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  "body": "\n## Summary\n\nAn algebraic curve is an [[algebraic variety|HV6YBS0YDH]] of dimension one, such as the set of solutions of one polynomial equation in two unknowns.\nCurves are the first case of algebraic geometry beyond points, and their theory is the most complete.\nTheir main invariant is the [[genus|6W8TDV9VVN]].\nOver $\\C$, smooth projective curves are the same as compact [[Riemann surfaces|5K0N8QMV6Q]].\n\n## Explanations\n\n### L1 human\n\nA curve is a geometric object with one degree of freedom, cut out by polynomial equations.\nThe circle $x^2 + y^2 = 1$ and the cubic $y^2 = x^3 - x$ are curves in the plane.\nThe twisted cubic $\\set{(t, t^2, t^3)}$ is a curve in space that does not lie in any plane.\nOver the complex numbers each curve is a real surface, and a smooth projective connected curve looks like a sphere with some number of handles; that number is its genus.\n",
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