Rational points of universal curves

Source abstract

Suppose that k k is a field of characteristic zero and that g + n > 2 g+n>2 . The universal curve C C of type ( g , n ) (g,n) is the restriction of the universal curve to the generic point Spec ⁡ k ( M g , n ) \operatorname {Spec} k(\mathcal {M}_{g,n}) of the moduli stack M g , n \mathcal {M}_{g,n} of n n -pointed smooth projective curves of genus g g . In this paper we prove that if g ≥ 3 g \ge 3 , then its set of rational points C ( k ( M g , n ) ) C(k(\mathcal {M}_{g,n})) consists only of the n n tautological points. We then prove that if g ≥ 5 g\ge 5 and n = 0 n=0 , then Grothendieck’s Section Conjecture holds for C C when, for example, k k is a number field or a non-archimedean local field. When n > 0 n>0 , we consider a modified version of Grothendieck’s conjecture in which the geometric fundamental group of C C is replaced by its ℓ \ell -adic unipotent completion. We prove that if k k is a number field or a non-archimedean local field, then this modified version of the Section Conjecture holds for all g ≥ 5 g \ge 5 and n ≥ 1 n \ge 1 .