On accumulation points of volumes of log surfaces

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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$1/{42^2}$.

About the authors

Valery Anatol'evich Alexeev

University of Georgia

Wenfei Liu

Xiamen University

Email: wliu@xmu.edu.cn

References

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