Quasi algebraically closed fields, or C1 fields, are defined in terms of a
low degree condition. Namely, the field K is C1 if every degree d
hypersurface of the projective space PKn contains a K-point as
soon as d≤n. In this article we define a notion of low toric degree
generalizing this condition for hypersurfaces of simplicial projective split
toric varieties. This allows us to prove a particular case of the C1
conjecture of Koll\'{a}r, Lang and Manin : any smooth separably rationally
connected variety that can be embedded as such a hypersurface over a C1
field has a rational point. Our results are based on the fact that the ambient
toric varieties are Mori Dream Spaces : they are naturally endowed with
homogeneous coordinates and their Minimal Model Program works in all cases.Comment: 46 pages. This is the long version of the article, with quite a lot
of preliminaries aimed at non (toric) geometers. Changes from v1 : Statement
and proof of the core theorem simplified, section about decomposition of
toric rational contractions removed + minor correction