- Title : Special Lecture Series on Algebraic Stacks - Speaker : Jarod Alper (Columbia University) - Place : Math Science building room 404
- Abstract I will present a short lecture series on algebraic stacks. The philosophy will be to begin with a rigorous definition of an algebraic stack verifying the definition in several concrete examples. I will cover several of the most influential results on algebraic stacks and present the main idea of the proof omitting technical details.
Lecture 1: Stacks ✎Date : 19, March 2012 (Mon) 16:00 ~ 17:30 We intend to introduce rigorously the concept of a stack. The motivating examples will be the stack of smooth curves of genus g, the stack of vector bundles over a smooth curve and quotient stacks.
Lecture 2: Algebraic stacks ✎Date : 20, March 2012 (The) 16:00 ~ 17:30 We will define algebraic stacks. The relationship between algebraic stacks and groupoids will be discussed. We will then specialize the definition to discuss algebraic spaces and Deligne-Mumford stacks. The stack of smooth curves, the stack of vector bundles over a smooth curve and general quotient stacks are algebraic. We will prove that an algebraic stack is Deligne-Mumford if and only if the diagonal is unramified and we will apply this result to conclude that the stack of smooth curves is Deligne-Mumford.
Lecture 3: The Keel-Mori Theorem ✎Date : 21, March 2012 (Wed) 16:00 ~ 17:30 The Keel-Mori theorem states that every separated Deligne-Mumford stack admits a coarse moduli space. In this lecture, we will define the notion of coarse moduli space, prove the Keel-Mori theorem and discuss applications. In particular, we will apply this theorem to conclude that the moduli space of stable curves admits a projective moduli space.
Lecture 4: Artin's criterion ✎Date : 22, March 2012 (Thu) 13:00 ~ 14:30 Artin's criterion is a powerful result in the theory of algebraic stacks. It states vaguely that one can verify the algebraicity of a stack by verifying local deformation-theoretic properties of objects. This lecture will be a whirlwind tour of deformation theory, Schlessinger's criterion, and Artin's results on approximation and algebraization. Correspondence: 현동훈 교수(POSTECH, E-mail: dhhyeon@gmail.com Tel. 279-2326) |
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