Barth surface

For degree 6 surfaces in P3, David Jaffe and Daniel Ruberman (1997) showed that 65 is the maximum number of double points possible.

The Barth sextic is a counterexample to an incorrect claim by Francesco Severi in 1946 that 52 is the maximum number of double points possible.

The Barth sextic may be visualized in three dimensions as featuring 50 finite and 15 infinite ordinary double points (nodes).

Referring to the figure, the 50 finite ordinary double points are arrayed as the vertices of 20 roughly tetrahedral shapes oriented such that the bases of these four-sided "outward pointing" shapes form the triangular faces of a regular icosidodecahedron.

The 15 remaining ordinary double points at infinity correspond to the 15 lines that pass through the opposite vertices of the inscribed icosidodecahedron, all 15 of which also intersect in the center of the figure.