Let $f: X \to Y$ be a finite, surjective morphism of smooth, projective, irreducible varieties over $\mathbb{C}$ and let $y \in Y$.

  1. Can I find a smooth curve $C \subseteq Y$ with $y \in C$ such that $f^{-1}(C)$ is again a smooth curve?

  2. If not, can I find a curve $C \subseteq Y$ such that $y$ is a smooth point of $C$ and such that every $x \in f^{-1}(\{y\})$ is a smooth point of $f^{-1}(C)$?

  3. Can I find a curve $C \subseteq Y$ with the property of 1. or 2. with the additional property that $f$ is unramified along a Zariski dense subset of $f^{-1}(C)$?

Edit: As Yusuf recommended, I'd like to point out that I am especially interested in the case where $y$ lies in the branch locus. Also $y$ is not necessarily a smooth point of the underlying reduced scheme of the branch divisor.

  • $\begingroup$ Is $y \in Y$ allowed to be in the branch divisor $B$ of $f$? If not, 1 and 3 can be settled by taking $C$ to be a complete intersection of sufficiently ample divisors on $Y$ such that $C$ contains $y$ and intersects $B$ transversally (this follows from Bertini). $\endgroup$ Aug 13, 2014 at 11:37
  • $\begingroup$ Thank you. But in fact, the case where $y$ is in the branch locus is the case where I am interested in. $\endgroup$
    – Hans
    Aug 13, 2014 at 11:43
  • $\begingroup$ I figured that this might be the case. You are still okay if $y$ is in the smooth locus of the underlying reduced scheme $B_{red}.$ However, you may want to edit the question to emphasize your interest in the branch point case. $\endgroup$ Aug 13, 2014 at 11:45
  • $\begingroup$ Okay thanks again. At the moment, I can't see whether my problem reduces to such a situation. $\endgroup$
    – Hans
    Aug 13, 2014 at 12:11

1 Answer 1


Consider the map $\mathbb P^2 \to \mathbb P^2$ with equations $(xy,x^2-y^2,z^2)$. Any smooth curve passing through the point $(0,0,1)$ has an equation that looks locally linear, and its inverse image has an equation that looks locally quadratic.

  • 2
    $\begingroup$ This is correct. Jouanolou's theorem that I cited [Theoreme 6.3 (2) in Theoremes de Bertini et Applications] implies the smoothness of the inverse images of a Zariski-dense set of hyperplane sections, but the condition of passing through a fixed point is Zariski closed. So this theorem does not apply here. $\endgroup$ Aug 13, 2014 at 12:37
  • $\begingroup$ Thank you for this counterexample. Now, I have another question which is related to that. $\endgroup$
    – Hans
    Aug 13, 2014 at 14:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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