I am looking at various properties of finite group actions on projective (and usually rational) varieties. Given any such group action, my work takes me into one of two directions: I either look at the properties of the space you get when you quotient the variety out by this action, or I look at differrent transformations of the variety that preserve the action.
One of the things I've been looking into is the exceptionality of quotient singularities on hypersurfaces, a notion defined by V. Shokurov in his paper Complements on sufaces. Exceptional and weakly-exceptional singularities have some nice boundedness properties, making them fairly useful in singularity theory. I am trying to classify the weakly-exceptional quotient singularities (the larger of the classes, that includes the exceptional ones), so that their properties can be observed by comparing the singularity with a (potentially large) list. Such lists for dimensions up to 4 already exist (see below for my preprint on dimensions 3 and 4), but there are still plenty of directions the lists can be expanded into.
