Timeline for What is the minimal degree of a smooth curves which is not on a cubic surface in $P^3$?
Current License: CC BY-SA 3.0
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| Jun 6, 2013 at 15:34 | comment | added | Jérémy Blanc | Here is some partial answer: Lemma 2.3 in arxiv.org/abs/1106.3716 : Let $C$ be a smooth curve of genus $g$ and degree $d$ in $\mathbb{P}^3$. If $(2g-2)/3 < d < (19+g)/3$ then $C$ is contained in a cubic surface. | |
| May 27, 2013 at 20:31 | history | edited | Charles Staats | CC BY-SA 3.0 |
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| May 27, 2013 at 18:32 | comment | added | Nikita Kalinin | Thank you! but question is about any curve $C$, i.e. we fix a complex structure on a curve of genus $g$ and we are looking for minimal degree map of such a curve to $\mathbb C^3$. (sure, it is not known does there exist such a map) | |
| May 26, 2013 at 23:14 | history | edited | Charles Staats | CC BY-SA 3.0 |
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| May 26, 2013 at 21:52 | comment | added | Jérémy Blanc | Note that a general curve of degree $7$ and genus $g$ is not contained in a cubic, for $0\le g\le 2$ but is contained in a cubic if $g\ge 3$ (and even on a quadric if $g\ge 6$). See for example arxiv.org/abs/1106.3716 | |
| May 26, 2013 at 21:39 | comment | added | Charles Staats | For a more general solution to this sort of problem (that does not rely on a computer), see Hirschowitz, "Sur la postulation generique des courbes rationelles." | |
| May 26, 2013 at 20:49 | history | answered | Charles Staats | CC BY-SA 3.0 |