I am reading the paper "Canonical models of surfaces of general type" by E. Bombieri. In the last section of this paper, there is a statement saying that surfaces with $K^2=1$ and $p_g=0$ do not have pencils of genus $2$, and there is no proof. Is there a proof of this statement?
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2$\begingroup$ Perhaps it is worth including Bombieri's footnote from the paper: "Our proof is too long to be inserted here; we hope to return to this argument in another paper." $\endgroup$– Jack HuizengaCommented Nov 11, 2011 at 2:06
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2$\begingroup$ You are right. But I really want to see this proof and I can not find the "another paper". $\endgroup$– TongCommented Nov 11, 2011 at 4:11
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$\begingroup$ I think you should ask this question to Torsten Ekedahl (because of his paper "Canonical models of surfaces of general type in positive characteristic"). He is also active in MO. $\endgroup$– Damian RösslerCommented Nov 11, 2011 at 7:12
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2 Answers
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In fact it seems that the statement is not correct.
The paper [Calabri, Ciliberto, Mendes Lopes, Numerical Godeaux surfaces with an involution. Trans. Amer. Math. Soc. 359 (2007), no. 4] contains the classification of numerical Godeaux surfaces (i.e., minimal surfaces of general type with $K^2=1$ and $p_g=0$) that have an automorphism of order 2. The examples described in section 6 have a pencil of curves of genus 2 (cf. Remark 6.3).
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$\begingroup$ 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. $\endgroup$– TongCommented Nov 11, 2011 at 17:41
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$\begingroup$ 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. $\endgroup$– ritaCommented Nov 11, 2011 at 18:07
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$\begingroup$ 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}? $\endgroup$– TongCommented Feb 17, 2012 at 19:17
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$\begingroup$ 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. $\endgroup$– ritaCommented Feb 17, 2012 at 20:56
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$\begingroup$ 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? $\endgroup$– TongCommented Feb 18, 2012 at 15:24
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I do not know whether this result is true.
In fact G. Xiao in his book Surfaces fibrees en courbes de genre deux, page 66 claims that a surface of general type with $p_g=q=0$, $K^2=1$ and a pencil of curves of genus $2$ has been constructed by Oort and Peters in their paper A Campedelli surface with torsiongroup $\mathbb{Z}/2$. You should check their construction, since unfortunately I have not time to do it now.
Notice that Oort and Peter call "Campedelli surface" what is nowadays called "Numerical Godeaux surface" (i.e., a surface of general type with $p_g=q=0$, $K^2=1$); in fact, the name "Campedelli surface" is currently used for surfaces with $p_g=q=0$, $K^2=2$.
At any rate, the following is surely true:
if $S$ is a minimal surface of general type with $p_g=q=0$ and $K^2=1$, then the bicanonical pencil $|2K_S|$ cannot be composed with a pencil of curves of genus $2$.
See the paper by Catanese and Pignatelli Fibrations of low genus I, Section 5. The proof given there is not based on Bombieri's paper, but it uses a structure theorem for genus $2$ fibrations which involves vector bundles techniques.
answered Nov 11, 2011 at 10:23