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  "body": "\n## Summary\n\nA projective variety is an irreducible subset of projective space $\\mathbb{P}^n$ over an algebraically closed field cut out by homogeneous polynomial equations.\nProjective varieties are the algebraic analogue of compact spaces.\nOver $\\C$ they are compact in the usual topology, and images of projective varieties under morphisms are closed.\nMost classical algebraic geometry, such as plane curves, Bézout's theorem and intersection theory, takes place on projective varieties.\n\n## Explanations\n\n### L1 human\n\nIn projective space a point has homogeneous coordinates that are only defined up to scaling.\nSo an equation makes sense on points only if scaling does not change whether it holds, which is true for homogeneous polynomials such as $x^2 + y^2 - z^2$.\nThe zero set of $x^2 + y^2 - z^2$ in the projective plane is a conic.\nIn the chart $z = 1$ it is the circle $x^2 + y^2 = 1$, and the projective version adds no points at infinity over $\\R$ but two over $\\C$.\n",
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