= Rationality of a smooth projective genus-zero curve
{title2=$g(C)=0\Longrightarrow C\cong\mathbb P^1$}
Over an <algebraically closed field>, choose a point $P$ on the <smooth projective curve>. The <Riemann-Roch theorem> gives $\dim L(P)=2$: the complementary <canonical divisor> term has negative degree. A nonconstant function in $L(P)$ has exactly one simple pole, so it defines a degree-one <morphism of algebraic curves> to the <projective line>. A degree-one map between <smooth projective curves> is an <isomorphism>. The existence of a rational point is essential over a field which is not algebraically closed.
Back to article page
