Timeline for On a result about genus two pencils
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 18, 2012 at 17:11 | comment | added | rita | "pencil" is used both for "fibration onto a curve" and for "1-dimensional linear system". To the latter is associated a rational map to $P^1$. I don't remember Xiao's result, sorry. Try checking the paper where he proves that the bicanonical image is a surface if $K^2>1$. | |
| Feb 18, 2012 at 15:24 | comment | added | Tong | Thanks, rita! I think the original statement should be in the Xiao's Lecture Notes. But I can not find the statement in the book. I knew it from papers written by other people where they used "PENCIL" not "fibration". So this is why I was confused with the precise statement. Do you know the precise statement of Xiao's result? | |
| Feb 17, 2012 at 20:56 | comment | added | rita | If the surfaces is minimal of general type and $|C|$ is a pencil with base points such the general $C$ is smooth of genus 2 then the index theorem gives $K^2= 1$. On the other hand, I do not know whether one could have a pencil with base points and such that the general $C$ is singular but has geometric genus 2. | |
| Feb 17, 2012 at 19:17 | comment | added | Tong | I have another quick question, rita. It is about what I said before. I said that minimal surfaces of general type with $p_g=0$ and $K^2 \ge 3$ have no pencils of genus 2. Actually I do not know what the pencil means here. Does it mean a \emph{morphism} of fibration or just a \emph{rational map}? | |
| Nov 11, 2011 at 18:07 | comment | added | rita | For numerical Campedelli, you can look at [Calabri, Mendes Lopes, Pardini, Involutions on numerical Campedelli surfaces. Tohoku Math. J. (2) 60 (2008), no. 1, 1–22]. This is essentially a continuation of the paper I referred to in the answer and in 3.5 you can find examples of (numerical) Campedelli surfaces with a genus 2 pencil. | |
| Nov 11, 2011 at 17:43 | vote | accept | Tong | ||
| Nov 11, 2011 at 17:41 | comment | added | Tong | Thanks, rita. I remember that minimal surfaces of general type with $p_g=0$ and $K^2 \ge 3$ do not have genus two pencil. I just want to know the cases that when $K^2$ is smaller, which correspond the Numerical Godeaux or Campedelli surfaces. I will check the paper you mentioned. | |
| Nov 11, 2011 at 14:53 | history | answered | rita | CC BY-SA 3.0 |