Editing Kummer surface (section)

=== Singular quartic surfaces and the double plane model ===
Let <math>K\subset\mathbb{P}^3 </math> be a quartic surface with an ordinary double point ''p'', near which ''K'' looks like a quadratic cone. Any projective line through ''p'' then meets ''K'' with multiplicity two at ''p'', and will therefore  meet the quartic ''K'' in just two other points.   Identifying the lines in <math>\mathbb{P}^3</math> through the point ''p'' with <math>\mathbb{P}^2</math>, we get a double cover from the blow up of ''K'' at ''p'' to <math>\mathbb{P}^2</math>; this double cover is given by
sending ''q''&nbsp;≠&nbsp;''p''&nbsp;↦&nbsp;<math>\scriptstyle\overline{pq}</math>, and any line in the [[tangent cone]] of ''p'' in ''K'' to itself. The [[ramification locus]] of the double cover is a plane curve ''C'' of degree 6, and all the nodes of ''K'' which are not ''p'' map to nodes of&nbsp;''C''.

By the [[genus degree formula]], the maximal possible number of nodes on a sextic curve is obtained when the curve is a union of <math>6</math> lines, in which case we have 15 nodes. Hence the maximal number of nodes on a quartic is 16, and in this case they are all simple nodes (to show that <math>p</math> is simple project from another node). A quartic which obtains these 16 nodes is called a Kummer Quartic, and we will concentrate on them below.

Since <math>p</math> is a simple node, the tangent cone to this point is mapped to a conic under the double cover. This conic is in fact tangent to the six lines (w.o proof). Conversely, given a configuration of a conic and six lines which tangent to it in the plane, we may define the double cover of the plane ramified over the union of these 6 lines. This double cover may be mapped to <math>\mathbb{P}^3</math>, under a map which [[blowing up|blows down]] the double cover of the special conic, and is an isomorphism elsewhere (w.o. proof).