Autoequivalences of blow-ups of minimal surfaces

Let $X$ be the blow-up of $P_C^2$ 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 $X$ has only standard autoequivalences, no non-trivial Fourier–Mukai partners, and admits no spherical objects. If $X$ is the blow-up of $P_C^2$ 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 $X$ is a blow-up of finitely many points in a minimal surface of non-negative Kodaira dimension which contains no $(-2)$-curves. Independently, we characterize spherical objects on blow-ups of minimal surfaces of positive Kodaira dimension.