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

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.