About toric varieties---properties of stabilizers - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:36:24Z http://mathoverflow.net/feeds/question/47670 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47670/about-toric-varieties-properties-of-stabilizers About toric varieties---properties of stabilizers zcqc 2010-11-29T12:42:16Z 2010-11-29T17:56:48Z <p>Let <em>P</em> be a normal variety over an algebraically closed field <em>k</em>, <em>G</em> a torus over <em>k</em> acting on <em>P</em>, assume that the stabilizer of the generic point of <em>P</em> is reduced (resp. connected or both), is it ture then that the stabilizer of every point of <em>P</em> is reduced (resp. connected or both)? And WHY?</p> <p>What will happen if we assume moreover that <em>P</em> has finitly many <em>G</em>-obits?</p> http://mathoverflow.net/questions/47670/about-toric-varieties-properties-of-stabilizers/47700#47700 Answer by Francesco Polizzi for About toric varieties---properties of stabilizers Francesco Polizzi 2010-11-29T17:56:48Z 2010-11-29T17:56:48Z <p>For "connected" the answer is surely <strong>no</strong>, as the following example shows. </p> <p>Let $k=\mathbb{C}$, and consider the action of $\mathbb{C}^*$ on $\mathbb{C}^3-\{0\}$ defined as</p> <p>$\lambda \cdot (x,y,z):=(\lambda x, \lambda y, \lambda^2 z)$.</p> <p>Then the general point has trivial stabilizer, whereas the points on the line $x=y=0$ have a non-connected stabilizer isomorphic tO $\mathbb{Z}/2\mathbb{Z}$. The quotient is actually the weighted projective space $\mathbb{P}(1,1,2)$, which has a singularity of type $\frac{1}{2}(1,1)$ at the point $[0:0:1]$.</p> <p>For "reduced" the answer is <strong>yes</strong> when $k$ has characteristic $0$, since every group scheme in characteristic zero is reduced. </p> <p>When $k$ has characteristic $p >0$, I suspect that there are counterexamples. For instance, if $p=2$ it seems to me that, repeating the previous construction with $k$ instead of $\mathbb{C}$, the points lying on the line $x=y=0$ have stabilizer isomorphic to the non-reduced group scheme $\mu_2:=\textrm{Spec }k[t]/(t^2-1)$. </p>