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... 
 A: 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
A: 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.
A: 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.
A: 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: A very elementary treatment may be found in the arXiv as arXiv:1110.3350v1 [math.HO].
A: There is a nice chapter on projective geometry in Hilbert's "Geometry and Imagination". 
