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The projectivity of the moduli space of stable curves. III: The line bundles on \(M_{g,n}\), and a proof of the projectivity of \(\bar M_{g,n}\) in characteristic 0. (English) Zbl 0544.14021
[For part II see the preceding review.]
In this paper we study some basic line bundles on the stacks \(M_{g,n}\) and their behaviour under pull back by the contraction morphism and the clutching morphism. This enables us to compute the selfintersection of the divisor at infinity. Combining this with a result of Arakelov we prove that \(\bar M_{g,n}\) is a projective variety in characteristic 0. Here \(\bar M_{g,n}\) is the sheafification of \(M_{g,n}\) or the coarse moduli space of n-pointed stable curves.

14H10 Families, moduli of curves (algebraic)
14D20 Algebraic moduli problems, moduli of vector bundles
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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