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Nonexistence of degree two rational multisections of conic bundles over the plane
Abstract
We prove that a standard conic bundle X → ℙ2ℂ whose discriminant is very general of degree at least 18 admits no rational multisections of degree two. This is the first step towards proving a conjecture of Iskovskikh that there are conic bundle threefolds that are not unirational, since to prove that X is not unirational, it suffices to show that there are no rational multisections of any degree. Proving Iskovskikh's conjecture would provide the first example of a rationally connected variety that is not unirational.
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BibTeX
@misc{diller2026nonexistence,
title = {{Nonexistence of degree two rational multisections of conic bundles over the plane}},
author = {Jeffrey Diller and Lena Ji and Eric Riedl},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00185},
eprint = {2609.00185},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00185},
}
RIS
TY - UNPB
TI - Nonexistence of degree two rational multisections of conic bundles over the plane
AU - Jeffrey Diller
AU - Lena Ji
AU - Eric Riedl
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00185
UR - https://arxiv.org/abs/2609.00185
AB - We prove that a standard conic bundle X → ℙ^2_ℂ whose discriminant is very general of degree at least 18 admits no rational multisections of degree two. This is the first step towards proving a conjecture of Iskovskikh that there are conic bundle threefolds that are not unirational, since to prove that X is not unirational, it suffices to show that there are no rational multisections of any degree. Proving Iskovskikh's conjecture would provide the first example of a rationally connected variety that is not unirational.
ER -