15

Yes, and in fact something much more general is true Let $X$ and $Y$ be affine varieties defined over a field $k$. If the $k$-points of $X$ are Zariski dense, $X$ is reduced, and $f: X_{\mathbb C} \to Y_{\mathbb C}$ sends $X(k)$ to $Y(k)$, then $f$ is defined over $k$.
This was inspired by comments of Martin Brandenburg, Jef L, Andy Putman, and Piotr ...

8

This follows from Theorem 6 in Steinberg's "Some consequences of the elementary relations in $SL_n$" in "Finite groups — coming of age", Contemporary Math. 45 Amer. Math. Soc. (sorry, I could only find a Google books link), together with the remarks at the end of the proof. Restricting $f$ to $\Gamma = SL_n(\mathbb{Z})$, the theorem yields a polynomial map $...

5

Angelo's conjectural statement is proven in Fogarty, J.; Norman, P. "A fixed-point characterization of linearly reductive groups." MR0485896
(not enough reputation to post this as comment, sorry)

4

They are the same as the usual highest weight vectors but the $G$-representation is realized in the coordinate ring of some affine variety $X$. The usual argument goes if $\alpha:G \times X \to X$ is an algebraic action and $f \in \mathcal{O}_{X}$ then $\alpha^*(f) \in \mathcal{O}_G \otimes \mathcal{O}_X$ satisfies $\alpha^{*}(f)(g,x) = \sum_{i=1}^{n} \phi_i(...

4

The following is a counterexample which can be defined for arbitrarily large $p$'s.
Consider $U=\left\{ \begin{pmatrix}1&a&b\\&1&a^p\\&&1\end{pmatrix}:a,b\in k\right\}\subseteq\mathrm{GL}_3(k)$ and take $u=u_\lambda=\begin{pmatrix}1&\lambda\\&1&\lambda\\&&1\end{pmatrix}$ with $0\ne \lambda\in\mathbb{F}_p$ (i.e. $\...

3

I don't think that there is a different natural construction of a group scheme associated to a finite group. You can assume that authors mean this construction unless otherwise stated.
Notice, however, that this construction can be also done in the functorial setup, and a little bit more general and elegant as well:
Let $S$ be any base scheme. If $G$ is ...

answered Feb 15 at 13:31

Martin Brandenburg

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2

The map $D\to X$ induces a map $D/G\to X/G$. Since $D\to T$ is a principal $G$-bundle the induced map $D/G\to T$ is an isomorphism (e.g. by checking locally over $T$ and reduce to the case of the trivial bundle) and so we get a map $T \to X/G$. This is the map they refer to.
For the second, in general the map $X\to X/G$ is not a principal bundle. In ...

2

The affirmative answer to the first two question is indeed well-known. The existence of the section $W$ boils down to the vanishing of $H^1(X,\mathbf G_a)$ on an affine variety $X$.
A generic $n-d$-dimensional subspace will intersect a $G$-orbit in $D$ points where $D$ is the degree of that orbit. So, the answer to the third question is clearly negative if ...

2

The first two questions have an affirmative answer. See my paper https://arxiv.org/abs/1712.03838 for a constructive proof. The results are even true for connected solvable groups, also in positive characteristic. In the paper there are further references for special cases that should cover your setting. Your third questions is not covered, though, since the ...

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