Here are some videos made by students in the Univeristy of Oregon Math Department’s summer undergraduate research program, mentored by me and Corey Brooke, illustrating some aspects of the geometry of surfaces.
This video by Isabella Harker shows the conic fibration on a cubic surface:
On the left we see the fixed surface and the rotating plane. On the right we just see the part of the surface that lies on one side of the plane, with the points where they intersect highlighted.
Famously, a smooth cubic surface contains 27 lines, at least if we’re willing to work in complex projective space — some of the lines might be at infinity, and some might be defined over C but not R.
If we slice the surface with a plane, we’ll usually get a smooth cubic curve. But if the plane contains one of the lines, we’ll get that line and a conic section: a circle, ellipse, parabola, hyperbola, or occasionally a pair of lines.
If we fix one line and look at the whole “pencil” of planes that contain it, we see that those residual conic sections sweep out the whole surface: every point on the surface lies on exactly one conic. We call this a “conic fibration” of the surface.
The surface is “KM01” from Oliver Labs’ collection, given
8x3 − 24xy2 + 6x2z +6y2z − 17z3 − 9x2 − 9y2 -24z2 − 6z + 4 = 0.
All 27 lines are real, but it’s not as special as the Clebsch cubic. You can buy a model from Labs’ Shapeways shop.
Harker produced the video in MATLAB, adapting an idea from a blog post by Mike Garrity; her code is available on her GitHub page. We also did a lot of preliminary messing around in CalcPlot3D and Mathpix’s 3D Grapher.
Here’s another version of the video, pausing when the conic degenerates to a pair of lines.
While surfaces of degree 3 always contain lines and conics, the Noether–Lefschetz theorem implies that surfaces of degree 4 and higher almost never do. If you have a 1-parameter family of quartic (that is, degree 4) surfaces, you expect only finitely many of the surfaces to contain lines or conics. This video by Karl Richter shows such a 1-parameter family, pausing to highlight some lines and conics when they pop up.
The family is given by
cos(t) · f(x, y, z) + sin(t) · g(x, y, z) = 0,
f(x, y, z) = 2x3y + 2x2y2 + 2xy3 + y4 + y3z − x2z2 + 2xyz2 + y2z2 + yz3 − 2z4 + x3 + 2xy2 + 2y3 + 2xyz + 2xz2 + yz2 + xy + z2 − x
g(x, y, z) = 2x4 + 3xy3 − 2x2yz + xy2z − 2y3z − x2z2 + 2xyz2 + 2xz3 − 2yz3 − 2x3 − 2x2y + 3xy2 + y3 + xyz + 2y2z − xz2 + yz2 + 2z3 − 3x2 − xy − 3y2 + 2yz − 3z2 − y − 2z + 3,
and t runs from -45° to 135°.
Finding all the lines in the family amounts to finding real solutions to 5 polynomial equations in 5 variables. We used a fantastic new piece of software called msolve. Actually there are 26 values of t where the surface contains a real line (and a further 294 if we work over C) but Richter just picked a few nice-looking ones.
Finding all the conics would have meant solving 9 equations in 9 variables, but that was too much even for msolve. Instead we planted some pre-chosen conics at t = 0° and t = 90°, and gave up on finding the others. Fun question: why do they always come in pairs?
For a precise statement and proof of the Noether–Lefschetz theorem, see Voisin’s book Hodge Theory and Complex Algebraic Geometry, §II.3.3. A higher-dimensional analogue, involving surfaces on cubic fourfolds, plays a big role in my research.
Here’s another version of the video, showing the whole family of surfaces with no pauses:
For a quartic surface that does contain a line, we can again take the pencil of planes containing the line and get an interesting fibration on the surface, as shown in this video by Isabella Harker:
Now the fibers are not conic sections, but elliptic curves. Despite the name, elliptic curves are different from ellipses: they're defined by quadratic cubic equations like y2 = x3 − x, whereas ellipses are described quadratic equations like x2 + 2y2 = 1. They come up when you try to calculate the circumference of an ellipse.
The surface is given by
−x3y + x3z + xy2z + 2y3z + 2x2z2 + xyz2 + 2xz3 + yz3 + 2z4 + y3 + 2xyz + y2z + 2xz2 + yz2 + xy + 2xz + 2yz + z2 + z = 0,
and the line is the x-axis. You can go spin it around if you want.
Here’s another version of the video, pausing at the two nodal fibers — over C there are actually 24 nodal fibers, but only two of them are defined over R — and again where the discriminant of the curve almost has a double root and we almost get a cuspidal fiber.
This work was funded in part by NSF grant no. DMS-2039316, and benefited from access to the University of Oregon high performance computing cluster, Talapas. We also thank Ben Young for computer time.