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?



On a result about genus two pencils
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). 


