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


*

*$P^{gp}$ is torsion-free: If $x \in P^{gp}$ and $n\cdot x = 0$ then $x = 0$.

*$P$ is cancellative: If $x + y = x + y'$, then $y = y'$. This is equivalent to saying that the map $P \rightarrow P^{gp}$ is injective.

*$P$ 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 $x \in P$.
A: Sloppy stab at a reformulation:
I think we are assuming P is commutative?  (\sigma^\vee \cap M certainly is.)  If so, then isn't P spanned by all linear combinations of the generators?  But then there are relations, too, etc.  So the question might be, does P embed as a submonoid of a free abelian group over Z?  (I don't know.)  If so, convexity of the corresponding domain is clear, and finite generatedness means it's cut out by a finite number of conditions.  Your lattice is the Z-span of P and your cone is then the (dual of the) R-convex hull.
A: Picking up some of Eric Zaslow's reformulation: Assume $P$ is commutative, saturated, and cancellative, as well as finitely generated.  The answer to your question is affirmative if and only if the "groupification" $P^{gp}$ of $P$ is torsion-free.  (As mentioned in Dustin's answer, saturated means that for all $p$ in $P^{gp}$, $np \in P$ implies $p\in P$.  Cancellative means $p_1+q=p_2+q$ implies $p_1=p_2$.) 
All this amounts to $P$ being embeddable as a sub-monoid of ${\Bbb Z}^n$ for some $n$.  Then take the subgroup of ${\Bbb Z}^n$ spanned by $P$.  This is isomorphic to some ${\Bbb Z}^m$; take the dual of the convex hull of $P$ in ${\Bbb R}^m$ and you've got your cone $\sigma$, just as Eric says.  When $P$ is saturated, it is equal to $\sigma^\vee \cap M$; otherwise, this gives the saturation of $P$, corresponding to the integral closure of $k[P]$.
Depending on what references you use, when $P^{gp}$ is torsion-free, $P$ is called either integral or toric.  (See, e.g., the toric variety notes on M. Mustata's webpage versus the log geometry notes on Danny Gillam's  webpage; both sources are worth looking at.)  It seems the latter terminology is more standard in the log geometry world, where "integral" sometimes just means "cancellative".
