Noether–Lefschetz theorem
= Noether–Lefschetz theorem
{c}
{title2=$\operatorname{Pic}(X)=\mathbb Z[\mathcal O_X(1)]$}
{wiki}
A <very general> smooth degree-$d$ surface $X\subset\mathbb P^3_{\mathbb C}$ with $d\geq4$ has <Picard group> generated by its hyperplane class. The exceptional surfaces lie in a countable union of proper closed loci. A proof is given by https://publications.ias.edu/sites/default/files/noether.pdf[Griffiths and Harris, "On the Noether-Lefschetz Theorem"].