Editing Projective variety (section)

=== Projective curves ===
{{Further|Algebraic curve}}
Projective schemes of dimension one are called ''projective curves''. Much of the theory of projective curves is about smooth projective curves, since the [[Singular point of an algebraic variety|singularities]] of curves can be resolved by [[Normalization of an algebraic variety|normalization]], which consists in taking locally the [[integral closure]] of the ring of regular functions. Smooth projective curves are isomorphic if and only if their [[Function field of an algebraic variety|function fields]] are isomorphic. The study of finite extensions of

:<math>\mathbb F_p(t),</math>

or equivalently smooth projective curves over <math>\mathbb F_p</math> is an important branch in [[algebraic number theory]].<ref>{{citation|author=Rosen|first=Michael|title=Number theory in Function Fields|year=2002|publisher=Springer}}</ref>

A smooth projective curve of genus one is called an [[elliptic curve]]. As a consequence of the [[Riemann–Roch theorem]], such a curve can be embedded as a closed subvariety in <math>\mathbb{P}^2</math>. In general, any (smooth) projective curve can be embedded in <math>\mathbb{P}^3</math> (for a proof, see [[Secant variety#Examples]]). Conversely, any smooth closed curve in <math>\mathbb{P}^2</math> of degree three has genus one by the [[genus formula]] and is thus an elliptic curve.

A smooth complete curve of genus greater than or equal to two is called a [[hyperelliptic curve]] if there is a finite morphism <math>C \to \mathbb{P}^1</math> of degree two.<ref>{{harvnb|Hartshorne|1977|loc=Ch IV, Exercise 1.7.}}</ref>