In *Birational Geometry of Algebraic Varieties*, Kollar and Mori write that for a line bundle "being big is essentially the birational version of being ample" (page 67). Recall that a line bundle $L$ on a projective variety $X$ of dimension $d$ is *big* if

$$ \limsup_{n \to \infty } \dfrac{H^0(X,L^n)}{n^d} \neq 0.$$

In other words, the rate of growth of the spaces of global sections is as big as possible. Big line bundles tend to exhibit behavior analogous to ample line bundles. I will give a couple of examples. In what follows, let $X$ be a variety over the complex numbers and let $L$ be a line bundle on $X$.

- Suppose $X$ is normal. If $L$ is ample, some power of $L$ defines an embedding in a projective space. Analogously, if $L$ is big, some power of $L$ defines a map

$$ \varphi_m: X \dashrightarrow H^0(X,L^m)$$

that is birational onto its image (*Positivity in Algebraic Geometry I*, page 139).

- If $L$ is ample, some power of $L$ is globally generated. On the other hand, if $L$ is big, some positive power of $L$ is generically globally generated; that is, the natural map

$$ H^0(X,L^m) \otimes \mathcal{O}_{X} \rightarrow L^m$$

is generically surjective (*Positivity in Algebraic Geometry I*, page 141).

Now, to get to my question, recall that if $L$ is ample, there exists a natural number $m$ such that the multiplication maps

$$ H^0(X,L^a) \otimes H^0(X,L^b) \rightarrow H^0(X,L^{a+b}) $$

are surjective for $a,b \geq m$ (*Positivity in Algebraic Geometry I*, page 32).

**Question**: Do big line bundles have a property analogous to the surjectivity of multiplications maps?

It is not clear to me what this property should be, but I would hope that these multiplication maps have eventually high rank in some suitable sense.