Extension of rational map from normal variety to algebraic group

Question

Assume $k$ is an algebraically closed field, $X$ is a normal variety over $k$, $G$ is an algebraic group over $k$. I heard that Weil proved any rational map from $X$ to $G$ which defined over a codimension $>1$ open subset can be extended to the whole $X$, and the idea was to consider the diagonal. What is the whole proof? I think this generalizes the case $G=\Bbb G_a$ (well-known fact about extension of rational functions) and there are counterexamples without the assumption on $G$(e.g $\Bbb A^2-0 \rightarrow \Bbb P^1$).

Answer 1

I'm not too convinced by the statement but, here's something that may be relevant. I'm paraphrasing from page 151-152, Chapter 8 section f of James Milne's ``Algebraic Groups: the theory of group schemes of finite type over a field". The proofs are given in a further reference:

MILNE, J. S. 1986. Abelian varieties, pp. 103–150. In Arithmetic geometry (Storrs, Conn., 1984). Springer-Verlag, Berlin.

1. Every rational map from a normal variety to a complete variety is defined on an open subset whose complement has codimension $\geq 2$.

2. Every rational map from a smooth variety to a connected group variety is defined on an open set whose complement is either empty or has pure codimension 1.

3. Combining 1)+2), every rational map from a smooth variety $X$ to an abelian variety $A$ is defined on all of $X$.

Maybe some part of, or some combination of 1), 2), 3) answers some part of your question.