Cones, monoids, and the space of (very) ample divisors

Question

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.

Answer 1

Regarding your second question, take $A$ and $X$, where $A$ is a general principally polarized Abelian surface and $X$ is a very general surface of degree $\geq 4$ in $\mathbb{P}^3$.

Then the Néron-Severi of both these varieties is cyclic of rank $1$, generated by the class $\Theta$ of a Theta divisor in the former case and by the class $H$ of a hyperplane section in the latter case.
Therefore $\textrm{NS}(A) \cong \textrm{NS}(X) \cong \mathbb{Z}$.

However, the very ample monoids are

$M_A:=\{n \Theta | n \geq 3 \}\cup \{0\} \quad \textrm{and} \quad M_X:=\{n H | n \geq 1 \} \cup \{0 \}$,

respectively.

These are not isomorphic monoids since $M_A$ is not cyclic whereas $M_X$ is cyclic.

EDIT. In order to give an example with the Picard groups instead of the Néron-Severi group, take a general $K3$ surface $Y$ which is a double cover of $\mathbb{P}^2$ branched along a sextic curve. Therefore the Picard group of $Y$ is generated by $D$, where $D$ is the pull-back in $Y$ of a line $\ell \subset \mathbb{P}^2$. Therefore

$\textrm{Pic}(Y) \cong \textrm{Pic}(X) \cong \mathbb{Z}$,

and consequently the ample cones are also isomorphic. However, the very ample monoid of $Y$ is

$M_Y:=\{n D | n \geq 2 \}\cup \{0\}$,

which is not cyclic, hence not isomorphic to $M_X$.