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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|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055980/k0559801.png" />-surface]]; it is determined within the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055980/k0559802.png" />-surfaces by the condition that it contains 16 irreducible rational curves.
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{{TEX|done}}{{MSC|14J25}}
 
 
====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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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.
  
 
====Comments====
 
====Comments====
A quartic surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055980/k0559803.png" /> has at most 16 double points (as has the Kummer surface).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055980/k0559804.png" />-torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055980/k0559805.png" />, taking the involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055980/k0559806.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055980/k0559807.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055980/k0559808.png" />, taking the quotient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055980/k0559809.png" /> divided out by this involution, and resolving the sixteen double points of this surface.
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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====
 
====References====
<table><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>
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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) {{ZBL|0716.14025}}</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>
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</table>

Latest revision as of 12:27, 12 November 2023

2020 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.

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

[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) Zbl 0716.14025
[3] F. Enriques, "Le superficie algebraiche" , Bologna (1949)
[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://encyclopediaofmath.org/index.php?title=Kummer_surface&oldid=13944
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article