Jumps, folds, and singularities of Kodaira moduli spaces

Abstract

For any integer $k$ we construct an explicit example of a twistor space which contains a one--parameter family of jumping rational curves, where the normal bundle changes from $O(1)+O(1)$ to $O(k)+O(2-k)$. For $k>3$ the resulting anti--self--dual Ricci-flat manifold is a Zariski cone in the space of holomorphic sections of $O(k)$. In the case $k=2$ we recover the canonical example of Hitchin's folded hyper-Kahler manifold, where the jumping lines form a three--parameter family. We show that in this case there exist normalisable solutions to the Schrodinger equation which extend through the fold.

Final version. To appear in the Journal of the London Mathematical Society