Timeline for Lifting a linear surface from a curve to the ambient surface
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 17, 2019 at 13:41 | comment | added | meh | @user52991 No, that can never happen as K-3 is regular ! I don't want to speak to Aprodu, but the gyst of his comment should be that curves have way more special linear systems than most K-3 surfaces have. | |
| Jun 17, 2019 at 13:32 | comment | added | user52991 | @aginensky, now I think I understand. For K3 surfaces with elliptic fibration the restriction map of Picard group from $X$ to $E$ is surjective isn't it? | |
| Jun 13, 2019 at 15:12 | comment | added | meh | @user52991 hence the constant use of 'most' in this conversation | |
| Jun 13, 2019 at 14:19 | comment | added | user52991 | @aginensky, Aprodu is sAying that the Linear system cannot be lifted if the K3 surface has no elliptic curves. An elliptic K3 surface has elliptic curves. So that is not the reason I guess. | |
| Jun 13, 2019 at 13:22 | comment | added | meh | I also think that there are other restrictions on the existence of elliptic curves on K-3 surfaces, but exact theorems escape me at the moment. | |
| Jun 13, 2019 at 13:21 | comment | added | meh | @user52991 , again as in the Polizzi comment, on most K-3 surfaces, that would mean that the surface has an elliptic fibration and most K-3 surfaces aren't elliptic. | |
| Jun 13, 2019 at 3:13 | comment | added | user52991 | I have edited the question to include for most $|A|$. | |
| Jun 13, 2019 at 3:12 | history | edited | user52991 | CC BY-SA 4.0 |
added 15 characters in body
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| Jun 13, 2019 at 3:11 | comment | added | user52991 | @aginensky, What about when he says if $X$ contains no elliptic curve. I did not understand that bit? | |
| Jun 12, 2019 at 14:25 | comment | added | meh | @ Polizzi, of course you are correct. Omitted from the quote OP shared is 'for most |A|" . I think your comment pretty much explains what Aprodu is trying to say. | |
| Jun 12, 2019 at 12:52 | comment | added | Francesco Polizzi | I do not completely understand this statement. What if $\mathrm{Pic}(X)$ is generated by a very ample line bundle $C$ and $A=\mathcal{O}_X(C)|_C$? In this case $A$ can be clearly lifted to $X$ , one lifting being $\mathcal{O}_X(C)$. Or am I missing something? | |
| Jun 12, 2019 at 12:37 | history | asked | user52991 | CC BY-SA 4.0 |