BCAM Course | Approaching singularity theory via derived categorical techniques

Abstract

The interplay between derived categories and singularity theory raises the question: Why are these categories central to modern algebraic geometry? This question arises from the observation that various singularities in birational geometry are naturally characterized using the language of derived categories. Examples include rational singularities, Du Bois singularities, and derived splinters. This lecture series approaches the question from a particular perspective, blending a notion of 'generation' for derived categories introduced by Bondal--Van den Bergh and approximability for triangulated categories à la Neeman. In particular, we cover the foundational concepts and their connections to commutative algebra, then highlight recent developments in the study of singularities using generation in derived categories, and conclude by mentioning potential future directions for further applications