View source for Kummer surface

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.

====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>



====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).

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.

====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>