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  "body": "\n## Definition\n\nLet $k$ be an algebraically closed field and let $G$ be a [[reductive group|2XS84PWE3S]] over $k$.\nLet $G$ [[act|RW85C34GQW]] on a [[projective variety|A4H4CZ2V35]] $X$.\nLet $L$ be an [[ample line bundle|RRK8P6TXG2]] on $X$ with a $G$-linearization, that is, a lift of the action of $G$ to $L$.\nWrite $R = \\bigoplus_{m \\ge 0} H^0(X, L^{m})$ and let $R^G$ be its [[ring of invariants|6VTR5EBNYC]].\n\n1. A point $x \\in X$ is *semistable* if there are $m \\ge 1$ and an invariant section $s \\in H^0(X, L^m)^G$ with $s(x) \\ne 0$.\n2. A semistable point $x$ is *stable* if its orbit $G x$ is closed in the semistable locus $X^{ss}$ and its stabilizer $G_x$ is finite.\n3. A point that is not semistable is *unstable*.\n\nThe *GIT quotient* is the projective variety\n$$X /\\!\\!/_L\\, G = \\operatorname{Proj} R^G,$$\nbuilt with the [[Proj construction|A39XQHCVNW]], together with the morphism $\\pi \\colon X^{ss} \\to X /\\!\\!/_L\\, G$ induced by the inclusion $R^G \\subseteq R$.\n\n## Remarks\n\nThe ring $R^G$ is finitely generated because $G$ is reductive, so the quotient is a projective variety.\nThe map $\\pi$ is a good categorical quotient: two semistable points have the same image exactly when the closures of their orbits meet in $X^{ss}$.\nOn the stable locus $X^s$, which is open, $\\pi$ restricts to a geometric quotient $X^s \\to X^s / G$ whose fibres are single orbits.\n\nExample: let $G = \\mathbb{G}_m$ act on $X = \\mathbb{P}^1$ by $t \\cdot [x : y] = [t x : t^{-1} y]$, linearized on $L = \\mathcal{O}(1)$ by the weights $1$ and $-1$.\nA monomial $x^a y^b$ of degree $m = a + b$ is invariant exactly when $a = b$.\nSo the invariant sections are the multiples of powers of $xy$, and the semistable points are those with $xy \\ne 0$.\nThey form a single orbit, which is closed in $X^{ss}$, and the stabilizer of $[1 : 1]$ is $\\set{\\pm 1}$, so these points are stable and the quotient is a point.\n\nNon-example: the point $[1 : 0]$ is unstable, since every invariant section of positive degree is a multiple of $xy$ and vanishes there.\n\nFor an affine variety $X = \\operatorname{Spec} A$ with the trivial linearization, the quotient is $X /\\!\\!/ G = \\operatorname{Spec} A^G$.\n",
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