Let $K$ be an algebraically closed field.
We have different important theorems to study morphism of variety, in particular there is a clear and constructive description of the morphism between affine variety
Let $X \subseteq \mathbb{A}^n_K$ and $Y \subseteq \mathbb{A}^m_K$ be affine varieties (my definition of affine variety don’t require the irreducibility). We know that a morphism $f: U \subseteq X\to Y$ with $U$ open is exactly of the form $f=(\phi_1, \dots , \phi_m):U \to Y$ where $\phi_i \in \mathcal{O}_X(U)$ for all $i$.
There is a clear and constructive description of the morphism between projective spaces
$f: \mathbb{P}ⁿ \to \mathbb{P}^m $ is of the form $$x \to (f_0(x):\dots : f_m(x))$$ with $f_0,\dots , f_m \in K[x_0,\dots ,x_n]$ homogeneous polynomials of the same degree such that $V_p(f_0,\dots , f_m)=\emptyset$. Furthermore if $n>m$ then $f$ is constant.
So I am curious to know if there exist theorems that can describe general morphism of the form
- $f: X \subseteq \mathbb{P}^n \to Y \subseteq \mathbb{P}^m$
- $ f: X \subseteq \mathbb{P}^n \to Y \subseteq \mathbb{A}^m$
- $f: X \subseteq \mathbb{A}^n \to Y \subseteq \mathbb{P}^m$
where $X, Y$ are affine or projective varieties.