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Notes on the construction of the moduli space of curves. (English) Zbl 0990.14008
Ellingsrud, Geir (ed.) et al., Recent progress in intersection theory. Based on the international conference on intersection theory, Bologna, Italy, December 1997. Boston, MA: Birkhäuser. Trends in Mathematics. 85-113 (2000).
This is an expository paper, in which the author describes Gieseker’s construction of the (coarse) moduli scheme of stable curves of genus \(g \geq 3\). In the first section of the paper the author discusses the existence of a coarse moduli scheme of smooth curves. In section 2, groupoids and their morphisms are defined and the concept of stack is introduced, followed by the definition, characterization and properties of a Deligne-Mumford stack. The third section is devoted to the study of stable curves and the groupoid of stable curves, which is shown to be a Deligne-Mumford stack over Spec \(\mathbb Z\); the section ends with a discussion of the irreducibility of the moduli stack. In the last section the author defines the moduli space of a Deligne-Mumford stack and proves that a geometric quotient of a scheme by a group is the moduli space of the quotient stack, and after a discussion of the method of geometric invariant theory for constructing geometric quotients for actions of reductive groups, the author presents Gieseker’s construction of the coarse moduli space.
For the entire collection see [Zbl 0935.00036].

14H10 Families, moduli of curves (algebraic)
14L24 Geometric invariant theory
14D22 Fine and coarse moduli spaces
14M17 Homogeneous spaces and generalizations
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