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.