# Isomorphism outside of negative curves against the canonical

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 non-positive 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 $$2$$.

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.)
• Nice counterexample! Thanks. Do you think we can do this in dimension 3 with $F$ also being some curves? Commented Sep 5, 2022 at 16:23
• 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 $C_-$. Commented Sep 6, 2022 at 11:37