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 \({\mathbb {P}}^{5}\) along the image of Segre threefold \({\mathbb {P}}^{1}\times {\mathbb {P}}^{2}\) and its hyperplane section.