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Izv. RAN. Ser. Mat., 2019, Volume 83, Issue 4, Pages 5–25 (Mi izv8842)  

This article is cited in 1 scientific paper (total in 1 paper)

On accumulation points of volumes of log surfaces

V. A. Alexeeva, W. Liub

a Department of Mathematics, University of Georgia, Athens, USA
b School of Mathematical Sciences, Xiamen University, China

Abstract: 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, volume, accumulation points.

Funding Agency Grant Number
National Science Foundation DMS-1603604
National Natural Science Foundation of China 11501012
11771294
The Recruitment Program for Young Professionals
The first author's work was partially supported by NSF, grant DMS-1603604. The second author was partially supported by NSFC (no. 11501012, no. 11771294) and Recruitment Program for Young Professionals.


DOI: https://doi.org/10.4213/im8842

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English version:
Izvestiya: Mathematics, 2019, 83:4, 657–675

Bibliographic databases:

UDC: 512.774.15+512.774.2
MSC: Primary 14J29; Secondary 14J26, 14R05
Received: 13.07.2018

Citation: V. A. Alexeev, W. Liu, “On accumulation points of volumes of log surfaces”, Izv. RAN. Ser. Mat., 83:4 (2019), 5–25; Izv. Math., 83:4 (2019), 657–675

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Alexeev V., Liu W., “Log Surfaces of Picard Rank One From Four Lines in the Plane”, Eur. J. Math., 5:3, SI (2019), 622–639  crossref  isi
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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