An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so these cones sometimes can be used to distinguish non-isomorphic varieties.

Another interesting class of divisors are the *very* ample divisors.
Each ample divisor has a multiple that is very ample, so there is no such thing as the "cone" of very ample divisors: by tensoring with $\mathbb{R}$ or $\mathbb{Q}$ and looking only at rays, we lose the information of when an ample divisor becomes very ample. Thus one must look at the *monoid* of very ample divisors. Though it seems to be difficult in general, there are some criteria in some cases for what power of an ample line bundle one must take for it to become very ample. However, I am curious about the following:

**Are there applications to studying the structure of the very ample monoid, similar to those of the ample cone appearing in the theory of surfaces and higher-dimensional geometry?**

A more specific question is:

**What is an example of two varieties such that their Picard groups are isomorphic and their ample cones coincide, but their very ample monoids do not?**

And also:

**For a variety to be a Mori dream space imposes rather strict conditions on the cones associated to the variety, e.g., polyhedrality of the effective and nef cones. Does it impose further restrictions on the monoids of basepoint-free divisors, very ample divisors, etc., beyond the restrictions imposed on the cones themselves?**

Basically, I understand what information is lost about a *specific* divisor by passing from it to the ray it spans, but I would like to know examples of the information that is lost about the *space* of divisors in this process. This is admittedly somewhat vague, but any suggestions in this direction would be appreciated.

anyvariety of Picard number at least 2 (no matter how nice), it is not finitely generated. $\endgroup$