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The Stacks Project Expository Collection (SPEC) compiles expository articles in advanced algebraic geometry, intended to bring graduate students and researchers up to speed on recent developments in the geometry of algebraic spaces and algebraic stacks. The articles in the text make explicit in modern language many results, proofs, and examples that were previously only implicit, incomplete, or expressed in classical terms in the literature. Where applicable this is done by explicitly referring to the Stacks project for preliminary results. Topics include the construction and properties of important moduli problems in algebraic geometry (such as the Deligne-Mumford compactification of the moduli of curves, the Picard functor, or moduli of semistable vector bundles and sheaves), and arithmetic questions for fields and algebraic spaces.
List of contents
List of contributors; Preface; 1. Projectivity of the moduli of curves Raymond Cheng, Carl Lian and Takumi Murayama; 2. The stack of admissible covers is algebraic Elsa Corniani, Neeraj Deshmukh, Brett Nasserden, Emanuel Reinecke, Nawaz Sultani and Rachel Webb; 3. Projectivity of the moduli space of vector bundles on a curve Jarod Alper, Pieter Belmans, Daniel Bragg, Jason Liang and Tuomas Tajakka; 4. Boundedness of semistable sheaves Haoyang Guo, Sanal Shivaprasad, Dylan Spence and Yueqiao Wu; 5. Theorem of the Base Raymond Cheng, Lena Ji, Matt Larson and Noah Olander; 6. Weil restriction for schemes and beyond Lena Ji, Shizhang Li, Patrick McFaddin, Drew Moore and Matthew Stevenson; 7. Heights over finitely generated fields Stephen McKean and Soumya Sankar; 8. An explicit self-duality Nikolas Kuhn, Devlin Mallory, Vaidehee Thatte and Kirsten Wickelgren; 9. Tannakian reconstruction of coalgebroids Yifei Zhao.
About the author
Pieter Belmans is Assistant Professor of Mathematics at the University of Luxembourg. He studies algebraic geometry and noncommutative algebra from the point-of-view of derived categories. He developed the infrastructure that runs the Stacks project.Wei Ho is Associate Professor of Mathematics at the University of Michigan. Her research interests are primarily in arithmetic geometry, number theory, and algebraic geometry. She first became involved with the Stacks project during her postdoc at Columbia University.Aise Johan de Jong is Professor at Columbia University. He has worked at Harvard University, Princeton University, and MIT. Currently he spends most of his research time advising his graduate students and working on the Stacks project. He received the 2022 AMS Leroy P. Steele Prize for Mathematical Exposition for his work on the Stacks project.
Summary
SPEC collects expository articles on the geometry of algebraic spaces and stacks. The articles bring researchers up to speed on recent developments and include detailed accounts in modern language of many results previously unavailable or only implicit in the literature, and where applicable use the Stacks project for preliminary material.
