Timeline for Multiplication maps for big line bundles
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 26, 2020 at 1:57 | vote | accept | Roberto Nunez | ||
| Aug 25, 2020 at 4:28 | comment | added | Hacon | And, yes, I used that the limit and lim sup agree (but this is easy to prove) | |
| Aug 25, 2020 at 4:27 | comment | added | Hacon | The statement is true for all $a,b$ sufficiently big, but more annoying to state/prove. Essentially, we have proven the statement for ll $a,b$ divisible by $m$. Since $L$ is big, we can assume that $H^0(cL)>0$ for all $c\geq m$. So in general we write $a=km+a'$ with $m\leq a'<2m$, $b=jm+b'$ for $m\leq b'<2m$ and then $V_{a,b}\supset V_{km,jm}$ and we still have $\dim V_{km,jm}/h^0((a+b)L)>(1-\epsilon)$ for $j,m\gg 0$. | |
| Aug 24, 2020 at 23:39 | comment | added | Roberto Nunez | Thank you. This is exactly the sort of statement I had hoped would be true. I have a couple of quick questions. 1. Since $a$ and $b$ should clear the denominators of $A$ for the argument to work, we could still expect the dimension of $V_{a,b}$ to be quite small even for arbitrarily big $a$ and $b$. Is that correct? 2. Does your argument use the fact that the volume is a limit rather than just a limsup? Thanks again. | |
| Aug 24, 2020 at 15:20 | history | answered | Hacon | CC BY-SA 4.0 |