k-schemes examples I'm trying to get a feel for k-schemes, and I was wondering what would be some good concrete examples to play with that would illustrate the importance of their rational points?
Thanks in advance
 A: I think the thing to keep in mind is the following: if $X/k$ is a variety, then $X(k)$ reflects arithmetic, whereas $X(\overline{k})$ reflects geometry. 
For example, take the elliptic curve $E/\mathbb{Q}$ defined by
$$E:y^2z=x^3+z^3$$
Then, one can check that $E(\mathbb{Q})=\{[0:1:0]\}$. This is reflecting the arithmetic of $\mathbb{Q}$--there are no solutions to the Diophantine equation
$$y^2=x^3+1$$
over $\mathbb{Q}$. 
But, while over $\mathbb{Q}$ the curve $E$ doesn't have many rational points, it has many, many points over $\overline{\mathbb{Q}}$. More concretely, if one thinks about $E/\mathbb{C}$ then $E(\mathbb{C})$ has the structure of a torus! This toroidal structure, the biholomorphism type of the Riemann surface $E(\mathbb{C})$, is measuring something inherently geometric.
As another example to mull over, consider the following two conics in $\mathbb{P}^2_{\mathbb{R}}$:
$$C_1:y^2+x^2+z^2=0\qquad C_2:y^2+x^2-z^2=0$$
These two conics are very different when thought about over $\mathbb{R}$. In particular:
$$C_1(\mathbb{R})=\varnothing$$
, whereas 
$$C_2(\mathbb{R})=\mathbb{R}\cup\{\infty\}=\mathbb{P}^1_\mathbb{R}(\mathbb{R})$$
Once again, the lack of rational points in the first case is an arithmetic statement about the field $\mathbb{R}$--it lacks a square root of $-1$. What makes this example more interesting is that even though $C_1\not\cong C_2$ (they don't even have the same number of $\mathbb{R}$-points) they are geometrically the same:
$$(C_1)_{\mathbb{C}}\cong (C_2)_{\mathbb{C}}\cong \mathbb{P}^1_{\mathbb{C}}$$
(in fact, you can check that already over $\mathbb{R}$, $C_2\cong \mathbb{P}^1_{\mathbb{R}}$). Thus, the two curves geometrically are the same object, but it's the arithmetic captured by their $\mathbb{R}$-points that makes them interesting to study separately. 
Remark: If you have any background in number theory, the last example has a nice generalization involving something you might already be familiar with. Namely, one can try and study varieties over $k$ which become $\mathbb{P}^1$ over $\overline{k}$. These naturally give classes in the Brauer group $\mathrm{Br}(k)$. In fact, they actually give elements of $\mathrm{Br}(k)[2]$--what you get from such objects are essentially the quaternion algebras over $k$. And, if you extend from $\mathbb{P}^1$ to $\mathbb{P}^n$, then looking at the interesting arithmetic structure of varieties over $k$ geometrically equal to some projective space (spaces somehow made special/distinct by their rational points) then one recovers all of $\mathrm{Br}(k)$--an object of obvious arithmetic importance.
If you'd like, a nice exercise is to use this fact together with the computation $\mathrm{Br}(\mathbb{R})=\{\mathbb{R},\mathbb{H}\}$ (the Hamiltonian quaternions) to show that the only varieties over $\mathbb{R}$ which are geometrically some projective space are 
$$\{C_1\}\cup\left\{\mathbb{P}^n_{\mathbb{R}}:n\in\mathbb{N}\right\}$$
