Togliatti, Labs, Endraß, Barth, Sarti — the surfaces holding the world records for singular points
These are the supermodels of algebraic geometry. Spiky, symmetric, and famous — they turn up on conference posters, book covers, and the occasional postage stamp, and they exist because of a competition that has been running for a hundred and sixty years.
The question is simple enough to state. A surface in space is the zero set of a polynomial. Most such surfaces are smooth, but some pinch to a point here and there — those pinch points are called nodes. So: for a polynomial of degree d, how many nodes can a surface possibly have? Not typically, not on average, but at the absolute limit.
Write μ(d) for that maximum. Kummer settled degree 4 in 1864 with the answer 16. Togliatti found 31 nodes on a quintic in 1940. Barth built a sextic with 65 in 1996 and a degree-10 surface with 345 the same year. Sarti reached 600 in degree 12. And then the trail goes cold: from degree 8 upward, nobody knows the answer. There is a proven ceiling and a best-known construction, and a gap in between that has resisted everyone for decades.
Climb the ladder from Cayley’s 4-nodal cubic to Sarti’s 600-nodal dodecic. Each surface is extracted live by marching cubes; raise the quality setting for the high-degree ones, where the spikes get fine enough to slip between sample points. Drag to rotate.
Icosahedral symmetry and the golden ratio in every coefficient. 50 nodes are visible here; 15 more sit in the plane at infinity.
mu(6) = 65, proven maximal (Jaffe & Ruberman 1997)
A node is the mildest singular point there is: the surface pinches to a cone, locally looking like x² + y² = z². The node counts quoted are counts over the complex numbers, and only real points can be drawn — the Kummer quartic carries 16 nodes over C, of which this real model shows 12. What the pictures do show honestly is the symmetry: the Barth and Sarti surfaces wear their icosahedral five-fold symmetry openly.
A surface is smooth at a point if it has a well-defined tangent plane there — equivalently, if the gradient of the defining polynomial is not zero. Where the gradient vanishes, the surface is singular, and the mildest way that can happen is a node, also called an ordinary double point. Near a node the surface looks like the cone x² + y² = z²: two sheets touching at a single sharp point, like the tips of two ice-cream cones pressed together.
Nodes are the cheapest singularities, which is precisely why the competition is about them. Nastier singularities “use up” more of the surface’s budget, so if you want to cram in as many singular points as possible, plain nodes are what you want. The competition is a packing problem: how many pinch points can a polynomial of degree d support before it runs out of room?
One honesty note about the pictures. Node counts are counted over the complex numbers, but only real points can be drawn. Sometimes these agree — Barth deliberately built his sextic so that all 65 nodes are real. Sometimes they do not: the Kummer quartic has 16 nodes over C, of which a real model shows 12.
Best-known construction against proven ceiling, degree by degree. To the left of the dashed line the two curves touch and the problem is finished. To the right they separate, and the shaded band is pure ignorance — the true value of μ(d) is somewhere inside it, and no one can say where.
Hover a degree to read its story. Filled dots are settled records; hollow dots are merely the best anyone has managed so far, with the small amber dot marking the proven ceiling above them.
Progress on μ(d) comes from two directions that have never quite met. From above, Miyaoka’s bound (1984) says a degree-d surface has at most (4/9)·d(d−1)² nodes, proved by techniques from the theory of surfaces of general type. For degree 8 that gives 174.2, so at most 174. From below, people build actual surfaces — and the best degree-8 construction, by Endraß in 1997, achieves 168. The gap between 168 and 174 has stood ever since.
The constructions are not random searches. Almost every record holder is built from symmetry: Barth’s sextic and decic and Sarti’s dodecic are invariants of the icosahedral group, which is why they look like spiky twenty-sided dice. Imposing a large symmetry group collapses the search space from millions of coefficients to a handful, and the nodes then arrive in whole orbits at a time. Labs found his 99-nodal septic in 2004 by an entirely different route: searching over finite fields, then lifting the answer back to characteristic zero.
The exact values are known only for degree 6 and below: μ(3) = 4, μ(4) = 16, μ(5) = 31, μ(6) = 65. Everything past that is an open problem in a subject where you can see the objects with your own eyes.
A one-parameter family of quartics. Start below μ² = 1 and the surface is smooth and rounded. Push the parameter up and the surface pinches — twelve nodes appear at once and then drift apart. This is what “crossing the discriminant” looks like from the inside.
Past μ² = 1 the surface has pinched: at each yellow beacon two sheets meet at a single point, exactly like the tip of a cone. These twelve nodes sit where the four tetrahedral planes cut each other. The full Kummer quartic has 16 nodes over the complex numbers — the remaining four are not real, so no picture in ordinary space can show them.
Nodes are not accidents; they are what you get for free when a family of surfaces crosses a wall. The set of singular quartics forms a hypersurface — the discriminant — in the space of all quartics, and this slider drives a straight line through it. Each crossing is a moment where the surface must pinch, and building record surfaces means engineering a polynomial that crosses as deeply into the discriminant as possible.