I am trying to understand when a morphism defined in an open dense subset of a variety can be extended to the whole variety.
For curves, it is known that if $f:C \to C’$ is a rational morphism from the curve $C$ to the curve $C’$ then f can be uniquely extended to the whole curve $C$.
On the other hand, for higher dimensional varieties the situation is more complicated, for instance,one can not extend a morphism from $\mathbb{A}^2\setminus(0,0) \to \mathbb{P}^2$.
I would like to know if is there any “nice” condition on a morphism $f:U \subset X \to Y$ where $U$ is a open dense subset of a (smooth, if necessary) projective variety $X$, and $Y$ is a projective variety, that make possible to extend $f$ to the whole $X$.
Thank you in advance.