Fujita freeness for projectivized toric vector bundles

@misc{laface2026fujita,
  title = {{Fujita freeness for projectivized toric vector bundles}},
  author = {Antonio Laface},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28438},
  eprint = {2608.28438},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28438},
}

TY  - UNPB
TI  - Fujita freeness for projectivized toric vector bundles
AU  - Antonio Laface
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28438
UR  - https://arxiv.org/abs/2608.28438
AB  - Let X be a smooth projective toric variety of dimension n≥1 over an algebraically closed field of characteristic zero, let ℰ be a toric vector bundle of rank r≥2, and let π: Y=ℙ_X(ℰ)→ X be the projective bundle of one-dimensional quotients. Write an ample line bundle on Y as A=𝒪_Y(a)⊗π^*L, with a≥1. We record a blow-up argument proving that K_Y+mA is globally generated whenever an integer m satisfies ma≥ r and mδ(A)>n, where δ(A) is a positive integer obtained from the degrees of A on the invariant quotient sections over the torus-invariant curves of X. In particular, K_Y+mA is globally generated for m≥ n+1 and ma≥ r. Consequently every projectivized toric vector bundle satisfies Fujita's freeness conjecture. The uniform bound is sharp. We also formulate the result as a global-generation theorem for adjoint symmetric powers of ℰ and explain its relation with the Seshadri-constant results of Hering--Mustaţă--Payne and Fulger--Murayama. ChatGPT (OpenAI) was used to assist with mathematical discussion, language, and bibliographic searches.
ER  -