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!



relation between toric geometry and log geometry
Not all finitely generated monoids $P$ will come from the cone construction. You need to assume that:


1. $P^{gp}$ is torsion-free: If $x \in P^{gp}$ and $n\cdot x = 0$ then $x = 0$.
2. $P$ is cancellative: If $x + y = x + y'$, then $y = y'$. This is equivalent to saying that the map $P \rightarrow P^{gp}$ is injective.
3. $P$ is saturated: If $x \in P^{gp}$ and $x^n \in P$, then $x \in P$. Assuming the previous two proporties, this is equivalent to $k[P]$ being normal.


Here, $P^{gp}$ refers to the group formed by inverting all the elements of $P$.


If $P$ is finitely generated and satisfies 1, then $P^{gp}$ is a lattice, i.e. isomorphic to $\mathbb Z^r$ for some $r$, and this is the lattice $M$ from the cone construction. The dual lattice $N$ is $\textrm{Hom}(M, \mathbb Z)$, and $\sigma$ can be taken to be those $\lambda$ in $N \otimes\_{\mathbb Z} \mathbb R = \textrm{Hom}(M, \mathbb R)$ such that $\lambda(x) \geq 0$ for all $x \in P$.


