Geodesic
On accumulation points of volumes of log surfaces
Izvestiya. Mathematics , Tome 83 (2019) no. 4, pp. 657-675

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Let $\mathcal{C} \subset [0,1]$ be a set satisfying the descending chain condition. We show that every accumulation point of volumes of log canonical surfaces $(X, B)$ with coefficients in $ \mathcal{C} $ can be realized as the volume of a log canonical surface with big and nef $K_X+B$ and with coefficients in $\overline{\mathcal{C}} \cup \{1 \}$ in such a way that at least one coefficient lies in $\operatorname{Acc} (\mathcal{C}) \cup \{1 \}$. As a corollary, if $\overline {\mathcal{C}} \subset \mathbb{Q}$, then all accumulation points of volumes are rational numbers. This proves a conjecture of Blache. For the set of standard coefficients $\mathcal{C}_2=\{1-1/{n} \mid n\in\mathbb{N} \} \cup \{1 \}$ we prove that the minimal accumulation point is between $1/{(7^2 \cdot 42^2)}$ and $1/{42^2}$.
Keywords: log canonical surfaces
Mots-clés : volume, accumulation points. 
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     author = {V. A. Alexeev and W. Liu},
     title = {On accumulation points of volumes of log surfaces},
     journal = {Izvestiya. Mathematics },
     pages = {657--675},
     publisher = {mathdoc},
     volume = {83},
     number = {4},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2019_83_4_a1/}
}
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V. A. Alexeev; W. Liu. On accumulation points of volumes of log surfaces. Izvestiya. Mathematics , Tome 83 (2019) no. 4, pp. 657-675. http://geodesic.mathdoc.fr/item/IM2_2019_83_4_a1/