I'm sorry if my questions are trivial, but i can't solve them
1) Let $X$ be an irreducible affine variety with the action of a torus $T$.
Is it always true that there exists a dense orbit for the action of $T$? Or do i need some additional hypothesis on $X$?
2) Given $X$ as in (1), but without the hypothesis that $X$ is irreducible, why should the closure of every $T$-orbit be irreducible?
Thank you very much
1) No, you must require that the dimension of the torus equal the dimension of $X$. Here's an easy counterexample: take any projective variety $Y$ and let $X$ be its affine cone. Then there is an action of $\mathbb C^\ast$ on on $X$, but the orbits are just lines and $\{0\}$.
2) A $T$-orbit is given by $O(T,x_0) = \{ t \cdot x_0 \in X \mid t \in T\}$ for some $x_0 \in X$. Thus there is a surjective map $T \to O(T,x_0)$ (the orbit of $x_0$). Since $T$ is irreducible, we must have that $O(T,x_0)$ is irreducible as well (the image of an irreducible topological space is irreducible). Taking closure does not change this.