Adjunctions and Applications

This lecture covers the topic of adjunctions and their applications in projective varieties, focusing on the concept of regular varieties and the valuative criterion of properness. The instructor discusses the properties of locally free modules and the criteria for a projective variety to satisfy the valuative criterion. The lecture also delves into the concept of closed immersions and the construction of normal bundles. Additionally, it explores concrete examples of regular hypersurfaces and their properties, such as birationality and the behavior of poles. The presentation concludes with a detailed examination of morphisms and their induced structures on various types of varieties.