All Fat Point Subschemes in \({\mathbb {P}}^2\) with the Waldschmidt Constant Less than 5 / 2

Abstract

Let \({\mathscr {A}}=m_1p_1+ \cdots +m_np_n\) be a fat point subscheme of \({\mathbb {P}}^2\), and let \(I({\mathscr {A}})\), which is called a fat point ideal, be its corresponding ideal in \({\mathbb {K}}[{\mathbb {P}}^2]\). In this note, we identify those fat point ideals in \({\mathbb {K}} [{\mathbb {P}}^2]\) for which their Waldschmidt constants are less than 5 / 2.

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Acknowledgements

We would like to thank the anonymous referee for her/his careful reading of this manuscript, valuable suggestions and making helpful remarks. These all helped to improve the manuscript. This paper was prepared based on a research project supported by K.N. Toosi University of Technology research council and Iran National Science Foundation (INSF) Grant No. 97008366.

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Correspondence to Hassan Haghighi.

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Haghighi, H., Mosakhani, M. All Fat Point Subschemes in \({\mathbb {P}}^2\) with the Waldschmidt Constant Less than 5 / 2. Bull. Malays. Math. Sci. Soc. 43, 3221–3228 (2020). https://doi.org/10.1007/s40840-019-00865-y

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Keywords

  • Configuration of points
  • Star configuration
  • Symbolic power
  • Waldschmidt constant
  • Fat points

Mathematics Subject Classification

  • Primary 14N20
  • 13A02
  • Secondary 14N05
  • 13F20