Fix an algebraically closed field $k$, an algebraic one-dimensional torus $G\_m$ and a non-singular scheme $X$ of finite type over $k.$


Let us define the following:


**Condition 1:** $X$ can be covered by $G\_m$-invariant quasi-affine open subschemes.


In the paper "Some theorems on actions of algebraic groups" (The Annals of Mathematics, Second Series, Vol. 98, No. 3 (Nov., 1973), pp. 480-497), Bialynicki-Birula constructs, roughly speaking, to any action of $G\_m$ on $X$, satisfying Condition 1, two canonical decompositions of $X$ into non-singular $G\_m$-invariant locally closed subschemes (Theorem 4.1).


Moreover, Bialynicki-Birula states that if $X$ is projective, Condition 1 is automatically satisfied (he cites Kambayashi, Projective representations of algebraic groups of 
transformations, Amer. J. Math. 88 (1966), 199-205.)


**Question:** Assume that $X$ is a non-singular quasi-projective scheme over $k.$ Under what extra assumptions, does $X$ satisfy Condition 1?



Bialynicki-Birula decomposition of a non-singular quasi-projective scheme.
It is a theorem of Sumihiro (Equivariant completion, Corollary 2) that a normal variety over an algebraically closed field with an action of a torus is covered by invariant affine open subsets.


Here, the normality hypothesis is necessary : the conclusion does not hold for the action of $\mathbb{G}\_m$ on $\mathbb{P}^1$ with $0$ and $\infty$ glued together transversally.
 The hypothesis that the group is a torus is also necessary. Indeed, the statement already fails for SL(2), even for a proper action. There is an example in Białynicki-Birula and Święcicka's paper "On complete orbit spaces of SL(2) actions II".


