**1.** Let $X$ be a smooth irreducible $\Bbb C$-variety,
on which the algebraic $\Bbb C$-group $G={\bf G}\_{a,{\Bbb C}}$
(the additive group) acts freely on the right:
$$ X\times \_{\Bbb C} G\to X,\quad (x,g)\mapsto x\cdot g.$$
Assume that there exists a surjective morphism
onto a smooth $\Bbb C$-variety $Y$
\begin{equation}\label{e:\*}
\varphi\colon X\to Y \tag{$\*$}
\end{equation}
whose fibres are the orbits of $G$ in $X$.
Then the morphism $\varphi$ is smooth, from which one can deduce that $\varphi$
induces a locally trivial fibre bundle (in the usual topology) of $C^\infty$-manifolds
$$\varphi\colon X(\Bbb C)\to Y(\Bbb C).$$



> 
> **Question 1.** Does it follow that $(\*)$
> is locally trivial in the *flat topology*, that is, a $Y$-*torsor* under $G$?
> In other words, is the morphism
> $$ X\times\_Y G\to X\times\_Y X,\quad (x,g)\mapsto (x,x\cdot g) $$
> an isomorphism of $\Bbb C$-varieties?
> 
> 
> 


**2.** Assume that $(\*)$ is a torsor.
Since $H^1({\Bbb C}(Y),G)=\{1\}$, we know that $(\*)$ admits a rational section.



> 
> **Question 2.** Does $(\*)$ admit a *regular* section? In other words, does there exist
> a regular map (morphism) $s\colon Y\to X$ such that $\varphi\circ s={\rm id}\_Y\,$?
> 
> 
> 



Taking quotient of a variety by the additive group
For 1, yes. In fact, any smooth morphism of varieties admits a section locally in the etale topology everywhere.


Proof: A generic hypersurface section is smooth of dimension one lower over any particular point. Repeat until the relative dimension is zero.


For 2, no. Any variety $Y$ with $H^1 (Y, \mathcal O\_Y) \neq 0$ is a counterexample, as every class gives a torsor $X$.


