Knudsen, Finn F. 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 Math. Scand. 52, 200-212 (1983). [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. Cited in 1 ReviewCited in 40 Documents MSC: 14H10 Families, moduli of curves (algebraic) 14D20 Algebraic moduli problems, moduli of vector bundles 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) Keywords:343.14008; line bundles; pull back by the contraction morphism; selfintersection of the divisor at infinity; coarse moduli space Citations:Zbl 0544.14020 PDFBibTeX XMLCite \textit{F. F. Knudsen}, Math. Scand. 52, 200--212 (1983; Zbl 0544.14021) Full Text: DOI EuDML