Full exceptional collections of line bundles on the blow-up of \({\mathbb {P}}^{5}\) along Segre threefold

We prove Kuznetsov’s conjecture on the fullness of exceptional collections in the line bundles case for the blow-up of \({\mathbb {P}}^{5}\) ${\mathbb{P}}^5$ along the image of Segre threefold \({\mathbb {P}}^{1}\times {\mathbb {P}}^{2}\) ${\mathbb{P}}^1\times {\mathbb{P}}^2$ and its hyperplane section.