Rouquier dimension of some blow-ups

Rapha"{e}l Rouquier introduced an invariant of triangulated categories which is known as Rouquier dimension. Orlov conjectured that for any smooth quasi-projective variety $X$ the Rouquier dimension of $D_{\mathrm{c}\mathrm{o}\mathrm{h}}^b(X)$ is equal to $\mathrm{d}\mathrm{i}\mathrm{m}X$. In this note we show that some blow-ups of projective spaces satisfy Orlov’s conjecture. This includes a blow-up of ${\mathbb{P}}^2$ in nine arbitrary distinct points, or a blow-up of three distinct points lying on an exceptional divisor of a blow-up of ${\mathbb{P}}^3$ in a line. In particular, our method gives an alternative proof of Orlov’s conjecture for del Pezzo surfaces, first established by Ballard and Favero.