Derived categories and semiorthogonal decompositions of projective varieties

Bounded derived categories of coherent sheaves play a crucial role in the study of the geometry of projective varieties. Their semiorthogonal decompositions build connections to other varieties and reflect many properties such as singularity and rationality. I will talk about the motivation to study these structures, illustrate the important examples of Bondal-Orlov and Burban, and report on my work about smooth complete intersections of three quadrics and nodal quintic del Pezzo threefolds that generalize their examples.