I've just started learning about projective plane and have trouble understand how to visualize the points and lines in the plane.
Specifically, for this lemma below, when its says "all points of $l$ are contained in $C$ or $l$ intersects $C$ in at most two points", what kind of mental image should I have?
Here are the definitions that we use. I tried to think of $l$ as a plane going through the origin, but it didn't seem to work.
On
You can go a long way visualizing these things in the Euclidean plane that you’re used to. The extra thing to consider is that there’s also a line at infinity: interactions with it can at times be hard to visualize. The space also wraps around itself in a way similar to old video games in which an object that goes off an edge of the screen comes back on the opposite edge. This reflects the fact that the line at infinity has only one intersection with any other line—if you move along that other line, once you reach its point at infinity, continuing along the line brings you back from the opposite direction.
The gist of this lemma is that for any conic and line, if the line isn’t part of the conic (which can only happen if the conic is degenerate: a pair of intersecting lines or a double line) then they intersect in at most two points. This is really nothing new. The same is true in Euclidean geometry. The main difference is that we also need to consider the line at infinity. So, imagine taking a point on the unit circle and stretching the circle by moving it along some line through the circle’s center. As we move the point farther and farther from the center, for a while we get ellipses, obviously with no intersections with the line at infinity. Eventually we reach the line at infinity. At this point, we’ve stretched the circle into a parabola. If we now keep going “past infinity,” we have two intersection points with the line at infinity. In our extended Euclidean plane view, the resulting conics are hyperbolas.
In the model described in Theorem 1.31 and Lemma 1.32, for nondegenerate conics we get the classic conic section construction: the surface defined by $\phi(x,y,z)=0$ is some cone with vertex at the origin, and we cut this cone with the plane $z=1$. The circle-stretching gedankenexperiment above corresponds directly to starting with the right circular cone $x^2+y^2=z^2$ and tilting it away from the vertical. Intersecting conics and lines correspond in this model to intersections of these cones with planes through their vertex—the origin. These planes can intersect only at the vertex—no intersection points, be tangent to the cone—one intersection point, or intersect the cone in a pair of lines—two points. In the degenerate case, the “cone” will consist of one or two planes through the origin, which opens up the possibility of the intersecting plane coinciding with one of them.
The other model mentioned by Arthur in his answer—the unit sphere with antipodes identified—has the conceptual advantage that there’s no distinguished line at infinity as there is in the two models above, so it can be a good way to visualize otherwise hard-to-imagine interactions with that particular line. In that model, lines are great circles—intersections of planes through the origin with the unit sphere. Nondegenerate conics will be intersections of the cones described above with the sphere, i.e., certain types on non self-intersecting closed curves on the surface of the sphere. Degenerate conics are one or two great circles. When visualized in this model, Lemma 2.6 seems almost trivial.
There are different ways to visualize this.
One is the direct way as described in the definition: Everything happens in $\Bbb R^3$, projective points are lines through the origin, and projective lines are planes containing the origin.
One is on a sphere centered at the origin. A projective point is a pair of antipodal points on the sphere, and a projective line is a great circle. This is the intersection between the above interpretation and the unit sphere.
One is to use the regular plane with an added infinity (which you can't draw directly). This infinity stretches all around the plane (so it is a line), and things that go off in one direction "wrap around" and come in from the other (typical example: the graph of $y = \frac1x$ plane consists of a single curve that wraps around and meets itself at vertical infinity and at horizontal infinity). This corresponds to taking the first example and intersecting it with the plane $z = 1$. The line at infinity corresponds to the $xy$ plane.
I am sure there are many others too, but these are, I believe, the most common visualisations of the projective plane. They each take some practice getting used to, so don't be discouraged if you're having difficulties to begin with.