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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Spivak,   "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish (1975) pp. 1–5</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Spivak, "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish (1975) pp. 1–5 {{MR|0394453}} {{MR|0394452}} {{MR|0372756}} {{ZBL|0306.53003}} {{ZBL|0306.53002}} {{ZBL|0306.53001}} </TD></TR></table>
  
 
An algebraic hypersurface is a subvariety of an algebraic variety that is locally defined by one equation. An algebraic hypersurface in the affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852017.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852018.png" /> is globally defined by one equation
 
An algebraic hypersurface is a subvariety of an algebraic variety that is locally defined by one equation. An algebraic hypersurface in the affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852017.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852018.png" /> is globally defined by one equation
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Carlson,   P. Griffiths,   "Infinitesimal variations of Hodge structure and the global Torelli problem" A. Beauville (ed.) , ''Algebraic geometry (Angers, 1979)'' , Sijthoff &amp; Noordhoff (1980) pp. 51–76</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Donagi,   "Generic Torelli for projective hypersurfaces" ''Compos. Math.'' , '''50''' (1983) pp. 325–353</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Carlson, P. Griffiths, "Infinitesimal variations of Hodge structure and the global Torelli problem" A. Beauville (ed.) , ''Algebraic geometry (Angers, 1979)'' , Sijthoff &amp; Noordhoff (1980) pp. 51–76 {{MR|0605336}} {{ZBL|0479.14007}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Donagi, "Generic Torelli for projective hypersurfaces" ''Compos. Math.'' , '''50''' (1983) pp. 325–353 {{MR|0720291}} {{ZBL|0598.14007}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
  
 
An analytic hypersurface is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852046.png" /> in a complex Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852047.png" /> that, in a neighbourhood of each of its points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852048.png" />, is defined by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852049.png" />, where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852050.png" /> is continuous with respect to the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852052.png" />, and, for each fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852053.png" />, is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852054.png" /> in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852055.png" /> which is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852056.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852057.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852058.png" />. In other words, an analytic hypersurface is a set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852059.png" /> that is locally the union of a continuous one-parameter family of complex-analytic surfaces of complex codimension one. For instance, if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852060.png" /> is holomorphic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852062.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852063.png" />, then the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852065.png" />, etc., are analytic hypersurfaces.
 
An analytic hypersurface is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852046.png" /> in a complex Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852047.png" /> that, in a neighbourhood of each of its points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852048.png" />, is defined by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852049.png" />, where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852050.png" /> is continuous with respect to the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852052.png" />, and, for each fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852053.png" />, is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852054.png" /> in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852055.png" /> which is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852056.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852057.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852058.png" />. In other words, an analytic hypersurface is a set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852059.png" /> that is locally the union of a continuous one-parameter family of complex-analytic surfaces of complex codimension one. For instance, if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852060.png" /> is holomorphic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852062.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852063.png" />, then the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852065.png" />, etc., are analytic hypersurfaces.
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====Comments====
 
====Comments====
Sometimes the phrase "analytic hypersurface" is also used for an [[Analytic set|analytic set]] of complex codimension 1, analogously to 3) above, cf. [[#References|[a1]]]. An analytic hypersurface as in 4) is also called a foliation by analytic varieties of codimension 1. The result concerning a twice-differentiable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852070.png" />, mentioned above, can be found in [[#References|[a2]]].
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Sometimes the phrase "analytic hypersurface" is also used for an [[Analytic set|analytic set]] of complex codimension 1, analogously to 3) above, cf. [[#References|[a1]]]. An analytic hypersurface as in 4) is also called a foliation by analytic varieties of codimension 1. The result concerning a twice-differentiable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852070.png" />, mentioned above, can be found in [[#References|[a2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Grauert,   R. Remmert,   "Theory of Stein spaces" , Springer (1977) (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.S. Vladimirov,   "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L. Kaup,   B. Kaup,   "Holomorphic functions of several variables" , de Gruyter (1983) (Translated from German)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1977) (Translated from German) {{MR|0513229}} {{ZBL|0379.32001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L. Kaup, B. Kaup, "Holomorphic functions of several variables" , de Gruyter (1983) (Translated from German) {{MR|0716497}} {{ZBL|0528.32001}} </TD></TR></table>

Revision as of 21:53, 30 March 2012

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 .


Comments

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

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 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) 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
How to Cite This Entry:
Hypersurface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hypersurface&oldid=23862
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article