This book focusses on a large class of objects in moduli theory and provides different perspectives from which compactifications of moduli spaces may be investigated. Three contributions give an insight on particular aspects of moduli problems. In the first of them, various ways to construct and compactify moduli spaces are presented. In the second, some questions on the boundary of moduli spaces of surfaces are addressed. Finally, the theory of stable quotients is explained, which yields meaningful compactifications of moduli spaces of maps. Both advanced graduate students and researchers in algebraic geometry will find this book a valuable read.
Foreword.- 1: Perspectives on moduli spaces.- The GIT Approach to constructing moduli spaces.- Moduli and periods.- The KSBA approach to moduli spaces.- Bibliography.- 2: Compact moduli of surfaces and vector bundles.- Moduli spaces of surfaces of general type.- Wahl singularities.- Examples of degenerations of Wahl type.- Exceptional vector bundles associated to Wahl degenerations.- Examples.- Bibliography.- 3: Notes on the moduli space of stable quotients.- Morphism spaces and Quot schemes over a fixed curve.- Stable quotients.- Stable quotient invariants.- Wall-crossing and other geometries.- Bibliography.