We demonstrate curved modifications with lengths of up to 2 mm within borosilicate glass produced by single 1030 nm picosecond laser shots with an Airy beam profile. Plasma ignition in the side lobes of the beam as well as surface damage prove to be the crucial limitations for confined bulk energy deposition on a curved trajectory. A combined experimental and numerical analysis reveals optimum laser parameters for confined bulk energy deposition. This way we achieved single pass cutting of a 525 $\mu$m thick glass sheet with a well defined convex edge down to a bending radius of 774 $\mu$m.