An application of wall-crossing to Noether-Lefschetz loci

Abstract

Consider a smooth projective 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda (such as ${\mathbb{P}}^3$, the quintic threefold or an abelian threefold). 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.

Referee's corrections. To appear in Quarterly Journal of Mathematics, special issue dedicated to the memory of Sir Michael Atiyah. 19 pages, appendix by Claire Voisin