Smooth completion

In algebraic geometry, the smooth completion (or smooth compactification) of a smooth affine algebraic curve X is a complete smooth algebraic curve which contains X as an open subset. Smooth completions exist and are unique over a perfect field.

An affine form of a hyperelliptic curve may be presented as ${\displaystyle y^{2}=P(x)}$ where ${\displaystyle (x,y)\in \mathbb {C} ^{2}}$ and P(x) has distinct roots and has degree at least 5. The Zariski closure of the affine curve in ${\displaystyle \mathbb {C} \mathbb {P} ^{2}}$ is singular at the unique infinite point added. Nonetheless, the affine curve can be embedded in a unique compact Riemann surface called its smooth completion. The projection of the Riemann surface to ${\displaystyle \mathbb {C} \mathbb {P} ^{2}}$ is 2-to-1 over the singular point at infinity if ${\displaystyle P(x)}$ has even degree, and 1-to-1 (but ramified) otherwise.

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