Phantoms and Exceptional Collections on Rational Surfaces

This thesis is concerned with exceptional collections on smooth projective surfaces. Any rational surface over an algebraically closed field admits a full exceptional collection. We extend certain classification results regarding exceptional collections, previously known for del Pezzo surfaces, to the blow-up of the projective plane in 9 very general points. More generally, we obtain a classification result for numerically exceptional collections of maximal length on smooth projective surfaces $X$ with \chi(\mathcal{O}_X)=1 and K_X² + rk(\mathsf{K}₀^\mathrm{num}(X)) = 12. In contrast to the case of 9 points, on the blow-up in 10 general points we construct an exceptional collection of maximal length which is not full. As a consequence, the orthogonal complement of this collection is a universal phantom category. This disproves a conjecture of Kuznetsov and a conjecture of Orlov.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Bielefeld

Year

2024

Author dc:creator

• Krah, Johannes

Repository record source_url

https://pub.uni-bielefeld.de/record/2994518

OAI identifier oai:identifier

oai:pub.uni-bielefeld.de:2994518