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Sharp Quadratic Bounds for Anticanonical Volumes of Weak Fano Surfaces
Abstract
We prove an explicit lower bound of optimal quadratic order for the anticanonical volume of every complex ε-log canonical weak del Pezzo surface, with no restriction on its Picard number or its singularities. More precisely, the volume is at least min{cexc,ε2/3032064}, where cexc>0 is an explicit absolute constant. Quotients of the projective plane show that the exponent two is optimal. We also prove cubic upper bounds for the cyclic determinant mass and for the Picard number of a smooth rational log Calabi--Yau pair. We give geometric examples showing that these two auxiliary exponents are optimal in their respective settings.
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BibTeX
@misc{bie2026sharp,
title = {{Sharp Quadratic Bounds for Anticanonical Volumes of Weak Fano Surfaces}},
author = {Pinxian Bie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.06845},
eprint = {2609.06845},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06845},
}
RIS
TY - UNPB
TI - Sharp Quadratic Bounds for Anticanonical Volumes of Weak Fano Surfaces
AU - Pinxian Bie
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.06845
UR - https://arxiv.org/abs/2609.06845
AB - We prove an explicit lower bound of optimal quadratic order for the anticanonical volume of every complex ε-log canonical weak del Pezzo surface, with no restriction on its Picard number or its singularities. More precisely, the volume is at least min{c_exc,ε^2/3032064}, where c_exc>0 is an explicit absolute constant. Quotients of the projective plane show that the exponent two is optimal. We also prove cubic upper bounds for the cyclic determinant mass and for the Picard number of a smooth rational log Calabi--Yau pair. We give geometric examples showing that these two auxiliary exponents are optimal in their respective settings.
ER -