Timeline for Surfaces in $\mathbb{P}^3$ with isolated singularities
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 14, 2012 at 12:09 | comment | added | rita | Just a remark about the final question. If the minimal model of $X\subset \mathbb P^3$ contains no rational curves, then no rational curve of $X'$ is exceptional for the map $X'\to X$. So $X$ contains some rational curve. | |
| Dec 14, 2012 at 11:20 | history | edited | Dmitri Panov | CC BY-SA 3.0 |
deleted 281 characters in body
|
| Dec 14, 2012 at 10:07 | comment | added | Dmitri Panov | Sandor, sure I understand what you say. I will rewrite this "answer" so there are no doubts in this. | |
| Dec 14, 2012 at 0:50 | comment | added | Sándor Kovács | ps: a minimal surface by any definition may have a curve with negative self-intersection. For instance a (smooth) K3 surface may contain a $(-2)$-curve, yet it is minimal in any which way you like to define minimal. (In particular, it is the minimal resolution of the surface you get when you blow down that $(-2)$-curve). | |
| Dec 14, 2012 at 0:46 | comment | added | Sándor Kovács | @Dmitri: the question was not whether there exists a birational morphism to $\mathbb P^3$ whose image has the desired properties, but whether any surface is birational to such a surface in $\mathbb P^3$. The point being that your proof shows that for certain surfaces there is no such morphism, but as CX points out, there still could be a rational map that is not everywhere defined. | |
| Dec 13, 2012 at 23:38 | history | edited | Dmitri Panov | CC BY-SA 3.0 |
added 365 characters in body; added 20 characters in body
|
| Dec 13, 2012 at 20:59 | comment | added | algori | I think there may be a confusion between two meanings of "minimal" here: surfaces do admit minimal resolutions (i.e., resolutions through which all other factor) but those needn't be minimal (i.e., they may contain $-1$-curves). | |
| Dec 13, 2012 at 19:57 | comment | added | CYXU | If you blow up the original surface S, you will always have curves with negative self-intersections. The question is asking for BIRATIONAL! | |
| Dec 13, 2012 at 17:18 | history | answered | Dmitri Panov | CC BY-SA 3.0 |