Timeline for A variant on the Fujita invariant
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 2, 2023 at 21:37 | answer | added | Jackson Morrow | timeline score: 2 | |
| Mar 2, 2023 at 11:37 | comment | added | Daniel Loughran | 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? | |
| Mar 2, 2023 at 0:15 | comment | added | Jackson Morrow | 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. | |
| Mar 1, 2023 at 21:49 | history | asked | Daniel Loughran | CC BY-SA 4.0 |