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Difference between revisions of "Kummer surface"

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(Category:Algebraic geometry)
(MSC 14J25)
 
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An algebraic surface of the fourth order and class, reciprocal to itself, having 16 double points, of which 16 groups (each containing 6 points) are situated in one double tangent to the surface (i.e. a plane which is tangent to the surface along a conic section). The equation of degree 16 defining the double element yields a single equation of degree 6 and several quadratic equations. Discovered by E. Kummer (1864). The Kummer surface is a special kind of [[K3-surface|$K3$-surface]]; it is determined within the class of $K3$-surfaces by the condition that it contains 16 irreducible rational curves.
 
An algebraic surface of the fourth order and class, reciprocal to itself, having 16 double points, of which 16 groups (each containing 6 points) are situated in one double tangent to the surface (i.e. a plane which is tangent to the surface along a conic section). The equation of degree 16 defining the double element yields a single equation of degree 6 and several quadratic equations. Discovered by E. Kummer (1864). The Kummer surface is a special kind of [[K3-surface|$K3$-surface]]; it is determined within the class of $K3$-surfaces by the condition that it contains 16 irreducible rational curves.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Klein,  "Development of mathematics in the 19th century" , Math. Sci. Press  (1979)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.W.H.T. Hudson,  "Kummer's quartic surface" , Cambridge  (1905)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. Enriques,  "Le superficie algebraiche" , Bologna  (1949)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  F. Klein,  "Development of mathematics in the 19th century" , Math. Sci. Press  (1979)  (Translated from German)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  R.W.H.T. Hudson,  "Kummer's quartic surface" , Cambridge  (1905)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  F. Enriques,  "Le superficie algebraiche" , Bologna  (1949)</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Barth,  C. Peters,  A. van der Ven,  "Compact complex surfaces" , Springer  (1984)</TD></TR>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Barth,  C. Peters,  A. van der Ven,  "Compact complex surfaces" , Springer  (1984)</TD></TR>
 
</table>
 
</table>
 
[[Category:Algebraic geometry]]
 

Latest revision as of 16:13, 5 March 2018

2010 Mathematics Subject Classification: Primary: 14J25 [MSN][ZBL]

An algebraic surface of the fourth order and class, reciprocal to itself, having 16 double points, of which 16 groups (each containing 6 points) are situated in one double tangent to the surface (i.e. a plane which is tangent to the surface along a conic section). The equation of degree 16 defining the double element yields a single equation of degree 6 and several quadratic equations. Discovered by E. Kummer (1864). The Kummer surface is a special kind of $K3$-surface; it is determined within the class of $K3$-surfaces by the condition that it contains 16 irreducible rational curves.

References

[1] F. Klein, "Development of mathematics in the 19th century" , Math. Sci. Press (1979) (Translated from German)
[2] R.W.H.T. Hudson, "Kummer's quartic surface" , Cambridge (1905)
[3] F. Enriques, "Le superficie algebraiche" , Bologna (1949)


Comments

A quartic surface in $P^3$ has at most 16 double points (as has the Kummer surface).

From a modern point of view, Kummer surfaces are obtained by taking a $2$-torus $T$, taking the involution $\sigma$ on $T$ defined by $\sigma(x)=-x$, taking the quotient of $T$ divided out by this involution, and resolving the sixteen double points of this surface.

References

[a1] W. Barth, C. Peters, A. van der Ven, "Compact complex surfaces" , Springer (1984)
How to Cite This Entry:
Kummer surface. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Kummer_surface&oldid=34198
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article