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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.

Some admit icosahedral symmetry.

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.