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A Family of Surfaces of Degree Six Where Miyaoka’s Bound is Sharp

  • Mariana Ferreira
  • Dayane Lira
  • Jacqueline RojasEmail author
Article
  • 61 Downloads

Abstract

Let \(r_d\) be the maximum number of skew lines that a smooth projective surface of degree d (over the complex numbers) can have. It is known that \(r_3=6\), \(r_4=16\) (Schläfli in Q J Math Soc 2:55–65, 110–121, 1858; Nikulin in Math USSR Izv 9:261–275, 1975) and was proven by Miyaoka in 1975 that \(r_d\le 2d(d -2)\) if \(d\ge 4\) (Miyaoka in Math Ann 268:159–172, 1984). Up to now \(r_d\) remains unknown for \(d\ge 5\). However, the lower bound \(d(d-2)+2\) was found by Rams (Proc Am Math Soc 133(1):11–13, 2005) which was improved by Boissière and Sarti (Ann Scuola Norm Sup Pisa Cl Sci 5:39–52, 2007), who showed that \(d(d-2)+4\le r_d\) for \(d \ge 5\) and odd. In this work, we take the family of degree d smooth surfaces \(\mathcal{R}_d\) in \({\mathbb {P}}^3\) (cf. (1)), considered by Boissière and Sarti (2007) and study \(r(\mathcal{R}_d)\), the maximum number of skew lines that \(\mathcal{R}_d\) can have. In fact, we prove that \(r(\mathcal{R}_d)\! =\! d(d-2)+4\) if \(d \ge 5\) and odd. Otherwise, we prove that \(r(\mathcal{R}_6)\ge 48\), which implies that Miyaoka’s bound is sharp for \(d=6\), i.e. \(r_6=48\). Still in the even case, we show that \(\mathcal{R}_d\) contains \(d(d-2)+4\) skew lines and we improve the Miyaoka’s bound for the family \(\mathcal{R}_d\) if d is even (Theorem 4.11).

Keywords

Skew lines Miyaoka’s bound Boissière–Sarti’s lower bound 

Mathematics Subject Classification

14N05 14N20 

Notes

References

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Copyright information

© Sociedade Brasileira de Matemática 2019

Authors and Affiliations

  1. 1.CCEN-Departamento de MatemáticaUFPB Cidade UniversitáriaJoão PessoaBrazil

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