# Degree Bound in Bend and Break Lemmas

I have read the sections on the Bend & Break Lemmas in Koll\'ar-Mori and Debarre and have the following question. (See below for background and what I do know.)

Question: I would like to know if the following is true: if $$X$$ is a normal (projective) variety and $$-K_X$$ is $$\mathbb{Q}$$-Cartier and ample, for the generic point $$x \in X$$, can one find a rational curve $$C$$ such that $$-K_X \cdot C \le \dim X + 1$$?

On a related note, if this is false, I would also like to know if it is true for such varieties $$X$$ with terminal singularities. The reason I wonder if it is true in this case is because terminal singularities appear on minimal models of smooth varieties and we know the statement is true in that case.

Background: On a (smooth) Fano variety $$X$$, through every point $$x\in X$$, there is a rational curve $$C$$ such that $$0 < -K_X \cdot C \le \dim X + 1$$. However, if $$X$$ is singular, the situation differs. Theorem 3.6 in Debarre's Higher-Dimensional Algebraic Geometry implies that, if $$-K_X$$ is ample and $$X$$ is normal, there exists a rational curve $$C$$ through every point $$x\in X$$ such that $$0 < -K_X \cdot C \le 2 \dim X$$. I would like to understand why the bound on the degree changes. In my mind, I could see it coming from singular points where $$K_X$$ is not Cartier, so one doesn't expect the same behavior, or from some finer difference that I do not understand.

So, what I would like to know is: if $$x$$ is contained in the smooth locus of $$X$$, can one use the same Bend-and-Break argument to reduce the degree and find a curve $$C$$ with $$-K_X \cdot C \le \dim X + 1$$? We can still find curves containing that point in the smooth locus of $$X$$, so can produce a rational curve through that point, so it seems like we can use the same trick (passing to characteristic $$p$$ and increasing the degree with the Frobenius) to find curves of lower degree. Perhaps, though, the problem comes when one tries to produce a rational curve--if it passes through the singular locus of $$X$$, the same argument will not work. I do not have enough experience in this area to know.

The bound $$-K_X \cdot C \le \dim X + 1$$ can be guaranteed if $$X$$ has local complete intersection singularities, and the curve to which you are applying bend and break intersects the smooth locus of $$X$$; see [Kollár 1996, Thm. II.5.14 and Rem. II.5.15]. The reason is that you need certain lower bounds on dimensions of deformation spaces; see [Kollár 1996, Thm. II.1.3].
• Thank you! I checked the reference and followed the remark to find that the bound $-K_X \cdot C \le \dim X + 1$ is even guaranteed when $X$ is a finite quotient of a hypersurface singularity, so in particular all terminal threefolds have this bound. – be928 Nov 9 '18 at 0:14
• @be928 Glad to help! I think those extra references still have the bound $-K_X \cdot C \le 2\dim X$, though. For example, the main theorem in Cone theorems and cyclic covers has $-K_X \cdot C \le 2\dim X$. – Takumi Murayama Nov 9 '18 at 3:24