A variant on the Fujita invariant Let $X$ be a Fano variety over $\mathbb{C}$. Let $D$ be a divisor on $X$. Recall that the Fujita invariant of $D$ is defined to be
$$a(D) = \inf \{ t \in \mathbb{R} : K_X + tD \text{ is effective} \}.$$
This appears in both birational geometry and Manin's conjecture on rational points of bounded height.
The following variant has arisen in my work: Let $D$ be a divisor on $X$. Then define
$$a'(D) = \sup \{ t \in \mathbb{R} : -K_X - tD \text{ is effective} \}.$$
If $\mathrm{Pic}(X) \cong \mathbb{Z}$ then it is easy to see that $a(D) = a'(D)$. However I can't see a simple relation between $a(D)$ and $a'(D)$ in general; for example $a'(D) > 0$ for any effective divisor $D$, but $a(D) > 0$ if and only if $D$ is big.
So my questions are as follows:
Is there any relation between $a(D)$ and $a'(D)$?
Has the invariant $a'(D)$ appeared anywhere in the literature before?
 A: Here is an expansion on my comment. For a field $K$, let $X$ be a normal projective $K$-variety, let $\mathscr{L}$ be a big line bundle on $X$, and let $D$ be a nonzero effective Cartier divisor on $X$. The beta constant of $\mathscr{L}$ along $D$ (or the asymptotic volume of $\mathscr{L}$ along $D$) is defined to be
$$
\beta(\mathscr{L},D) = \int_0^{\infty}\frac{\text{vol}(\mathscr{L} - tD)}{\text{vol}(D)}\, dt
$$
This constant appears in works of Ru--Vojta and Grieve, Section 1.10. My comment was meant to allude to the fact that one can prove that
$$
\beta(\mathscr{L},D) = \int_0^{a'(D)}\frac{\text{vol}(\mathscr{L} - tD)}{\text{vol}(D)}\, dt
$$
where
$$
a'(D) = \sup\{ t\in \mathbb{R} : \mathscr{L} - tD \text{ is effective}\}.
$$
See e.g., Remark 2.4 of DeVleming's notes or one can use the same argument as in McKinnon--Roth, beginning of Section 4. (In the McKinnon--Roth setting, their $\beta_x(\mathscr{}L) $ is just $\beta(\pi^*\mathscr{L},E)$ where $\pi\colon \widetilde{X}\to X$ is the blow-up at a point $x\in X$ with exceptional divisor $E$.) I'm not sure about the relationship between $a(D)$ and $a'(D)$ as above, but you may be able to find something in the linked notes of DeVleming.
