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In general, the only definition I know of GIT quotient is $Proj$ of the invariant ring. The obvious statements one can make about the rational map $Proj\ R\to Proj\ R^G$ are that it collapses $G$-orbi …

Let $SL_2$ act on ${\mathbb A}^2$. This has two orbits, $\{\vec 0\}$ and its complement, and the latter is not affine.

If you're willing to quotient by a nonreductive group, then $M_n//B$ will get you the $GL(n)$ flag manifold. (People are usually afraid to do so, worrying that the ring of invariants won't be Noetheri …

There are many GIT quotients, since to define one requires a choice of $G$-line bundle, so a pair of naturals for each $F$. There's an obvious democratic choice -- $(a,b) = (1,1)$ for every $F$ -- b …

$n=1$ already tells you that the anticanonical divisor is going to be nicer than the canonical, in that it's effective. There, the divisor given by the three coordinate lines is anticanonical. Next s …

Let me advertise the polytopal point of view. Start with $T^k$ acting on ${\mathbb A}^n$ linearly, i.e. we have a map $T^k \to T^n$. The $T^n$-moment polytope of ${\mathbb A}^n$ is just the orthant $ …