Let X be an affine toric variety. The total coordinates on X provide a canonical presentation $$\overline{X}\to X$$ of X as a quotient of a vector space $$\overline{X}$$ by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.

We discuss the problem of stable conjugacy of finite subgroups of Cremona groups. We compute the stable birational invariant H 1(G, Pic(X)) for cyclic groups of prime order.

Here we give an explicit polynomial bound (in term of $K_X^2$ and not depending on the prime $p$) for the order of the automorphism group of a minimal surface $X$ of general type defined over a field of characteristic $p>0$.

Here we give an upper polynomial bound (as function of $K_{X^2}$ but independent on $p$) for the order of a $p$-subgroup of $Aut{\left(X\right)}_{red}$ with $X$ minimal surface of general type defined over the field $K$ with $char\left(K\right)=p>0$. Then we discuss the non existence of similar bounds for the dimension as $K$-vector space of the structural sheaf of the scheme $Aut\left(X\right)$.