Autoequivalences of blow‐ups of minimal surfaces

Let be the blow‐up of in a finite set of very general points. We deduce from the work of Uehara [Trans. Amer. Math. Soc. 371 (2019), no. 5, 3529–3547] that has only standard autoequivalences, no non‐trivial Fourier–Mukai partners, and admits no spherical objects. If is the blow‐up of in 9 very general points, we provide an alternate and direct proof of the corresponding statement. Further, we show that the same result holds if is a blow‐up of finitely many points in a minimal surface of non‐negative Kodaira dimension which contains no ‐curves. Independently, we characterize spherical objects on blow‐ups of minimal surfaces of positive Kodaira dimension.