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On the geometry of Deligne-Mumford stacks. (English) Zbl 1169.14001

Abramovich, D. (ed.) et al., Algebraic geometry, Seattle 2005. Proceedings of the 2005 Summer Research Institute, Seattle, WA, USA, July 25–August 12, 2005. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4702-2/hbk; 978-0-8218-4057-3/set). Proceedings of Symposia in Pure Mathematics 80, Pt. 1, 259-271 (2009).
Algebraic stacks are objects which generalize schemes from the viewpoint of fibered categories. As it has turned out, over the past decades, stacks are especially well adapted to the study of classification problems in algebraic geometry via geometric invariant theory.
In the present survey article, the author summarizes some general structure results about the particular class of Deligne-Mumford (DM) stacks of finite type over a field of characteristic 0, together with their applications to certain moduli spaces. After a brief introduction to stacks in general, a concrete description of smooth Deligne-Mumford stacks in dimension 1 is given, with special emphasis placed on orbifold curves over \(\mathbb{C}\) and their role in the classical moduli theory of vector bundles on compact Riemann surfaces. This is followed by a more detailed discussion of smooth Deligne-Mumford stacks, orbifolds, and quotient stacks of arbitrary dimension over a base field \(k\). Then the study turns to possibly singular Deligne-Mumford stacks, culminating in a new characterization of those stacks that are isomorphic to the stack quotient of a quasi-projective scheme by a reductive algebraic group acting linearly. This leads to the suggestion that a Deligne-Mumford stack over a field of characteristic 0 should be called “(quasi-)projective” if it admits a (locally) closed embedding into a smooth proper Deligne-Mumford stack with projective coarse moduli space. In this context, the class of (quasi-)projective Deligne-Mumford stacks is completely characterized in different ways. More precisely, it is outlined that a separated Deligne-Mumford stack with quasi-projective coarse moduli space over a field of characteristic 0 is quasi-projective (in the above sense) if and only if it enjoys several equivalent, well-studied properties, namely: being a quotient stack, satisfying the so-called resolution hypothesis, admitting a finite flat covering by a scheme, or possessing a generating sheaf. At the end of the paper, these general structure results are illustrated by various examples of concrete moduli stacks known from the recent literature.
Throughout the entire article, a basic reference is the fundamental work “Brauer Groups and Quotient Stacks” by D. Edidin, B. Hassett, A. Kresch and A. Vistoli [Am. J. Math. 123, No. 4, 761–777 (2001; Zbl 1036.14001)], together with numerous other, mostly very recent original research papers by various authors.
For the entire collection see [Zbl 1158.14003].

MSC:

14A20 Generalizations (algebraic spaces, stacks)
14D20 Algebraic moduli problems, moduli of vector bundles
14F22 Brauer groups of schemes
14L24 Geometric invariant theory
14H60 Vector bundles on curves and their moduli
14F35 Homotopy theory and fundamental groups in algebraic geometry

Citations:

Zbl 1036.14001
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