# .

In algebraic geometry, a Barth surface is one of the complex nodal surfaces in 3 dimensions with large numbers of double points found by Wolf Barth (1996). Two examples are the Barth sextic of degree 6 with 65 double points, and the Barth decic of degree 10 with 345 double points.

Barth sextic

$$4(\phi^2x^2-y^2)(\phi^2y^2-z^2)(\phi^2z^2-x^2)-(1+2\phi)(x^2+y^2+z^2-w^2)^2w^2=0.$$

Barth decic

$$8(x^2-\phi^4y^2)(y^2-\phi^4z^2)(z^2-\phi^4x^2)(x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2)+(3+5\phi)(x^2+y^2+z^2-w^2)^2[x^2+y^2+z^2-(2-\phi)w^2]^2w^2 =0,$$

where $$\phi$$ is the golden ratio and w a parameter.

For degree 6 surfaces in P3, Jaffe & 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.

Endrass surface
Sarti surface
Togliatti surface
List of algebraic surfaces

References

Barth, W. (1996), "Two projective surfaces with many nodes, admitting the symmetries of the icosahedron", Journal of Algebraic Geometry 5 (1): 173–186, MR 1358040.
Jaffe, David B.; Ruberman, Daniel (1997), "A sextic surface cannot have 66 nodes", Journal of Algebraic Geometry 6 (1): 151–168, MR 1486992.

Barth sextic
Barth decic
Weisstein, Eric W., "Barth Sextic", MathWorld.
Weisstein, Eric W., "Barth Decic", MathWorld.
animations of Barth surfaces