Question

Let P be a normal variety over an algebraically closed field k, G a torus over k acting on P, assume that the stabilizer of the generic point of P is reduced (resp. connected or both), is it ture then that the stabilizer of every point of P is reduced (resp. connected or both)? And WHY?

What will happen if we assume moreover that P has finitly many G-obits?

Answer 1

For "connected" the answer is surely no, as the following example shows.

Let $k=\mathbb{C}$, and consider the action of $\mathbb{C}^*$ on $\mathbb{C}^3-\{0\}$ defined as

$\lambda \cdot (x,y,z):=(\lambda x, \lambda y, \lambda^2 z)$.

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 $[0:0:1]$.

For "reduced" the answer is yes when $k$ has characteristic $0$, since every group scheme in characteristic zero is reduced.

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 $\mu_2:=\textrm{Spec }k[t]/(t^2-1)$.