Birational modifications of surfaces via unprojections

We describe elementary transformations between minimal models of rational surfaces in terms of unprojections. These do not fit into the framework of Kustin–Miller unprojections as introduced by Papadakis and Reid, since we have to leave the world of projectively Gorenstein varieties. Also, our unprojections do not depend on the choice of the unprojection locus only, but need extra data corresponding to the choice of a divisor on this unprojection locus.