Canonical morphism associated to the fibered product of schemes and its fiber. After working a little bit I could show that given a fibered product of schemes $X \times_S Y$ any point of the underlying topological (denoted by $\vert X \times_S Y \vert$ space can be identified with a tuple $(x,y,s,\mathfrak{p})$ where $x \in X,y \in Y$ are points lying over $s$ and $\mathfrak{p}$ a prime ideal of 
$$k(x) \otimes_{k(s)} k(y).$$
Therefore we have a canonical morphsim 
$$\pi:\vert X \times_S Y \vert \to \vert X \vert \times_{\vert S \vert} \vert Y \vert$$
$$(x,y,s,\mathfrak{p}) \mapsto (x,y,s)$$ 
Clearly the fiber is given by $\pi^{-1}(x,y,s)= \operatorname{Spec}\left( k(x) \otimes_{k(s)} k(y) \right)$. I am asked to give examples of fibers of $\pi$ such that : $\pi^{-1}(p)$ is infinite and $\pi^{-1}(p)$ is disconnected.
I am having problems finding the right fields and since I am quite new to algebraic geometry I don't have the geometric intuition that is probably necessary to construct these examples of fibres. However in the disconnected case I know that I should find fields such that the tensor product has a non trivial idempotent or equivalent the tensor product is isomorphic to the product of two rings.
EDIT: I could find the disconected fibre via
$$ \mathbb{C} \otimes_{\mathbb{R}} \mathbb{C}\cong \mathbb{C} \times \mathbb{C}$$.
 A: Here are the required examples, in which $k$ is a field and $x,y$ are indeterminates over $k$:
I) Infinite fibre
Take $X=\operatorname {Spec}k(x), Y=\operatorname {Spec}k(y), S=\operatorname {Spec}k$.
Then  $X \times_S Y=\operatorname {Spec}(k(x) \otimes_{k} k(y))=\operatorname {Spec}(T^{-1}k[x,y])$ where $T\subset k[x,y]$ is the multiplicative set consisting of the non-zero polynomials of the form $f(x)g(y)$.
The set $\vert X \times_S Y\vert=\vert \operatorname {Spec}(T^{-1}k[x,y])\vert$ is the fibre of the unique point of $\vert X \vert \times_{\vert S \vert} \vert Y \vert$, and it is clearly infinite since it contains the prime ideals $\mathfrak p=\langle y-f(x)\rangle \subset T^{-1}k[x,y]\quad (f(x)\in k[x])$
II) Disconnected fibre
Take $X=Y=\operatorname {Spec}(\mathbb C\otimes_\mathbb R \mathbb C)$ and $  S=\operatorname {Spec}\mathbb R$.
Then  $\vert X \times_S Y\vert =\vert \operatorname {Spec}(\mathbb C\times \mathbb C)\vert$, a discrete two-point space, which is the disconnected unique fibre of $\vert X \times_S Y\vert \to \vert S\vert$
