Action of \$SL_{n+1}\$ on couples of linear spaces in \$\mathbb{P}^n\$. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:29:46Z http://mathoverflow.net/feeds/question/77363 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77363/action-of-sl-n1-on-couples-of-linear-spaces-in-mathbbpn Action of \$SL_{n+1}\$ on couples of linear spaces in \$\mathbb{P}^n\$. IMeasy 2011-10-06T14:45:16Z 2011-10-06T15:53:34Z <p>Let \$SL_{n+1}\$ act on \$\mathbb{P}^n\$ in the natural way. Suppose I take two linear subspaces \$\mathbb{P}^m\$ and \$\mathbb{P}^{n-m}\$, with \$m &lt; n\$, that intersect in one point. Is the action of \$SL_{n+1}\$ transitive on the set of such couples of linear subspaces?</p> http://mathoverflow.net/questions/77363/action-of-sl-n1-on-couples-of-linear-spaces-in-mathbbpn/77368#77368 Answer by Anton Fonarev for Action of \$SL_{n+1}\$ on couples of linear spaces in \$\mathbb{P}^n\$. Anton Fonarev 2011-10-06T15:53:34Z 2011-10-06T15:53:34Z <p>Sure it does. Denote your variety by \$X\$. Consider this action on the corresponding vector space \$V\$. Then your projective subspaces correspond to linear subspaces \$V_1\$ and \$V_2\$ of dimension \$m+1\$ and \$n-m+1\$ that intersect transversely. There is an obvious projection from the variety \$Y\$ of all frames in \$V\$ to \$X\$ (take the linear hull of the first \$m+1\$ and the last \$n-m+1\$ vectors). Now, up to a multiplication of all the vectors in the frame by a common scalar, \$SL_{n+1}\$ acts transitively on \$Y\$. Thus, it acts transitively on \$X\$.</p>