Timeline for Compactifying morphisms and ample line bundles
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 27, 2018 at 16:52 | comment | added | Omprokash Das | Thank you @SándorKovács! I see what's going on there now. | |
| Mar 27, 2018 at 6:22 | comment | added | Sándor Kovács | I guess you might need to take a power of $L$, although I am not sure if it is really necessary. It definitely makes your life easier. In any case $\overline X$ is projective over $\overline Y$ so it comes with a natural relatively very ample line bundle. If $L$ is also relatively very ample then from the fact that you constructed $\overline X$ as the closure of a quasi-projective embedding of $X$ induced by $L$ tells you that that line bundle restricts to $L$ on $X$. The point is to think relatively over $Y$. | |
| Mar 27, 2018 at 4:29 | comment | added | Omprokash Das | @SándorKovács, I am sure I am being very stupid here, but how do I extend L to a relatively ample line bundle $\hat{L}$ on $X$? Given that $\iota:X\hookrightarrow\overline{X}$, only thing I can think of is $\hat{L}:=\iota_*L$. | |
| Mar 27, 2018 at 4:20 | comment | added | Sándor Kovács | From your argument you also get that you can extend $L$ to $\overline X$ as a relatively ample line bundle, say $\widehat L$. Then take a very ample line bundle $M$ on $Y$ and $\overline L:=\widehat L\otimes f^*M$ will do the trick. | |
| Mar 27, 2018 at 3:50 | history | asked | Omprokash Das | CC BY-SA 3.0 |