Timeline for Special fibre of the modular curve $X(N)$
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 18, 2016 at 14:18 | comment | added | ACL | One needs to agree on the definition of the genus. For a proper irreducible curve $X$, you have arithmetic genus $1-\chi(X,\mathscr O_X)$, and the geometric genus (genus of normalization); for a reducible curve, I don't know how it is defined. In a proper flat family, the Euler characteristic of the fibers do not change (EGA III), and smooth implies flat. | |
| Apr 18, 2016 at 9:41 | comment | added | Adel BETINA | I find in EGA III that in our case the genus is locally constant, thence the genus is constant in the case where $X$ is geometrically irreducible. But i don't understand why the smoothness implies that the genus is constant. Can you give me a reference? | |
| Apr 16, 2016 at 17:51 | vote | accept | Adel BETINA | ||
| Apr 15, 2016 at 17:07 | history | answered | ACL | CC BY-SA 3.0 |