Stacks Project Expository Collection

Description

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. Contents Contributors Preface 1 Projectivity of the moduli of curves -- Raymond Cheng, Carl Lian, Takumi Murayama Introduction 1.1 Stable curves 1.2 Nakai–Moishezon criterion for ampleness 1.3 Positivity of invertible sheaves 1.4 Nef locally free sheaves 1.5 Ampleness lemma 1.6 Nefness for families of nodal curves 1.7 Projectivity of the moduli of curves References [18] 2 The stack of admissible covers is algebraic -- Elsa Corniani, Neeraj Deshmukh, Brett Nasserden, Emanuel Reinecke, Nawaz Sultani, Rachel Webb 2.1 Introduction 2.2 Setup 2.3 Admissible 𝐺-covers References [6] 3 Projectivity of the moduli space of vector bundles on a curve -- Jarod Alper, Pieter Belmans, Daniel Bragg, Jason Liang, Tuomas Tajakka Introduction 3.1 The moduli stack of all vector bundles 3.2 Semistability 3.3 Good moduli spaces 3.4 Determinantal line bundles References [15] 4 Boundedness of semistable sheaves -- Haoyang Guo, Sanal Shivaprasad, Dylan Spence, Yueqiao Wu 4.1 Introduction 4.2 Preliminaries 4.3 Bogomolov’s inequality 4.4 Restriction to hypersurfaces and Bogomolov’s inequality 4.5 Boundedness of torsion-free sheaves References [3] 5 Theorem of the Base -- Raymond Cheng, Lena Ji, Matt Larson, Noah Olander Introduction 5.1 The Picard functor 5.2 Components of the Picard functor 5.3 Castelnuovo–Mumford regularity 5.4 Boundedness 5.5 Finiteness of cycles modulo numerical equivalence 5.6 Alterations in families 5.7 Theorem of the Base 5.8 Examples of Picard schemes References [12] 6 Weil restriction for schemes and beyond -- Lena Ji, Shizhang Li, Patrick McFaddin, Drew Moore, Matthew Stevenson 6.1 Weil restriction of schemes 6.2 Basic properties of Weil restriction 6.3 Geometric connectedness 6.4 Examples 6.5 Weil restriction of algebraic spaces 6.6 Olsson’s theorem References [9] 7 Heights over finitely generated fields -- Stephen McKean, Soumya Sankar 7.1 Introduction 7.2 Generalized global fields 7.3 Heights 7.4 Analytic background 7.5 Arithmetic intersection theory and Arakelov theory 7.6 Moriwaki heights References [16] [34] 8 An explicit self-duality -- Nikolas Kuhn, Devlin Mallory, Vaidehee Thatte, Kirsten Wickelgren 8.1 Introduction 8.2 Commutative algebra preliminaries 8.3 The explicit isomorphism References 9 Tannakian reconstruction of coalgebroids -- Yifei Zhao 9.1 Introduction 9.2 Tannakian adjunction 9.3 Reconstruction 9.4 The embedding problem References [7]