# Are hypersurfaces robustly simply connected?

(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.

• 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 $\mathbb{P}^{n-1}$. – Jason Starr Sep 23 '15 at 12:54
• 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. – Jason Starr Sep 23 '15 at 14:39
• 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 $d \geq 2$. – Jason Starr Sep 23 '15 at 14:47
• @JasonStarr Any argument really has to use $n \geq 3$ because it's dramatically false for $n =2$. – Will Sawin Sep 23 '15 at 15:28
• Yes, also the technique I was proposing would apply to any projective variety, not just hypersurfaces. So that is another reason it cannot work. – Jason Starr Sep 23 '15 at 15:32