Cubic Surfaces as Blowup

A classical result in algebraic geometry, probably first shown by A. Clebsch in the 19th century, says that any smooth complex cubic surface arises as the blowup (see no. 09 of our advent calendar) of the plane in six distinct points. A. Clebsch also already explained that the cubic surface admits a singularity if three of the points are on a common line or if all six are on a common conic; otherwise the six points are called points in general position. Our film starts with the smooth cubic surface corresponding to the six points in pentagon-symmetric position shown in the leftmost image below (the so-called Clebsch Cubic) and ends with the one with 4 singularities (the so-called Cayley Cubic).