Let $X$ be a smooth projective complex variety and let us suppose that the closure of the union of curves $C$ on $X$ that are nonpositive against the canonical divisor is a closed subset $F\subsetneq X$. Then, I guess that every birational map $X \dashrightarrow X$ is an automorphism of $X\setminus F$. Does someone has a reference for this? I am happy with the case where the codimension of $F$ is at least $2$.
1 Answer
Perhaps with some more careful justification the following will give a counterexample?
Suppose that $S$ is a surface of general type with an automorphism $f\colon S\to S$ and a point $p\in S$ such that $f(p)\neq p$. Then let $X$ be the surface obtained by first blowing up $p\in S$ with exceptional divisor $E$, and then blowing up two more points along $E$ so that $E^2=3$ and $K_XE=1$. Now the set $F\subset X$ is the union of the last two exceptional $(1)$curves, but the induced birational map $f\colon X\dashrightarrow X$ contracts $E$ to the point $f(p)$.
(This doesn't satisfy your codimension $\geq 2$ hypothesis, but I suppose the principle it illustrates is that $K$positive curves can become $K$negative after following some steps of the MMP.)

$\begingroup$ Nice counterexample! Thanks. Do you think we can do this in dimension 3 with $F$ also being some curves? $\endgroup$ Commented Sep 5, 2022 at 16:23

$\begingroup$ I would guess so. I am not sure how to go about writing down an explicit example, but I could imagine a terminal threefold $X$ with two curves $C_,C_+\subset X$ with $\pm KC_\pm > 0$ and a birational map $X\dashrightarrow X$ which flips $C_$ to $C_+$ and antiflips $C_+$ to $C_$. $\endgroup$ Commented Sep 6, 2022 at 11:37