Bialynicki-Birula decomposition of a non-singular quasi-projective scheme. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:12:21Z http://mathoverflow.net/feeds/question/94620 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94620/bialynicki-birula-decomposition-of-a-non-singular-quasi-projective-scheme Bialynicki-Birula decomposition of a non-singular quasi-projective scheme. Francesco Sala 2012-04-20T10:32:55Z 2012-04-22T08:23:51Z <p>Fix an algebraically closed field $k$, an algebraic one-dimensional torus $G_m$ and a non-singular scheme $X$ of finite type over $k.$</p> <p>Let us define the following:</p> <p><strong>Condition 1:</strong> $X$ can be covered by $G_m$-invariant quasi-affine open subschemes.</p> <p>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).</p> <p>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.)</p> <p><strong>Question:</strong> Assume that $X$ is a non-singular quasi-projective scheme over $k.$ Under what extra assumptions, does $X$ satisfy Condition 1?</p> http://mathoverflow.net/questions/94620/bialynicki-birula-decomposition-of-a-non-singular-quasi-projective-scheme/94633#94633 Answer by Olivier Benoist for Bialynicki-Birula decomposition of a non-singular quasi-projective scheme. Olivier Benoist 2012-04-20T12:41:24Z 2012-04-22T08:23:51Z <p>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.</p> <p>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".</p> http://mathoverflow.net/questions/94620/bialynicki-birula-decomposition-of-a-non-singular-quasi-projective-scheme/94634#94634 Answer by Angelo for Bialynicki-Birula decomposition of a non-singular quasi-projective scheme. Angelo 2012-04-20T12:43:56Z 2012-04-20T12:43:56Z <p>Every normal variety with an action of a torus is covered by invariant affine open subsets. This is proved in Hideyasu Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28.</p>