Bounding degree of rational curves in the exceptional locus

Question

Let $f\colon \mathbb{P}^2\dashrightarrow\mathbb{P}^2$ be a rational map ($\mathrm{deg}(f)$ may be high), let $\Gamma\subset\mathbb{P}^2\times\mathbb{P}^2$ be the closure of the graph. Let $x\in\mathbb{P}^2$ be a point in the source, then the fiber $\Gamma|_x$ can be viewed as a union of rational curves in the target $\mathbb{P}^2$. Do we know if $\Gamma|_x\subset\mathbb{P}^2$ necessarily have a component of low degree (say, degree 1)?

Answer 1

The answer is no. Let us consider the case where $f$ is birational. In this case, for each point $x\in \mathbb{P}^2$ such that $f$ is not defined, the intersection of $\Gamma$ with $x\times \mathbb{P}^2$ contains all curves of $\mathbb{P}^2$ that are contracted by the inverse $f^{-1}$. The degree of these curves can be arbitrary large.

As an example, you can consider some so-called "Halphen map", that is a birational map of $\mathbb{P}^2$ that preserves a pencil of curves of genus $1$ (a simple case is given by a pencil given by two general cubics). You blow-up the $9$ points $p_1,\ldots,p_9$ of intersection and consider a translation given by $E_1-E_2$, where $E_1$ and $E_2$ are the exceptional divisors of $p_1$ and $p_2$ (which are two sections of the pencil). If you consider this map $f$ on $\mathbb{P}^2$, the degree of $f^n$ grows quadratically with $n$ and there are exactly $9$ points not defined for $f^n$, namely $p_1,\ldots,p_n$. There are exactly $9$ curves contracted by $f^{-n}$ and these are irreducible of degree that is also growing. For $n$ large, the degree of the curves contracted is then unbounded.

There is an explicit family given in Proposition 6.4 of https://algebra.dmi.unibas.ch/blanc/articles/degencremona.pdf where the degree of $f^n$ is $36n^2+1$, and where $f$ contracts $7$ curves of degree $12n^2$, one of degree $12n^2-6n$ and one of degree $12n^2+6n$.