On accumulation points of volumes of log surfaces

V. A. Alexeev ; W. Liu

Izvestiya. Mathematics , Tome 83 (2019) no. 4, pp. 657-675.

Résumé

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 $1/{42^2}$.

Détail

Keywords: log canonical surfaces
Mots-clés : volume, accumulation points.

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