holds:

a) the canonical class $ K _ {W} $ is equal to $ ( m - n - 1 ) H _ {W} $ where $ H _ {W} $ is the class of a hyperplane section of $ W $;

b) the cohomology groups $ H ^ {i} ( W, {\mathcal O} _ {W} ) = 0 $ for $ i \neq 0, n - 1 $, and

$$ \mathop{\rm dim} _ {k} H ^ {n - 1 } ( W, {\mathcal O} _ {W} ) = \ \frac{( m - 1) \dots ( m - n) }{n! } ; $$

c) if $ n \geq 3 $, the fundamental group (algebraic or topological if $ k = \mathbf C $) $ \pi _ {1} ( W) = 0 $;

d) if $ n \geq 4 $, the Picard group $ \mathop{\rm Pic} ( W) \simeq \mathbf Z $ 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 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 $ S $ in a complex Euclidean space $ \mathbf C ^ {n} $ that, in a neighbourhood of each of its points $ \zeta \in S $, is defined by an equation $ f _ \zeta ( z, t) = 0 $, where the function $ f _ \zeta ( z, t) $ is continuous with respect to the parameter $ t \in ( - \epsilon , \epsilon ) $, $ \epsilon > 0 $, and, for each fixed $ t $, is holomorphic in $ z $ in a neighbourhood $ U _ \zeta \ni \zeta $ which is independent of $ t $; moreover, $ \sum | \partial f/ \partial z _ {j} | \neq 0 $ for all $ ( z, t) \in U _ \zeta \times ( - \epsilon , \epsilon ) $. In other words, an analytic hypersurface is a set in $ \mathbf C ^ {n} $ that is locally the union of a continuous one-parameter family of complex-analytic surfaces of complex codimension one. For instance, if a function $ f $ is holomorphic in a domain $ D \subset \mathbf C ^ {n} $ and $ \mathop{\rm grad} f \neq 0 $ in $ D $, then the sets $ | f | = 1 $, $ \mathop{\rm Re} f = 0 $, etc., are analytic hypersurfaces.

A twice-differentiable hypersurface $ S $ in $ \mathbf R ^ {2n} = \mathbf C ^ {n} $ is an analytic hypersurface if and only if its Levi form vanishes identically on $ S $ or if $ S $ 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 $ S \subset \mathbf R ^ {2n} $, mentioned above, can be found in [a2].

References

[a1]	H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1977) (Translated from German) MR0513229 Zbl 0379.32001
[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) MR0716497 Zbl 0528.32001