Nonexistence of phantoms when the canonical or anticanonical bundle has a nonzero section

Prove that every smooth projective surface S over an algebraically closed field of characteristic zero with a nonzero global section of either the canonical bundle ω_S or the anticanonical bundle ω_S^∨ has no phantom admissible subcategories in the bounded derived category of coherent sheaves on S.

Background

The paper studies the existence of phantom subcategories in derived categories of smooth projective surfaces, motivated by examples and nonexistence results in the literature. The main theorem proves nonexistence of phantoms on certain rational surfaces admitting a smooth anticanonical divisor with injective restriction on Picard groups, such as blowups of the projective plane at very general points lying on a smooth cubic curve.

Building on Pirozhkov’s methods for del Pezzo surfaces, the authors propose a broader conjectural criterion ensuring the absence of phantoms: the presence of a nonzero section of either the canonical or anticanonical line bundle. This conjecture extends the scope beyond the specific geometric setups treated in the main theorem.

References

Conjecture. Let $S$ be a smooth projective surface over an algebraically closed field $k$ of characteristic zero. Assume $H0(S,\cL) \neq 0$ for $\cL$ either $\omega_S$ or $\omega_S\vee$. Then $S$ has no phantoms.

— Non-existence of phantoms on some non-generic blowups of the projective plane  (2405.01683 - Borisov et al., 2024) in Introduction, Conjecture (following Theorem 1)