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  "body": "\n## Summary\n\nThe genus of a smooth projective connected curve is the dimension of its space of regular differential $1$-forms.\nIt is the main invariant of a curve.\nCurves of genus $0$ are conics, and curves of genus $1$ with a rational point are elliptic curves.\nCurves of genus at least $2$ have finitely many automorphisms.\nOver $\\C$ it equals the topological [[genus|NJ787GZS1N]] of the associated compact Riemann surface, the number of handles.\nFor a singular curve, one takes the genus of its normalization, called the geometric genus.\n\n## Explanations\n\n### L1 human\n\nOver the complex numbers, a smooth projective connected curve is a closed surface, a sphere with some handles.\nThe number of handles is the genus.\nThe projective line is a sphere, of genus $0$.\nA smooth plane cubic such as $y^2 = x^3 - x$ is a torus, of genus $1$.\nThe genus controls almost everything about the curve, for instance through the [[Riemann–Roch theorem|WMVAT9MBZ1]].\n",
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