An Application of Wall-Crossing to Noether–Lefschetz Loci

Consider a smooth projective 3-fold $X$ satisfying the Bogomolov–Gieseker conjecture of Bayer–Macrì–Toda (such as ${\mathbb{P}}^3$, the quintic 3-fold or an abelian 3-fold). Let $L$ be a line bundle supported on a very positive surface in $X$. If $c_1(L)$ is a primitive cohomology class, then we show it has very negative square.