Lower semi-continuity of the Waldschmidt constants

Abstract

In this paper, we study the Waldschmidt constant of a generalized fat point subscheme $Z=m_1{p}_1+\cdots +m_r{p}_r$ of ${\mathbb{P}}^2,$ where $p_1,\dots ,p_r$ are essentially distinct points on ${\mathbb{P}}^2,$ satisfying the proximity inequalities. Furthermore, we prove its lower semi-continuity for $r\le 8.$ Using this property, we also calculate the Waldschmidt constants of the fat point subschemes $Z=p_1+\cdots +p_5$ giving weak del Pezzo surfaces of degree 4.

Keywords:

• Fat point subschemes
• symbolic powers
• Waldschmidt constant
• weak del Pezzo surfaces

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F20
• 14C20
• 14J26

Acknowledgments

I would like to thank my thesis advisor, Yongnam Lee, for his encouragement and great support. Also I thank Brian Harbourne for his incentive lectures at KAIST while this work was carried out.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

The author was supported by Basic Science Research Program through the NRF of Korea (2016930170).