A new phantom on a rational surface

TL;DR

This paper constructs a universal phantom subcategory on a rational surface, providing new counterexamples to existing conjectures and introducing a novel co-connective DG-algebra with a phantom derived category.

Contribution

It introduces a new phantom subcategory on a rational surface and constructs a co-connective DG-algebra with a phantom derived category, challenging previous conjectures.

Findings

Counterexample to Kuznetsov's conjecture

Counterexample to Orlov's conjecture

New co-connective DG-algebra with phantom derived category

Abstract

We construct a universal phantom subcategory on the blow-up of the complex projective plane in 11 general points. This phantom subcategory is the orthogonal complement of a non-full exceptional collection of line bundles of maximal length. It provides a new counterexample to a conjecture of Kuznetsov and to a conjecture of Orlov. The first counterexample was constructed by Krah [Invent. Math. {\bf 235} (2024),1009--1018]. As an application, we construct a new co-connective DG-algebra whose derived category is a phantom.