A new phantom on a rational surface
Shihao Ma, Yirui Xiong, Song Yang
TL;DR
The paper constructs a new universal phantom subcategory on the blow-up $Y=\mathrm{Bl}_{11}\mathbb{P}^{2}$ at 11 general points, realized as the right orthogonal to a numerically (but not entirely) exceptional collection of length 14. It provides a semi-orthogonal decomposition $\mathrm{D}^{\mathrm{b}}(Y)=\langle \mathcal{A}_{Y},\mathcal{O}_{Y},\mathcal{O}_{Y}(D_{1}),\dots,\mathcal{O}_{Y}(D_{11}),\mathcal{O}_{Y}(F),\mathcal{O}_{Y}(2F)\rangle$, with $D_i=K_Y-E_i$ and $F=3K_Y-H$, where $\mathcal{A}_{Y}$ is a phantom and, by a Lefschetz-type Chow motive, universal. Building on Krah’s earlier example, the authors compute a numerically exceptional collection of maximal length on $Y$ and show it is not full; this leads to the phantom subcategory. They further demonstrate that the associated cohomological DG-algebra $\mathrm{RHom}(\mathcal{T},\mathcal{T})$ attached to a strong generator $\mathcal{T}$ of the phantom is co-connective, hence the phantom category is realized as the derived category of a co-connective DG-algebra, yielding both new counterexamples and structural insight into the derived categories of rational surfaces.
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.