PreprintarXiv licencearXiv
Effective nonvanishing of K-trivial regular surfaces over imperfect fields
Abstract
For any regular projective surface X over an arbitrary field k of characteristic p>0 with KX≡0, we prove that 60KX∼0 . This uniform bound is optimal. To show that the factor $5$ is necessary, we construct an example in characteristic $5$ for which KX has exact order five. If X is geometrically normal, then 12KX∼0. Our main focus is the geometrically non-normal case, where we obtain more precise bounds according to the Albanese morphism. If it is nontrivial, then p=2 or $3$, and we prove 12KX∼0 and 6KX∼0, respectively; both bounds are optimal. If it is trivial, then KX∼0 for p≥7, while 5KX∼0, 3KX∼0, and 4KX∼0 for p=5,3,2, respectively.
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BibTeX
@misc{chen2026effective,
title = {{Effective nonvanishing of $K$-trivial regular surfaces over imperfect fields}},
author = {Jingshan Chen and Chongning Wang and Lei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09542},
eprint = {2610.09542},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09542},
}
RIS
TY - UNPB
TI - Effective nonvanishing of K-trivial regular surfaces over imperfect fields
AU - Jingshan Chen
AU - Chongning Wang
AU - Lei Zhang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09542
UR - https://arxiv.org/abs/2610.09542
AB - For any regular projective surface X over an arbitrary field k of characteristic p>0 with K_X≡0, we prove that 60K_X∼0 . This uniform bound is optimal. To show that the factor $5$ is necessary, we construct an example in characteristic $5$ for which K_X has exact order five. If X is geometrically normal, then 12K_X∼0. Our main focus is the geometrically non-normal case, where we obtain more precise bounds according to the Albanese morphism. If it is nontrivial, then p=2 or $3$, and we prove 12K_X∼0 and 6K_X∼0, respectively; both bounds are optimal. If it is trivial, then K_X∼0 for p≥7, while 5K_X∼0, 3K_X∼0, and 4K_X∼0 for p=5,3,2, respectively.
ER -