In algebraic geometry, theoretical arguments are often based on the existence of smooth models. In practice, smooth models may be computed using the constructive resolution algorithms of Villamayor or Bierstone-Milman. But any algorithm relying on a resolution then suffers from the high computational complexity of those general algorithms.

In this talk we introduce formal desingularizations as a weak version of resolutions. We also show how they can be computed using the method of Jung–Abhyankar, a specialized resolution procedure for surfaces. As an application, and an indication of usefulness, we demonstrate how to compute the graded module associated to the direct image of the canonical sheaf.