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
by

8*x*^{3} − 24*xy*^{2} + 6*x*^{2}*z* +6*y*^{2}*z* − 17*z*^{3} − 9*x*^{2} − 9*y*^{2} -24*z*^{2} − 6*z* + 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,

where

*f*(*x*, *y*, *z*) = 2*x*^{3}*y* + 2*x*^{2}*y*^{2} + 2*x**y*^{3} + *y*^{4} + *y*^{3}*z* − *x*^{2}*z*^{2} + 2*x**y**z*^{2} + *y*^{2}*z*^{2} + *y**z*^{3} − 2*z*^{4} + *x*^{3} + 2*x**y*^{2} + 2*y*^{3} + 2*x**y**z* + 2*x**z*^{2} + *y**z*^{2} + *x**y* + *z*^{2} − *x*

*g*(*x*, *y*, *z*) = 2*x*^{4} + 3*xy*^{3} − 2*x*^{2}*yz* + *xy*^{2}*z* − 2*y*^{3}*z* − *x*^{2}*z*^{2} + 2*xyz*^{2} + 2*xz*^{3} − 2*yz*^{3} − 2*x*^{3} − 2*x*^{2}*y* + 3*xy*^{2} + *y*^{3} + *xyz* + 2*y*^{2}*z* − *xz*^{2} + *yz*^{2} + 2*z*^{3} − 3*x*^{2} − *xy* − 3*y*^{2} + 2*yz* − 3*z*^{2} − *y* − 2*z* + 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 *y*^{2} = *x*^{3} − *x*, whereas ellipses are described quadratic
equations like *x*^{2} + 2*y*^{2} = 1.
They come up when you try to calculate the circumference of an
ellipse.

The surface is given by

−*x*^{3}*y* + *x*^{3}*z* + *x**y*^{2}*z* + 2*y*^{3}*z* + 2*x*^{2}*z*^{2} + *x**y**z*^{2} + 2*x**z*^{3} + *y**z*^{3} + 2*z*^{4} + *y*^{3} + 2*x**y**z* + *y*^{2}*z* + 2*x**z*^{2} + *y**z*^{2} + *x**y* + 2*x**z* + 2*y**z* + *z*^{2} + *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.