Silverman, arithmetic of EC, I1.9 no nonconstant morphisms $P^m \to P^n$ for m>n This topic goes about problem 9 of the first chapter of Silverman, arithmetic of EC:
If $m>n$, prove that there are no nonconstant morphisms $P^m \to P^n$.
A solution can be found for example at Why is $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$ constant if dim $\phi(\mathbb{P}^n)<n$?,
but since it is in the beginning of the book, 
I think there should exist also an easier solution too, shouldn't it?
There was a hint given in the book: the dimension theorem.
 A: A morphism $\phi: \mathbb{P}^m \rightarrow \mathbb{P}^n$ can be given by $\phi =(F_0: \dotsc : F_n)$, where the $F_i$ are homogeneous polynomials of same degree in $m$ variables (See Remark 3.2 on the same chapter of Silverman).
The only problem is that the $F_i$ might have common zeros in $\mathbb{P}^m$. Say $P$ is one such point, then $\phi(P) = (0: \dotsc: 0)$ which is impossible and means $\phi$ is not regular at $P$. All we need to know now is that $$\bigcap\limits_{i=0}^n V(F_i) \neq \emptyset$$ 
in $\mathbb{P}^m$.
That is the hint of the problem, which is Hartshorne's  Theorem 7.2 on Chapter 1, but if you prefer you can have a look at Corollary 1.7, Theorem 1.22 on Shafarevich's Basic Algebraic Geometry, Volume I. Those theorems imply that $$dim \left(\bigcap\limits_{i=0}^n V(F_i) \right) \ge m-n > 0$$
And therefore the set of common zeros of the $F_i$ is non-empty and $\phi$ cannot be regular everywhere unless the degrees of the $F_i$ are $0$.
A: You are consider in fact $\bigcap_{i=1}^nV(F_i)$ instead of $\bigcap_{i=0}^nV(F_i)$. And you can just use the hint (no more), because using an induction over n, you get that if $X$ is an irreducible component of $\bigcap_{i=0}^nV(F_i)$, then $\dim\bigcap_{i=0}^nV(F_i)\geq \dim X\geq \dim\bigcap_{i=1}^nV(F_i)+\dim V(F_n)-m=m-n-1\geq 0$, so that $\bigcap_{i=0}^nV(F_i)\neq\emptyset$ by the last statement of the theorem on Hartshorne
