Why are projective spaces and varieties preferable?

Question

I am reading Hartshorne's Algebraic Geometry and it seems to me that projective spaces and varieties are prefferable. I don't know why.

In a more elementary stage of mathematics, when we try to find solutions to given equations, in $\mathbb R$ or $\mathbb C$, are we thinking in a more affine way? I really find it easier to understand things or to draw pictures in affine spaces, since it "looks" much more like the space we are living in.

So why do people, for example Hartshorne, always transform problems in affine spaces to problems in projective spaces? Why do they prefer projective spaces and varieties? What books are more basic and thorough on picturing varieties in projective spaces?

Answer 1

Imagine some student comes and says to you "Why do we have to work with complex numbers? It's really much easier to draw functions from $\mathbb R \to \mathbb R$" and for me real numbers are more intuitive. What would you answer? If I was this student, a motivating answer for me would be something like "The equation $x^2 + 1 = 0$ shows to us that if we want to solve algebraic equations, we need to work with $\mathbb C$ and more generally with algebraically closed fields." Nustellensatz doesn't work in non-algebraically closed fields.

Now back to your problem, do you know Bézout Theorem ?

If $\mathcal C_1, \mathcal C_2$ are two curves of degree $d,e$ in $\mathbb P^2$, then they have $d\cdot e$ points of intersection (counted with multiplicities).

Bézout's theorem is obviously false for $\mathbb A^2$ : take two parallel lines. Projective space allowed us to simplify lot of situations, and is the good framework if you want to do interesection theory for example. Homogenous coordinates are incredibly elegant and useful!. If you need a motivation to work only in projective space, you can take a look at the elegant book of Benoit Kloeckner (in French) or the book of Pierre Samuel.

If you need more motivating examples, "Un bref aperçu de la géométrie projective" (but I'm sure plenty of other books do actually contain the same stuff) gives for several theorems (Pappos for sure and maybe Pascal) a first affine proof and after that a shortened and more elegant projective proof. I think it's very instructive to read a bit about it !

Answer 2

Looking at problems in the projective setting is a kind of compactification. Compactness is a strong property which makes life easy lots of the time.

I will make this a community wiki answer and start a list of statements that make working with projective varieties particularly interesting:

• The image of a morphism $f:X\to Y$ is closed if $X$ is projective. In general, the image of a morphism of varieties is not a variety.
• As N.H. already mentioned, intersections in projective space behave "well" in the sense of Bézout's Theorem.

Answer 3

Short answer: morphisms defined on projective variety to other schemes/varieties are proper under reasonable condition, and in particular closed in the topological sense.

Cancellation theorem (as in Vakil's rising sea) says that given $\phi:X\mapsto Y$, $\psi: Y\mapsto S$, $\pi:X\mapsto S$ such that $\psi\circ\phi=\pi$, $\pi$ is proper, $\psi$ is separated, then $\psi$ is proper.

Now, a scheme/variety morphism $f:X\mapsto Y$ is proper means it's of finite type, universally closed, and separated. Hence if $f$ is proper, then it's closed.

Now, by fundamental theorem of elimination theory, we know that $\mathbb{P}^n_A\mapsto \text{Spec}(A)$ is proper. Closed embeddings are automatically proper. Proper are closed under composition.

Therefore all projective scheme/projective varieties are proper over $\text{Spec}(A)$. Let $S=\text{Spec}(A)$, especially in the case when $A=k$, you can see that suddenly we have a lot of closed morphism. Just take a separated scheme and a projective variety. Any morphism between these two scheme/varieties are proper, hence in particular closed.

A closed morphism is so much nicer and friendlier than an arbitrary morphism, isn't it?

After all, properness is the AG's analogue of compactness.