TITLE: Are rationally connected varieties robustly simply connected?
QUESTION [17 upvotes]: Let $X$ be a smooth projective rationally connected variety over $\mathbb C$. Let $C$ be a curve class in $X$ such that rational curves equivalent to $C$ connect every two points.
Let $f: Y \to X$ be a finite morphism from a normal variety $Y$ to $X$, generically of degree $d$. Let $D$ be the branch divisor of $f$. (This is the pushforward of the ramification divisor, which is well-defined because the ramification divisor depends only on the codimension 1 points, which are regular.)

Do we have the inequality
  $$ C \cdot D \geq 2 d - o(d) ?$$

Here the $o(1)$ goes to $0$ as $d$ goes to $\infty$ for any fixed $X, C$. 
Of course we know that rationally connected varieties are simply connected, so $D \neq 0$, and $C$ is ample, so $C \cdot D >0$. This question asks whether we can show a much stronger inequality, so that $X$ isn't even close to being simply connected.
I can show a positive answer in some special cases, like projective space: Pull back any cover to the space of pairs of a line and a point on that line. It remains connected because the space of pairs of a line through a point is simply connected. Stein factorize the map from the pullback to the Grassmanian of lines in $\mathbb P^n$. We obtain a ramified cover of the Grassmanian, each point of which corresponds to a connected cover of a line. But the ramified cover of the Grassmanian is ramified only at the lines contained in $D$, which is a codimension $2$ subset, so it is unramified by purity, so it is a trivial cover as the Grassmanian is simply connected. Thus a general line has a connected inverse image is $Y$, so by Riemann-Hurwitz its intersection with $D$ must be at least $2d-2$.
But this argument uses in a couple places special facts about the Grassmanian of lines that I don't know how to generalize to an arbitrary rationally connected variety.
Of course for projective spaces with the line class, this lower bound is optimal, as can be seen by the cover adjoining the $d$th root of the ratio of two coordinates.

REPLY [4 votes]: For my own edification I'm rewriting Jason's answer so as to excise all mention of loops (as he suggests), and trying to get the best possible bound from this technique.
Let $M$ be a variety with a $\mathbb P^1$-bundles $U \to M$ and two sections $s_1, s_2$ with a map $f: U \to X$.  Let $Y \to X$ be a finite covering and let $x$ be a general unramified point. We want to bound the degree of the finite part of the Stein factorization of that map $U \times_X Y \to M$ by a constant $n$.
Once we do that, we get that a general fiber of that map has at most $n$ connected components and hence has branch divisor of degree at least $2d-2n$.
It is sufficient to bound it restricted to the fiber of $f \circ s_1$ over a general point $x_0$. But then because $s_1$ has a constant map to $X$, the covering $U_{x_0} \times_X Y$ restricted to $M_{x_0} \times_X Y$ by $s_1: M_{x_0} \to U_{x_0}$ is a finite union of copies of $M_{x_0}$. So we are taking the Stein factorization of a map with many sections - in fact one section in each connected component. Hence the finite part will have a section in each connected component, so because it is finite each connected component will actually be a section (maybe after pasing to an open set).  But now the fiber $U_{x_0} \times_X Y$ restricted to $M_{x_0} \times_X Y$ by $s_2$ has one connected component meeting each connected component of $U_{x_0} \times_X Y$.  So we may take $n$ to be an upper bound for the connected components, which we can take to be the degree of the Stein factorization of the map $f \circ s_2 : M_{x_0} \to X$.
So if $M$ is the space of all rational curves of class $\beta$ with two marked points, we may take $n$ to be the number of connected components of the space of curves of class $\beta$ between two general points. In fact we can do a little better - it's the index of $\operatorname{Gal}(M_{0,2}(X, \beta))$ in the image of $\operatorname{Gal}(X)\times \operatorname{Gal}(X)$, where by the Galois group of a variety I mean the Galois group of its function field.