PreprintCC BY 4.0arXiv
On ℙ1-bundles on ℙ3 that are small modifications of a Fano fourfold
Abstract
We find necessary conditions for the projectivization of a rank two vector bundle ℰ on ℙ3ℂ to be related to a smooth Fano 4-fold by a small ℚ-factorial modification (SQM). Furthermore, we fully describe the birational geometry of ℙℙ3(ℰ) when ℰ is stable, with c1=−1, c2=2. In this case, we prove that there exists a flip from X+ to ℙℙ3(ℰ), where X+ is smooth and Fano. We also determine the birational geometry and numerical invariants of X+.
Related
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BibTeX
@misc{pastorino2026ℙ1bundles,
title = {{On $\mathbbP^1$-bundles on $\mathbbP^3$ that are small modifications of a Fano fourfold}},
author = {Pier Roberto Pastorino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37608},
eprint = {2610.37608},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37608},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On ℙ^1-bundles on ℙ^3 that are small modifications of a Fano fourfold
AU - Pier Roberto Pastorino
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37608
UR - https://arxiv.org/abs/2610.37608
AB - We find necessary conditions for the projectivization of a rank two vector bundle ℰ on ℙ^3_ℂ to be related to a smooth Fano 4-fold by a small ℚ-factorial modification (SQM). Furthermore, we fully describe the birational geometry of ℙ_ℙ^3(ℰ) when ℰ is stable, with c_1=−1, c_2=2. In this case, we prove that there exists a flip from X^+ to ℙ_ℙ^3(ℰ), where X^+ is smooth and Fano. We also determine the birational geometry and numerical invariants of X^+.
ER -