Perspective and open problems on birational properties and singularities of moduli scheme of sheaves on surfaces

For complex projective smooth surface X, let M be the coarse moduli scheme of rank-two stable sheaves with fixed Chern classes. Grasping the birational structure of M, for example its Kodaira dimension, is a fundamental problem. However, in the case where (X) >0, the study of this problem has not necessarily been active in recent years. In this article we survey the study of this problem, especially for the case where (X) =1 and c₁=0. We will also survey some research on the structure of singularities of M, and a minimal model program of M. While explaining motivations, we raise several unsolved problems.