A family of surfaces of degree six where Miyaoka's bound is sharp (Q2279172)

In the paper under review, the authors study a classical question devoted to the number of skew lines contained in a smooth projective surface \(S\) of degree \(d\) in \(\mathbb{P}^{3}_{\mathbb{C}}\). Let us denote by \(\Phi(S)\) the set of all lines in the surface \(S\) and we define \[ r_{d} = \sup\bigg\{r(\Phi(S)):S\text{ is a smooth surface of degree }d\text{ in }\mathbb{P}^{3}_{\mathbb{C}}\bigg\}, \] where \(r(\cdot)\) denotes the maximal number of skew lines in \(\Phi(S)\). The main result of the paper provides a new lower and upper bound on the number of skew lines for a certain family of smooth surfaces \(\mathcal{R}_{d}\) of degree \(d \in \{5, 7, 8, 9,\dots\}\). Main Result. Let \(\mathcal{R}_{d}\) be the smooth surface of degree \(d\) in \(\mathbb{P}^{3}_{\mathbb{C}}\) defined by \(\pi(x_{0},x_{1}) - \phi(x_{2},x_{3}) = 0\), where \(\phi(u,v) = uv(u^{d-2} - v^{d-2})\) with \(d \in \{5,7,8,9,\dots\}\). Then we have the following: i) \(r(\Phi(\mathcal{R}_{d})) = 4 + d(d-2)\) if \(d\) is odd, ii) \(4+d(d-2) \leq r(\Phi(\mathcal{R}_{d})) \leq d(d-2)+d^{2}/2\) if \(d\) is even. It is worth mentioning that the Miyaoka bound on the number of skew lines contained in a smooth surface of degree \(d\) in \(\mathbb{P}^{3}_{\mathbb{C}}\) tells us that \(r_{d} \leq 2d(d-2)\), so the authors have shown that for surfaces \(\mathcal{R}_{d}\) Miyaoka's bound is not optimal. The second result tells us that the Miyaoka's bound is sharp for degree \(6\) surfaces in \(\mathbb{P}^{3}_{\mathbb{C}}\), and consequently \(r_{6}=48\).