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For a generically finite morphism $f:X\rightarrow Y$ of smooth projective surfaces over $\mathbb{C}$. Fix any integer $g$, denote $\mathcal{A}$ to be the set of smooth irreducible curve $C$ in $Y$ with geometric genus $g(C)\leq g$.

Q: Does the set of geometric genus $\{g(E)| E$ is an integral curve in $f^{-1}(C), \forall C\in \mathcal{A}$ and $C$ is a negative curve$\}$ have a bound?

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Does this work?

Take both $X$ and $Y$ to be $\mathbb{P}^1\times\mathbb{P}^1$, with the map $X\rightarrow Y$ corresponding to the product of a degree $2$ map $\mathbb{P}^1\rightarrow\mathbb{P}^1$ and the identity. Now for any $d$, $\mathcal{A}$ contains the generic curve of degree $(1,d)$ (which is automatically rational.) But its pullback to $X$ will be a curve of degree $(2,d)$, with genus $d-1$, which can be arbitrarily large.

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  • $\begingroup$ I have modified the original question, in Bold text. $\endgroup$
    – Feng Hao
    Commented Jun 28, 2017 at 14:36

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