Timeline for Global section of very ample line bundles and its value on stalks
Current License: CC BY-SA 3.0
7 events
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| Mar 3, 2015 at 22:53 | comment | added | meh | In the spirit of Karl and Sandor, I will also stick to algebraically closed fields. I'll go further and stick to curves because RR is so exact for a curve !. Certainly the number $t_0$ is bounded by $h^0(\mathcal{L})$. But you are asking for how many points are always guaranteed to be linearly independent. If $\mathcal{L} = K_X$ is very ample, this is the gonality of the curve X and it can be any integer between 3 and appx $\frac{g-1}{2} $ ( i forget the exact number the Riemann count gives). So even in this very well understood case, there isn't a great answer. | |
| Mar 3, 2015 at 21:07 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
added 47 characters in body
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| Mar 3, 2015 at 21:06 | comment | added | Sándor Kovács | Karl, you are right. I was only thinking about 0 or not zero. | |
| Mar 3, 2015 at 19:37 | comment | added | user43198 | @Kovacs: Sorry, I think I understand now. | |
| Mar 3, 2015 at 18:11 | comment | added | Sándor Kovács | To get a surjective map to a direct sum you need to get a single element that maps to the various choices. If the same point is repeated, then on the right hand side you can choose an element which has a component that is zero at that point and another component that is non-zero. Then to get surjectivity you would have to find a single section on the left hand side that maps to this element, but it could only map onto one of those components. | |
| Mar 3, 2015 at 17:57 | comment | added | user43198 | @Kovacs: Thank you very much for the answer. For the second question, I understand that for large $t_0$ this is not possible. But for small $t_0$, like less than $t$, is it possible? For your comment on distinct points I do not get it yet. As you say it is possible that a section is zero at point, but then there is another section that is non-zero at that point. Won't this suffice for surjectivity? | |
| Mar 3, 2015 at 17:53 | history | answered | Sándor Kovács | CC BY-SA 3.0 |