# Explaining the concept of projective space: notes for students

This is a question on teaching.

I am teaching at this moment a course in algebraic geometry for master students on a very basic level. Today (this was the fourth lecture) I discovered that only four out of 20 students have ever seen the definition of projective space.

I would like to ask you if you know some nice, short notes that explain what the projective spaces are and that give some simple but still not tautological statements about them. A nice example that comes to my mind is Desargue's theorem, but I would like to have more of such statements. Maybe there are some theorems from classical plane geometry that can be proven using projective geometry? Even though I know what is projective space for almost 20 years I find a bit hard to find a good way to introduce and motivate it...

• Chapter 4 of Jennings, Modern Geometry with Applications? Feb 7, 2012 at 23:21
• Hitchin has some notes on projective geometry on his web page: people.maths.ox.ac.uk/hitchin/hitchinnotes/hitchinnotes.html He gets to Desargues and Pappus in the first 15 pages. Feb 8, 2012 at 8:13
• Dan, thank you for this comment. In fact I know these notes, I even proposed them to my students. Maybe I should look more carefully there. My only problem, is that these notes take 22 pages. This is perfect if you give a course on projective geometry. But if you give a course on algebraic geometry and your notes are just 40 pages long, there is no place even for additional 10 pages... Feb 8, 2012 at 8:57
• One point here (of which I am sure you are already aware, but I could not help myself): it seems much less likely that you will find an actual short set of notes doing what you want, than that you will find a short subset of a longer passage/article/chapter. Feb 8, 2012 at 14:46
• I would have been very surprised if many of your students had heard of projective space. There is no standard course that teaches geometry in the Euclidean sense outside of 9th grade, or in the case of non-Euclidean geometry, at all. Occasionally some geometer on the faculty will opt to give such a course, but who knows? Apr 4, 2013 at 19:16

Here is a short presentation on projective geometry with applets and animated GIF's to illustrate the basic constructions. It's elementary, but it comes in handy since most students today don't have the foggiest idea of what projective geomety is about.

http://www.math.poly.edu/courses/projective_geometry/Inaugural-Lecture/inaugural.html

Appendix A of Rational Points on Elliptic Curves, by Silverman and Tate, may be helpful. This unfortunately deals only with the projective plane, not projective spaces in general, but a reasonably well-motivated definition is given in pages 220-224. Later sections of the appendix include an elementary proof of Bezout's Theorem.

• Charles, thanks for the answer. This appendix is definitely very good, I especially like the second part with the proof of Bezout's theorem there. But unfortunately this did not help me with introducing projective space... (the first half is a bit longish and does not seem to have what I am looking for) Feb 7, 2012 at 23:23

For some purposes it is convenient to use the definition $$\mathbb{C}P^{\;n-1} = \{ A \in M_n(\mathbb{C}) : A^2 = A^{\dagger} = A, \text{trace}(A) = 1\}$$ This avoids problems if your students are shaky about quotient constructions or confused about considering a line in $\mathbb{C}^n$ as a single point in $\mathbb{C}P^{\;n-1}$. Unfortunately this description is not very compatible with the structure as a complex algebraic variety.

• This equality may look puzzling on first sight. But all it says is that the space of lines is the same as the space of all orthogonal projections onto lines. Feb 8, 2012 at 9:57

To foster student intuitions about projective geometry I have found it useful to draw pictures of train tracks receding into the horizon at infinity. This association is familiar to anybody (except cave dwellers) and motivates nicely the introduction of ideal points at infinity via pencils of parallel lines. Doing it via homogeneous coordinates is technically more convenient for the lecturer but is more of a challenge to the student.

A very elementary treatment may be found in the arXiv as arXiv:1110.3350v1 [math.HO].

• Richard, thank you for the answer. Unfortunately, it is not what I am looking for since would like to have something really short. Say, five-ten pages at most... Feb 8, 2012 at 8:48
• rigtriv.wordpress.com/2007/12/18/projective-varieties might be more what you are seeking. Feb 13, 2012 at 19:02

There is a nice chapter on projective geometry in Hilbert's "Geometry and Imagination".