Genus bound for a nonplanar degree-five curve
= Genus bound for a nonplanar degree-five curve
{title2=$\deg C=5,\quad\dim\langle C\rangle\geq3\Longrightarrow g(C)\leq2$}
The embedding <line bundle> has at least four independent <global sections>. If it is nonspecial, the <Riemann-Roch theorem> gives $h^0=6-g$, so $g\leq2$. If it is special, the <Clifford inequality for curves> gives $h^0\leq3$, a contradiction. The <smooth projective curve> need not span its full ambient <projective space>.