The Cayley cubic, Clebsch diagonal, Barth sextic, and Kummer surface — rendered in full glory
Set a polynomial in x, y and z equal to zero and you get a surface. Some of the most beautiful ones are packed with as many singular points as their degree allows.
Pick a surface and rotate it. Find the Barth sextic’s 65 nodes and its icosahedral symmetry, and turn on the clipping plane to look inside.
Degree 6 — 65 ordinary double points, the maximum for a sextic. Icosahedral symmetry
An algebraic surface is the zero set V(f) of a polynomial f(x, y, z) = 0 in three-dimensional space. The degree of f controls the complexity of the surface: degree 1 gives planes, degree 2 gives quadrics (spheres, hyperboloids, cones), and higher degrees produce increasingly intricate shapes with folds, cusps, and self-intersections.
The Clebsch diagonal surface is a smooth cubic surface in CP3 on which all 27 lines are real — a rare and beautiful property. Its equation is symmetric in five variables (subject to their sum being zero), giving it the symmetry group S5. The surface can be realized as the blow-up of CP2 at six points in general position.
The Barth sextic is defined by a degree-6 polynomial involving the golden ratio φ. Its 65 nodes are arranged with icosahedral symmetry, and its construction relies on the fact that φ² = φ + 1. The Barth decic (degree 10) pushes the count even further, with 345 nodes — the current record for its degree, though whether it is the true maximum remains open. These surfaces demonstrate how algebraic constraints, symmetry, and number theory intertwine.
Switch between singularity types and compare their shapes near the orange dot at the origin. Brighter violet means closer to the singular point.
Two smooth branches crossing transversely — the simplest singularity
A central question in classical algebraic geometry is: how many singular points (nodes, cusps, or worse) can a surface of degree d possess? The answer involves beautiful combinatorics and has connections to lattice theory, coding theory, and even mathematical physics. Famous surfaces like the Cayley cubic, Clebsch diagonal surface, and Barth sextic push these bounds to their limits.
A point p on V(f) is singular if the gradient ∇f(p) = 0 — that is, all partial derivatives vanish simultaneously. At a smooth point, the gradient is nonzero and defines the tangent plane. The simplest singularity is an ordinary double point (node), where the surface looks locally like a cone x² + y² - z² = 0.
For a surface of degree d in CP3, the maximum number of nodes is bounded above by roughly (4/9)d(d-1)² (Miyaoka's bound). The Cayley cubic (d = 3) has 4 nodes, the Kummer quartic (d = 4) has 16 nodes, and the Barth sextic (d = 6) achieves the maximum of 65 nodes — a record that stood as a conjecture for decades before being confirmed.
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