Tjurina transform and resolutions of determinantal singularities

Transformation (or modifications) plays an important role in the study of singular varieties. There is the famous result by Hironaka that normalized blow-ups can be used to resolve singularities in characteristic 0, and also Spivakovsky’s result that normalized Nash transform can be used to resolve singular complex surfaces. In this talk we will focus on the Trurina transform, which is a transform of determinantal singularities defined by their linear structure. We will first discuss the Tjurina transform and its transpose for model determinantal singularities. See how they are related and how the are related to the Nash transform, and determine their homotopy type. We will then define the Tjurina transform (and its transpose) for general determinantal singulaities, show some basic properties and discuss how it differs from the model case. We will also discuss how in many cases the Tjurina transform can easily be computed, and show that often the Tjurina transform (or its transpose) is a complete intersection. Finally we will use the Tjurina transform to resolve some hypersurface singularities.