A refined version of the geometrically m-step solvable Grothendieck conjecture for genus 0 curves over finitely generated fields

In the present paper, we show a new result on the geometrically 2-step solvable Grothendieck conjecture for genus 0 curves over finitely generated fields. More precisely, we show that two genus 0 hyperbolic curves over a finitely generated field k are isomorphic as k-schemes (up to Frobenius twists) if and only if the geometrically maximal 2-step solvable quotients of their \'etale fundamental groups are isomorphic as topological groups over the absolute Galois group of k.