On accumulation points of volumes of log surfaces
- Authors: Alexeev V.A.1, Liu W.2
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Affiliations:
- University of Georgia
- Xiamen University
- Issue: Vol 83, No 4 (2019)
- Pages: 5-25
- Section: Articles
- URL: https://ogarev-online.ru/1607-0046/article/view/133776
- DOI: https://doi.org/10.4213/im8842
- ID: 133776
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Abstract
Let $\mathcal{C} \subset [0,1]$ be a set satisfying the descending chaincondition. We show that every accumulation point of volumes of log canonicalsurfaces $(X, B)$ with coefficients in $ \mathcal{C} $ can be realized asthe volume of a log canonical surface with big and nef $K_X+B$ and withcoefficients in $\overline{\mathcal{C}} \cup \{1 \}$ in such a way that atleast one coefficient lies in $\operatorname{Acc} (\mathcal{C}) \cup \{1 \}$.As a corollary, if $\overline {\mathcal{C}} \subset \mathbb{Q}$, then allaccumulation points of volumes are rational numbers. This proves a conjectureof Blache. For the set of standard coefficients$\mathcal{C}_2=\{1-1/{n} \mid n\in\mathbb{N} \} \cup \{1 \}$ we prove thatthe minimal accumulation point is between $1/{(7^2 \cdot 42^2)}$ and$1/{42^2}$.
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About the authors
Valery Anatol'evich Alexeev
University of Georgia
Wenfei Liu
Xiamen University
Email: wliu@xmu.edu.cn
References
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