Year: 2026
Type: preprint
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 $\min\{c_{exc},ε^2/3032064\}$, 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.
Source: arXiv (Cornell University)
Author: Bie, Pinxian
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212022389