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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 $p_g=q=0$, $K^2=2$.

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.

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.

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 based on Bombieri's paper, but it uses a structure theorem for genus $2$ fibrations which involves vector bundles techniques.

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 $p_g=q=0$, $K^2=2$.

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.

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 Bombieri's spirit, but is based on a structure theorem for genus $2$ fibrations which involves vector bundles techniques.

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 based on Bombieri's paper, but it uses a structure theorem for genus $2$ fibrations which involves vector bundles techniques.