Rouquier dimension of some blow-ups

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