4 deleted 7 characters in body

I do not know whether this result is actually true.

In fact G. Xiao in his book Surfaces fibrees en courbes de genre deux, page 66 says 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.

3 added 21 characters in body

I do not know whether this result is actually true.

In fact G. Xiao in his book Surfaces fibrees en courbes de genre deux, page 66 says 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 $p_g=q=0$, $K^2=2$.

What it

At any rate, the following is surely trueis that :

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.

2 added 2 characters in body

I do not know whether this result is actually true.

In fact G. Xiao in his book Surfaces fibrees en courbes de genre deux, page 66 says 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 $p_g=q=0$, $K^2=2$.

What it is surely true is that 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 in based on Bombieri's spiritpaper, but is based on it uses a structure theorem for genus $2$ fibrations which involves vector bundles techniques.

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