Non-projective complete log canonical surfaces
Abstract
We construct non-projective complete log canonical algebraic surfaces whose canonical divisor is semi-ample over any algebraically closed field other than the algebraic closure of a finite field. We provide a unified framework to construct such surfaces for any given non-negative Kodaira dimension, namely, zero, one, or two. Furthermore, we give a relatively elementary proof, within the framework of the minimal model theory, of the essentially known fact that any complete log canonical surface with Kodaira dimension minus infinity is projective. This fact confirms that our construction covers all possible Kodaira dimensions. As a byproduct, the proof shows that such a surface of Kodaira dimension minus infinity has at most one non-rational singular point.
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BibTeX
@misc{fujino2026nonprojective,
title = {{Non-projective complete log canonical surfaces}},
author = {Osamu Fujino and Nao Moriyama and Hiroshi Sato},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.05881},
eprint = {2606.05881},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.05881},
}
RIS
TY - UNPB
TI - Non-projective complete log canonical surfaces
AU - Osamu Fujino
AU - Nao Moriyama
AU - Hiroshi Sato
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2606.05881
UR - https://arxiv.org/abs/2606.05881
AB - We construct non-projective complete log canonical algebraic surfaces whose canonical divisor is semi-ample over any algebraically closed field other than the algebraic closure of a finite field. We provide a unified framework to construct such surfaces for any given non-negative Kodaira dimension, namely, zero, one, or two. Furthermore, we give a relatively elementary proof, within the framework of the minimal model theory, of the essentially known fact that any complete log canonical surface with Kodaira dimension minus infinity is projective. This fact confirms that our construction covers all possible Kodaira dimensions. As a byproduct, the proof shows that such a surface of Kodaira dimension minus infinity has at most one non-rational singular point.
ER -