A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2

Given a smooth projective toric variety $X$ of Picard rank 2, we resolve the diagonal sheaf on $X\times X$ by a linear complex of length $\dim <!--FUNCTION\; APPLICATION-->X$ consisting of finite direct sums of line bundles. As applications, we prove a new case of a conjecture of Berkesch, Ermana and Smith that predicts a version of Hilbert’s syzygy theorem for virtual resolutions, and we obtain a Horrocks-type splitting criterion for vector bundles over smooth projective toric varieties of Picard rank 2, extending a result of Eisenbud, Erman and Schreyer. We also apply our results to give a new proof, in the case of smooth projective toric varieties of Picard rank 2, of a conjecture of Orlov concerning the Rouquier dimension of derived categories.