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

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).