Timeline for What is the minimal degree of a smooth projective embedding of a hyperelliptic curve?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| May 3, 2011 at 0:15 | comment | added | Yusuf Mustopa | I should have said all the line bundles in $W^{3}_{\frac{3}{4}g+3}(C)$ give complete linear series \textit{of dimension 3}... | |
| May 3, 2011 at 0:11 | comment | added | Yusuf Mustopa | Regarding David's answer: The relevant reference is Theorem 1.8 on p.216 of Arbarello-Cornalba-Griffiths-Harris, which implies that if $C$ is a general curve of genus $g$ and $r \geq 3,$ then a general point of $G^{r}_{d}(C)$ corresponds to an embedding of $C.$ Since all the line bundles in $W^{3}_{\frac{3}{4}g+3}(C)$ give complete linear series--this is because $W^{4}_{\frac{3}{4}g+3}(C)$ is the empty set--we have that $W^{3}_{\frac{3}{4}g+3}(C) \cong G^{3}_{\frac{3}{4}g+3}(C)$ for a general curve $C,$ so the aforementioned theorem applies. | |
| May 2, 2011 at 23:27 | comment | added | Yusuf Mustopa | Thanks for catching that! The curve is indeed assumed to be nonplanar. Also, the bound is attained for all degrees 3 or greater. | |
| May 2, 2011 at 23:24 | comment | added | J.C. Ottem | I guess you mean non-plane curve here? | |
| May 2, 2011 at 23:16 | history | answered | Yusuf Mustopa | CC BY-SA 3.0 |