7
$\begingroup$

(This is a followup to my previous question on rationally connected varieties)

Let $X$ be a smooth hypersurface of degree $d$ in $\mathbb P^n$, $n \geq 3$.

Let $f: Y \to X$ be a finite morphism from a normal variety $Y$ to $X$, generically of degree $k$. Let $D$ be the branch divisor of $f$.

Do we have for some constant $c_{d,n}$ the inequality $$ H^{n-2} \cdot D \geq (c_{d,n}-o(1)) k?$$

As $n \geq 3$, by Lefschetz $X$ is simply connected, so clearly $D$ is nonzero and hence $H^{n-1} \cdot D \geq 1$. I'm asking if the inequality can be strengthened.

Jason Starr's positive answer to my previous question, plus the fact that Fano varieties are rationally connected, shows the answer is yes when $d < n+1$.

But I don't know anything about the Calabi-Yau case $d=n+1$ or the general type case, except that by pulling back covers from projective space you can see that $c_{d,n} \leq 2 d$ if it exists.

If the answer is yes, the only strategy I can see to prove it would be some Lefschetz type argument where you show that a cover extends. But it's certainly not the case that a ramified cover of a hypersurface in projective space always is a pullback of a ramified cover of projective space - then the branch divisor would always be a multiple of the hyperplane class.

$\endgroup$
16
  • $\begingroup$ How about the following? Fix, once and for all, a linear projection $g:X\to \mathbb{P}^{n-1}$ that is finite and flat. Denote the branch divisor of $g$ by $E_0$. The branch divisor of $g\circ f$ should be $E_0+g_*D$ (where $g_*$ is defined using norms as in Mumford's "Lectures on curves on an algebraic surface"). Now use the result for $\mathbb{P}^{n-1}$. $\endgroup$ Commented Sep 23, 2015 at 12:54
  • $\begingroup$ There is a mistake in my previous comment. Denoting by $D_f$ the branch divisor of $f$, the branch divisor of $g\circ f$ should be $g_*D_f + \text{deg}(f)\cdot E_0$. The formula has to be linear for precomposing $f$ with a finite 'etale morphism. $\endgroup$ Commented Sep 23, 2015 at 14:39
  • $\begingroup$ The degree of $E_0$ is $d(d-1)$. Your lower bound on the degree of a branch divisor in $\mathbb{P}^{n-1}$ is $$H^{n-2}\cdot(g_*D_f + \text{deg}(f)E_0) \geq 2\text{deg}(g\circ f).$$ Simplifying yields, $$H^{n-2}\cdot D_f \geq (2d -d(d-1))\text{deg}(f).$$ Unfortunately this is useless for $d \geq 2$. $\endgroup$ Commented Sep 23, 2015 at 14:47
  • $\begingroup$ @JasonStarr Any argument really has to use $n \geq 3$ because it's dramatically false for $n =2$. $\endgroup$
    – Will Sawin
    Commented Sep 23, 2015 at 15:28
  • $\begingroup$ Yes, also the technique I was proposing would apply to any projective variety, not just hypersurfaces. So that is another reason it cannot work. $\endgroup$ Commented Sep 23, 2015 at 15:32

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.