An application of wall-crossing to Noether-Lefschetz loci

TL;DR

This paper explores the application of wall-crossing techniques to the study of Noether-Lefschetz loci on certain smooth projective 3-folds, revealing properties of line bundles supported on positive surfaces.

Contribution

It demonstrates how wall-crossing methods can be applied to analyze Noether-Lefschetz loci in specific 3-folds satisfying the Bogomolov-Gieseker conjecture, providing new insights into their structure.

Findings

Line bundles supported on very positive surfaces have very negative square when their first Chern class is primitive.

Application of wall-crossing yields new constraints on the geometry of Noether-Lefschetz loci.

Results hold for 3-folds like $P^3$, quintic threefolds, and abelian threefolds.

Abstract

Consider a smooth projective 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture of Bayer-Macr\`{i}-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.