Cuspidal Sections and Birational Analogues

Abstract

As Grothendieck noticed in his letter to Faltings, a k-rational point in the boundary of a good compactification contributes a packet of sections, see Grothendieck ( Schneps, L., Lochak, P. (eds.), LMS Lecture Notes vol. 242, 1997), page 8, not necessarily accounted for by rational points of the variety. These sections will here be called cuspidal sections. In the general framework, cuspidal sections are best constructed by Deligne’s theory of tangential base points, see Deligne ( Mathematical Sciences Research Institute Publications, vol. 16, 1989). Cuspidal sections have been studied by Nakamura in the profinite setting (Nakamura, J. Reine Angew. Math. 405:117–130, 1990; J. Reine Angew. Math. 411:205–216, 1990; Math. Zeit. 206(4):617–622, 1991), while Esnault and Hai ( Adv. Math. 218(2):395–416, 2008) discussed the analogue in a tannakian setting.We will define packets of cuspidal sections and show in Proposition 250 that for hyperbolic curves over arithmetic base fields these packets are either empty or uncountable. Then we describe and extend Nakamura’s characterisation of cuspidal sections in terms of cyclotomically normalized pro-cyclic subgroups in order to cover the description foreseen by Grothendieck in his letter.We will also discuss the birational analogue of the section conjecture when smooth geometrically connected varieties are replaced by their function fields: by passing to the limit over all open subschemes, all sections in this birational setup are predicted to become cuspidal. For curves over finite extensions of ${\mathbb{Q}}_{p}$ the birational section conjecture holds by Koenigsmann’s Theorem, and we have even a 2-step nilpotent pro-p version due to Pop. Over an algebraic number field there are some partial results, see Theorem 265, and for curves over $\mathbb{Q}$ we even have a full group theoretic description of all birationally liftable sections in Theorem 269 with respect to certain ${GL }_{2}$-representations.For ease of exposition we will work in characteristic 0 only.