On Smirnov's approach to the ABC conjecture

We use algebraic geometry over pointed monoids to give an intrinsic interpretation for the compactification of the spectrum of the ring of integers of a number field $K$, for the projective line over algebraic extensions of ${\mathbb{F}}_1$ and for maps between them induced by elements of $K$, as introduced by Alexander Smirnov in his approach to the ABC conjecture.