Genus bound for a nonplanar degree-five curve ($\mathrm{deg}C=5,\dim ⟨C⟩\ge 3⟹g(C)\le 2$)

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 $g\le 2$. If it is special, the Clifford inequality for curves gives $h^0\le 3$, a contradiction. The smooth projective curve need not span its full ambient projective space.