Hello,

I'm trying to understand the relation between the points of view of log geometry (monoids) and toric geometry (fans).

Suppose that $k$ is a field and $P$ is a finitely generated monoid. Then $k[P]$ has a natural log structure and furthermore, any choice of generators $\mathbf N^r\to P$ induces a closed embedding $Spec(k[P])\subset\mathbf A^r$.

On the other hand, starting from a cone $\sigma$ satisfying some properties in a lattice $N\otimes\mathbf R$, where $N = \mathbf Z^r$, we obtain a monoid $P' = \sigma^\vee\cap M$, where $M = Hom(N,\mathbf Z)$ and $\sigma^\vee$ is the set of all $x\in M\otimes\mathbf R$ such that $x(\sigma) \geq 0$.

Question: if we start with $P$ (and a choice of generators as above), can one write a corresponding cone so as to recover $P$ by the construction in the previous paragraph?

Thanks!

saturated, cancellative and with the group completion torsion free in order to be able to recover it from the cone. "Saturated" means that if $p$ belongs to the group completion and $np\in P$ for some $n>0$ then $p\in P$. $\endgroup$