Complete algebraic curve

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In algebraic geometry, a complete algebraic curve is an algebraic curve that is complete as an algebraic variety.

A projective curve, a dimension-one projective variety, is a complete curve. A complete curve (over an algebraically closed field) is projective.^[1] Because of this, over an algebraically closed field, the terms "projective curve" and "complete curve" are usually used interchangeably. Over a more general base scheme, the distinction still matters.

A curve in ${\displaystyle \mathbb {P} ^{3}}$ is called an (algebraic) space curve, while a curve in ${\displaystyle \mathbb {P} ^{2}}$ is called a plane curve. By means of a projection from a point, any smooth projective curve can be embedded into ${\displaystyle \mathbb {P} ^{3}}$;^[2] thus, up to a projection, every (smooth) curve is a space curve. Up to a birational morphism, every (smooth) curve can be embedded into ${\displaystyle \mathbb {P} ^{2}}$ as a nodal curve.^[3]

Riemann's existence theorem says that the category of compact Riemann surfaces is equivalent to that of smooth projective curves over the complex numbers.

Throughout the article, a curve mean a complete curve (but not necessarily smooth).