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For any smooth connected linear algebraic group G over an algebraically closed field k, we describe the Picard group of the universal moduli stack of principal G-bundles over pointed smooth k-projective curves.

The aim of the present paper is to study the (abstract) Picard group and the Picard group scheme of the moduli stack of stable pointed curves over an arbitrary scheme. As a byproduct, we compute the Picard groups of the moduli stack of stable or smooth pointed curves over a field of characteristic different from two. define(1.0.3) P g,n (S) : Λ g,n P g,nis the base change morphism induced by φ. By construction, the homomorphisms {P rel g,n (S)} S define a natural transformation from the constant presheaf Λ g,n onto the relative Picard functor of f : M g,n → Spec Z; hence, by passing to the associated sheaves (for the fppf topology), we get a homomorphism of commutative group schemes, compatible with base change, that we call the tautological S-morphism:(1.0.5) Φ S g,n : Λ g,n S → Pic M S g,n /S . The main result of this paper is the following Main Theorem. Assume that S is a scheme over Spec Z[1/2] (or equivalently that the residue field k(s) of every s ∈ S has characteristic different from 2) or that g ≤ 5.(1) The homomorphism Φ S g,n of (1.0.5) is an isomorphism. (2) If S is connected, then there is an isomorphism (functorial in S)The above Theorem was proved for M 1,1 by Fulton-Ollson [FO10] over an arbitrary scheme S (not necessarily over Spec Z[1/2]). In Remark 6.6, we discuss what happens if the assumptions of the Main Theorem are not satisfied.The Main Theorem allows us to compute the Picard group of M k g,n and its open substack M k g,n parametrizing smooth n-pointed curves over a (not necessarily algebraically closed) field k if either char(k) = 2 or g ≤ 5.Main Corollary. Let k be a field. Assume that either the characteristic of k is different from 2 or that g ≤ 5.

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