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.

up vote 4 down vote accepted

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].

I don't know, however, if there have been improvements since then.

[Kollár 1996] J. Kollár. Rational curves on algebraic varieties. Ergeb. Math. Grenzgeb. (3), Vol. 32. Berlin: Springer-Verlag, 1996. doi: 10.1007/978-3-662-03276-3. mr: 1440180.

  • 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 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 at 3:24

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