# Difference between revisions of "Kummer surface"

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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> | + | <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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<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> | ||

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## 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