# 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 $f( x)$.

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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 $A _ {k} ^ {n}$ 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

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

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].