Topos points of quasi-coherent sheaves over monoid schemes

Let $X$ be a monoid scheme. We will show that the stalk at any point of $X$ defines a point of the topos $\Qc(X)$ of quasi-coherent sheaves over $X$. As it turns out, every topos point of $\Qc(X)$ is of this form if $X$ satisfies some finiteness conditions. In particular, it suffices for $M/M^\times$ to be finitely generated when $X$ is affine, where $M^\times$ is the group of invertible elements. This allows us to prove that two quasi-projective monoid schemes $X$ and $Y$ are isomorphic if and only if $\Qc(X)$ and $\Qc(Y)$ are equivalent. The finiteness conditions are essential, as one can already conclude by the work of A. Connes and C. Consani \cite{cc1}. We will study the topos points of free commutative monoids and show that already for $\mathbb{N}^\infty$, there are `hidden' points. That is to say, there are topos points which are not coming from prime ideals. This observation reveals that there might be a more interesting `geometry of monoids'.