Timeline for Ampleness under finite map
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 3, 2017 at 1:37 | comment | added | nfdc23 | If you're mainly interested in the case of a normalization map then please say so in the question! | |
| Aug 2, 2017 at 12:15 | comment | added | user111251 | yes f is normalisation | |
| Aug 2, 2017 at 10:05 | comment | added | nfdc23 | The converse holds (since $O_X$ is $Y$-ample: see EGA II 4.6.13(ii) and 5.1.6(a)(c')). If $f$ is flat or if $Y$ is normal then the equivalence is EGA II 6.6.3 (where "$g:X' \to X$" corresponds to your $f$ and where "$X\to Y$" corresponds to your map "$Y\to {\rm{Spec}}(k)$" for the implicit ground field $k$; see the end of 6.5.1 for meaning of "(II bis)" there). The point is making useful sense of "norm" of line bundles through $f$ (possible in such cases), and relating "norm" of $f^*(L)$ to a power of $L$. Passing to reduced schemes is harmless (EGA II, 4.5.14), but $f$ the normalization...? | |
| Aug 2, 2017 at 6:21 | comment | added | user111251 | does that follow from Hartshorne? | |
| Aug 2, 2017 at 0:29 | comment | added | Joe Silverman | For projective varieties, it's true that $L$ is ample if and only if $f^*L$ is ample (Hartshorne Exercise III.5.7). So is you question really about what happens for quasi-projective varieties? | |
| Aug 1, 2017 at 23:24 | history | asked | user111251 | CC BY-SA 3.0 |