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  "body": "\n## Summary\n\nGeometric invariant theory, or GIT, is a method for building quotients of algebraic varieties by actions of [[reductive groups|2XS84PWE3S]].\nIt replaces the orbit space, which is often badly behaved, by the spectrum or the Proj of a [[ring of invariants|6VTR5EBNYC]].\nTo get a good quotient one throws away the unstable points and keeps the semistable ones.\nDavid Mumford developed the theory to construct [[moduli spaces|FJZRW9P0K3]], such as the moduli of curves and of vector bundles.\n\n## Explanations\n\n### L1 human\n\nSuppose a group moves the points of a space around and you want a space whose points are the orbits.\nSome orbits are too small or too close to others, and the naive orbit space is then not even Hausdorff.\nGIT uses invariant polynomials to decide which orbits to keep: two points are identified when no invariant can separate them.\nFor example, the group of nonzero scalars acts on $\\C^{n+1}$ by scaling.\nWith a suitable linearization, the origin is the only unstable point, and removing it and taking orbits gives the projective space $\\mathbb{P}^n$.\n",
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