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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?

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    $\begingroup$ For your second question, the invariant $a'(D)$ is the upper limit of the integral in asymptotic volume constants $\beta(-K_X,D)$ (see e.g., the definition of $\gamma_{\text{eff}}$ in this paper of McKinnon--Roth). In more generality, one defines $$\beta(-K_X,D) = \int_0^{\infty} \frac{\text{vol}(-K_X - tD)}{\text{vol}(-K_X)} dt.$$ One can show that the upper limit of this integral is $a'(D)$ as you have defined it. If you would like more references on asymptotic volume constants (or beta constants), please let me know. $\endgroup$
    – Jackson Morrow
    Commented Mar 2, 2023 at 0:15
  • $\begingroup$ Thanks, this looks relevant. I'm a bit confused on a few points however. What do you mean by the "upper limit of this integral", since the integral is going to infinity? I also looked at the definition $\gamma_{\mathrm{eff}}$; I don't immediately see how it is related since the definition there involves a blow-up and also $-K_X$ doesn't appear. Can you add more details please, perhaps to an answer? $\endgroup$
    – Daniel Loughran
    Commented Mar 2, 2023 at 11:37

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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.

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    $\begingroup$ Thanks this make sense now. One simply uses that the volume of a divisor is non-zero if and only if it is big, and that the big cone is the interior of the pseudo-effective cone. I'll have a think about whether this is useful for me; in the mean time I'll leave the question open. $\endgroup$
    – Daniel Loughran
    Commented Mar 3, 2023 at 10:25

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