Rouquier dimension of some blow-ups

TL;DR

This paper investigates the Rouquier dimension of certain blow-ups of projective spaces, confirming Orlov's conjecture in these cases and providing alternative proofs for known results on del Pezzo surfaces.

Contribution

It demonstrates that specific blow-ups of projective spaces satisfy Orlov's conjecture, expanding the class of varieties where the conjecture holds and offering new proof techniques.

Findings

Blow-ups of projective spaces satisfy Orlov's conjecture.

Includes blow-up of P^2 at nine points and blow-up of P^3 along a line.

Provides alternative proof for del Pezzo surfaces.

Abstract

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 $D^b_{\mathrm{coh}}(X)$ is equal to $\mathrm{dim}\, X$. 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.